**3. Applications to financial options and numerical results via CNN**

Here the analysis of Section 2 is applied to some problems arising in option pricing theory. Some known pricing formulas are revisited in a more general setting and some new results are offered. We apply Cellular Neural Networks (CNN) approach [24] in order to obtain some numerical results. Let us consider a two-dimensional grid with 3 × 3 neighborhood system as it is shown on Figure 1.

**Figure 1.** 3 × 3 neighborhood CNN.

[htb] One of the key features of a CNN is that the individual cells are nonlinear dynamical systems, but that the coupling between them is linear. Roughly speaking, one could say that these arrays are nonlinear but have a linear spatial structure, which makes the use of techniques for their investigation common in engineering or physics attractive.

We will give the general definition of a CNN which follows the original one:

#### **Definition 1.** *The CNN is a*

8 Will-be-set-by-IN-TECH

2 .

*v*˜0(*ξ*, *η*)*G*(*τ*, *z*1, *z*2, *ξ*, *η*)*dξdη*+

<sup>4</sup>*πτ* [*e*<sup>−</sup> (*z*1−*ξ*)<sup>2</sup>

**3. Applications to financial options and numerical results via CNN**

Here the analysis of Section 2 is applied to some problems arising in option pricing theory. Some known pricing formulas are revisited in a more general setting and some new results are offered. We apply Cellular Neural Networks (CNN) approach [24] in order to obtain some numerical results. Let us consider a two-dimensional grid with 3 × 3 neighborhood system as

(1, 2)

✛ ✲ ✛ ✲

(2, 2)

✠❅

✠❅

❅❅❘❅❅■ ✒ ❄ ✻ ❅ ❅❅❘❅❅■ ✒

❅❅❘❅❅■✒ ✠

(3, 1) ✛ ✲ (3, 2) ✛ ✲ (3, 3)

[htb] One of the key features of a CNN is that the individual cells are nonlinear dynamical systems, but that the coupling between them is linear. Roughly speaking, one could say that these arrays are nonlinear but have a linear spatial structure, which makes the use of

❄ ✻

*∂ξ <sup>G</sup>*(*<sup>τ</sup>* <sup>−</sup> <sup>Θ</sup>, *<sup>z</sup>*1, *<sup>z</sup>*2, *<sup>ξ</sup>*, *<sup>η</sup>*)]*ξ*=0*dηd*Θ<sup>+</sup>

*∂η <sup>G</sup>*(*<sup>τ</sup>* <sup>−</sup> <sup>Θ</sup>, *<sup>z</sup>*1, *<sup>z</sup>*2, *<sup>ξ</sup>*, *<sup>η</sup>*)]*η*=0*dξd*Θ,

<sup>4</sup>*<sup>τ</sup>* <sup>−</sup> *<sup>e</sup>*<sup>−</sup> (*z*1+*ξ*)<sup>2</sup>

(1, 3)

(2, 3)

❄ ✻

❄ ✻ <sup>4</sup>*<sup>τ</sup>* ] <sup>×</sup> [*e*<sup>−</sup> (*z*2−*η*)<sup>2</sup>

<sup>4</sup>*<sup>τ</sup>* <sup>−</sup> *<sup>e</sup>*<sup>−</sup> (*z*2+*η*)<sup>2</sup>

<sup>4</sup>*<sup>τ</sup>* ].

*f*1(Θ, *ξ*, *η*)*G*(*τ* − Θ, *z*1, *z*2, *ξ*, *η*))*dξdηd*Θ+ (24)

<sup>0</sup> <sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>T</sup>*, *zj* <sup>≥</sup> 0, 1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> 2. Certainly, *<sup>ϕ</sup>*<sup>0</sup> <sup>=</sup> *<sup>π</sup>*

+ *τ* 0

> + *τ* 0

where the Green function *G*(*τ*, *z*1, *z*2, *ξ*, *η*) = <sup>1</sup>

 *τ* 0

> + ∞ 0

 ∞ 0 ˜ *<sup>g</sup>*˜1(Θ, *<sup>η</sup>*)[ *<sup>∂</sup>*

 ∞ 0 ˜ *<sup>g</sup>*˜2(Θ, *<sup>ξ</sup>*)[ *<sup>∂</sup>*

(1, 1)

(2, 1)

❄ ✻

❄ ✻ ❅ ❅❅❘❅❅■✒✠

techniques for their investigation common in engineering or physics attractive.

We will give the general definition of a CNN which follows the original one:

 ∞ 0

 ∞ 0 ˜˜

 ∞ 0 ˜

According to [21]:

it is shown on Figure 1.

**Figure 1.** 3 × 3 neighborhood CNN.

˜ *v*˜(*τ*, *z*) =


**Definition 2.** *An M* × *M cellular neural network is defined mathematically by four specifications:*


Now in terms of definition 2 we can present the dynamical systems describing CNNs. For a general CNN whose cells are made of time-invariant circuit elements, each cell *C*(*ij*) is characterized by its CNN cell dynamics :

$$\dot{\mathbf{x}}\_{\text{ij}} = -\mathbf{g}(\mathbf{x}\_{\text{ij}}, \boldsymbol{\mu}\_{\text{ij}}, \mathbf{I}\_{\text{ij}}^{\text{s}})\_{\text{\textdegree}} \tag{25}$$

where *xij* <sup>∈</sup> **<sup>R</sup>***m*, *uij* is usually a scalar. In most cases, the interactions (spatial coupling) with the neighbor cell *C*(*i* + *k*, *j* + *l*) are specified by a CNN synaptic law:

$$\begin{split} I\_{ij}^{s} &= A\_{ij,kl} \mathbf{x}\_{i+k,j+l} + \\ &+ \tilde{A}\_{ij,kl} \* f\_{kl}(\mathbf{x}\_{ij}, \mathbf{x}\_{i+k,j+l}) + \\ &+ \tilde{B}\_{ij,kl} \* u\_{i+k,j+l}(t). \end{split} \tag{26}$$

The first term *Aij*,*klxi*<sup>+</sup>*k*,*j*+*<sup>l</sup>* of (26) is simply a linear feedback of the states of the neighborhood nodes. The second term provides an arbitrary nonlinear coupling, and the third term accounts for the contributions from the external inputs of each neighbor cell that is located in the *Nr* neighborhood.

It is known [24] that some autonomous CNNs represent an excellent approximation to nonlinear partial differential equations (PDEs). The intrinsic space distributed topology makes the CNN able to produce real-time solutions of nonlinear PDEs. There are several ways to approximate the Laplacian operator in discrete space by a CNN synaptic law with an appropriate *A*-template:


$$A\_1 = (1, -2, 1)\_\prime$$


$$A\_2 = \begin{pmatrix} 0 \ 1 & 0 \\ 1 \ -4 \ 1 \\ 0 \ 1 & 0 \end{pmatrix}.$$

#### 10 Will-be-set-by-IN-TECH 40 Risk Management – Current Issues and Challenges Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach <sup>11</sup>

**Example 1 (Single-asset inside barrier options)** The case of single-barrier zero-rebate down-and-out options was already priced in [18], while the case with rebate is found in [22]. A simple method for obtaining analytical formulas for barrier options is the reflection principle that has a long history in Physics and is commonly used in Finance. Here we write down the pricing formula for a general payoff and rebate and study its analytical properties. Let us consider the following boundary value problem:

Thus, for fixed *τ*, 0 < *τ* < *<sup>σ</sup>*<sup>2</sup>

<sup>|</sup>*I*3(*τ*, *<sup>y</sup>*)<sup>|</sup> <sup>≤</sup>const.*e*2*αy*−2*τα*<sup>2</sup> <sup>√</sup>*<sup>τ</sup>*

2

<sup>|</sup>*I*1(*τ*, *<sup>y</sup>*)<sup>|</sup> <sup>≤</sup>const.*eα<sup>y</sup>* � <sup>+</sup><sup>∞</sup>

*limS*−→0<sup>+</sup> *<sup>u</sup>*(*t*, *<sup>S</sup>*) = *<sup>u</sup>*0(0)*e*−*r*(*T*−*t*)

<sup>|</sup>*I*1(*τ*, *<sup>y</sup>*)<sup>|</sup> <sup>≤</sup>const.*eα<sup>y</sup>* <sup>2</sup>

So *<sup>u</sup>* <sup>|</sup>Σ<sup>0</sup> , with <sup>Σ</sup><sup>0</sup> <sup>=</sup> �

free-boundary conditions:

of the parameters:

⎧ ⎨ ⎩

*Lu* = 0

*u*|*t*=*<sup>T</sup>* = *u*0(*S*1, *S*2) 0 ≤ *S*<sup>2</sup> ≤ *S*<sup>∗</sup>

*i*=1 *σ*2 *<sup>i</sup> S*<sup>2</sup> *i* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup>

*u*|*S*2=*S*<sup>∗</sup> = 0 0 ≤ *t* ≤ *T*

where *L* = *∂<sup>t</sup>* + ∑<sup>2</sup>

<sup>√</sup>*<sup>π</sup>* <sup>|</sup>*y*|*eα<sup>y</sup>* � *<sup>τ</sup>*

−*y* 2 √*τ e*−*θ*<sup>2</sup>

√*τ* √*π*|*y*|

*<sup>e</sup>*<sup>−</sup> *<sup>y</sup>*<sup>2</sup>

For this example our CNN model is the following:

*dSij*

0 < *t* < *T*, *S* = 0+�

<sup>√</sup>*τ*−*<sup>γ</sup>* yields

observe that:

<sup>|</sup>*I*1(*τ*, *<sup>y</sup>*)<sup>|</sup> <sup>≤</sup> max|*g*<sup>|</sup>

change *θ* = <sup>−</sup>*<sup>y</sup>* 2

<sup>2</sup> *T*, and *y* << −1, we have

<sup>4</sup>*<sup>τ</sup>* for *y* −→ −∞, *τ* fixed. Therefore we get:

, is uniquely determined by *u*0(0).

*U*(*T*) = *u*0(0). Evidently, *U*(*t*) = *u*0(0)*e*−*r*(*T*−*t*) is the only solution of that Cauchy problem.

1 2 *σ*2*S*<sup>2</sup>

where ∗ is the convolution operator [24], *M* ≤ *i*, *j* ≤ *M*. We shall consider this model with

*∂uij*(*x*, *t*)

*∂uij*(*x*, *t*)

*<sup>S</sup>*1*S*<sup>2</sup> <sup>+</sup> *<sup>r</sup>* <sup>∑</sup><sup>2</sup>

*u*0(*S*1, *S*∗) = 0. Assume that *σ*1, *σ*<sup>2</sup> > 0, *ρ*<sup>2</sup> < 1. Using the notation of Section 2 and

*dt* = +1,

*dt* <sup>=</sup> <sup>−</sup>1.

(*τ*−*γ*)3/2 exp(<sup>−</sup> *<sup>y</sup>*<sup>2</sup>

, 0 < *t* < *T*.

*dt* <sup>+</sup> *rSijA*<sup>1</sup> <sup>∗</sup> *Sij* <sup>+</sup>

*uij*(*x*, *t*) = *x* − *k*,

*uij*(*x*, *t*) = *k* − *x*,

*Si* <sup>+</sup> *ρσ*1*σ*2*S*1*S*2*∂*<sup>2</sup>

These are classical first-order contact free-boundary conditions for obstacle problems.

Based on the above CNN model (28) we obtain the following simulations for different values

**Example 2. (Multi-asset option with single barrier)** Analytic valuation formulas for standard European options with single external barrier have been provided in Heynen-Kat (1994), Kwok-Wu-Yu (1998) and Buchen (2001). Here we give a slightly more general formula in that we allow for any payoff and for both an internal and an external barrier. We confine ourselves to the case of an upstream barrier and zero rebate for simplicity of exposition. Consider the following boundary value problem in Ω = {(*t*, *S*1, *S*2); 0 < *t* < *T*, 0 < *S*1, 0 < *S*<sup>2</sup> < *S*∗}:

**Remark 4.** Assume that *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω). Then, putting *<sup>S</sup>* <sup>=</sup> 0, *<sup>U</sup>*(*t*) = *<sup>u</sup>*(*t*, 0), we get *<sup>U</sup>*�

*dθ*, that is

<sup>4</sup>*<sup>τ</sup>* , which implies that *limy*−→−<sup>∞</sup> *I*3(*τ*, *y*) = 0. Finally, we

<sup>4</sup>(*τ*−*γ*))*d<sup>γ</sup>* as *<sup>β</sup>* <sup>≤</sup> 0 implies 0 ≤ −*βγ* ≤ −*βτ*. The

Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

*ijA*<sup>2</sup> ∗ *Sij* − *r* = 0, (28)

*<sup>i</sup>*=<sup>1</sup> *Si∂Si* − *r* , *u*<sup>0</sup> is continuous and

(*t*) = *rU*,

41

<sup>−</sup>*<sup>y</sup> <sup>e</sup>*<sup>−</sup> *<sup>y</sup>*<sup>2</sup>

<sup>0</sup> *<sup>e</sup>*−*βτ*

$$\begin{cases} Lu = 0 \quad \text{in} \quad \Omega = \{(t, \mathcal{S}); 0 < t < T, 0 < \mathcal{S} < \mathcal{S}^\*\} \\ u|\_{t=T} = u\_0(\mathcal{S}), 0 \le \mathcal{S} \le \mathcal{S}^\* \\ u|\_{\mathcal{S} = \mathcal{S}^\*} = g(t), 0 \le t \le T \end{cases}$$

where *L* = *∂<sup>t</sup>* + *rS∂<sup>S</sup>* + <sup>1</sup> <sup>2</sup> *<sup>σ</sup>*2*S*2*∂*<sup>2</sup> *<sup>S</sup>* − *r* , *u*<sup>0</sup> and *g* are continuous and *u*0(*S*∗) = *g*(*T*). Using the notation of Section 2 and taking *α* = <sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>r</sup> <sup>σ</sup>*<sup>2</sup> , *β* = −*r* � 1 <sup>2</sup> <sup>+</sup> *<sup>r</sup> σ*2 � , *C* = √2 *<sup>σ</sup>* we straightforwardly obtain the following pricing formula (after changing to variables <sup>√</sup>*<sup>σ</sup>* 2 *λ* = ln *S*<sup>∗</sup> − *ξ*):

$$u(t,S) = \left(\frac{S}{S^\*}\right)^\alpha \frac{e^{\beta(T-t)}}{\sqrt{2\pi}\sigma} [\frac{1}{\sqrt{(T-t)}} \int\_0^{+\infty} u\_0(S^\*e^{-\frac{\tau}{\xi}})e^{a\tilde{\xi}} \times \tag{27}$$

$$\times [\exp(-\frac{[\ln(S/S^\*) + \xi]^2}{2\sigma^2(T-t)}) - \exp(-\frac{[\ln(S/S^\*) - \xi]^2}{2\sigma^2(T-t)})]d\tilde{\xi} +$$

$$+ \ln \frac{S^\*}{S} \int\_0^{T-t} \frac{g(T-s)}{(T-t-s)^{3/2}} e^{-\frac{\beta\sigma^2 s}{2}} \exp(-\frac{\ln^2(S/S^\*)}{2\sigma^2(T-t-s)})ds]$$

Let us study the properties of *u*(*t*, *S*) analytically. Without loss of generality we can assume *<sup>S</sup>*<sup>∗</sup> <sup>=</sup> 1 and therefore *<sup>e</sup>*−*β*(*T*−*t*)*u*(*t*, *<sup>S</sup>*) = *<sup>u</sup>*�(*t*, *<sup>S</sup>*) is written in the form *<sup>I</sup>*<sup>1</sup> <sup>+</sup> *<sup>I</sup>*<sup>2</sup> <sup>+</sup> *<sup>I</sup>*<sup>3</sup> with:

$$I\_{1}(\tau,y) = \frac{-y e^{ay}}{2\sqrt{\pi}} \int\_{0}^{\tau} \frac{\xi (T - \frac{2\gamma}{\tau^{2}})}{(\tau - \gamma)^{3/2}} e^{-\beta\gamma} \exp(-\frac{y^{2}}{4(\tau - \gamma)}) d\gamma$$

$$I\_{2}(\tau,y) = \frac{e^{ay}}{2\sqrt{\pi\tau}} \int\_{0}^{+\infty} \mu\_{0}(e^{-\xi}) e^{a\xi} \exp(-\frac{[y+\xi]^{2}}{4\tau}) d\xi$$

$$I\_{3}(\tau,y) = -\frac{e^{ay}}{2\sqrt{\pi\tau}} \int\_{0}^{+\infty} \mu\_{0}(e^{-\xi}) e^{a\xi} \exp(-\frac{[y-\xi]^{2}}{4\tau}) d\xi$$

where *y* = ln *S* and *τ* = *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> (*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*). We shall examine the asymptotics of *<sup>v</sup>*�(*τ*, *<sup>y</sup>*) = *<sup>u</sup>*�(*t*, *<sup>S</sup>*) for 0 < *τ* < *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> *<sup>T</sup>* (i.e. 0 <sup>&</sup>lt; *<sup>t</sup>* <sup>&</sup>lt; *<sup>T</sup>*) fixed and for *<sup>y</sup>* −→ −<sup>∞</sup> (i.e. *<sup>S</sup>* <sup>→</sup> <sup>0</sup>+). Put *<sup>h</sup>*(*ξ*) = *<sup>u</sup>*0(*e*−*<sup>ξ</sup>* ), *<sup>ξ</sup>* <sup>≥</sup> 0. Then:

$$I\_{2}(\tau, y) = \frac{e^{\tau n^{2}}}{2\sqrt{\pi \tau}} \int\_{0}^{+\infty} h(\xi) \exp(-\frac{[y + \xi - 2a\tau]^{2}}{4\tau}) d\xi = \frac{e^{\tau n^{2}}}{\sqrt{\pi}} \int\_{\frac{y - 2a\tau}{2\sqrt{\tau}}}^{+\infty} h(-y + 2a\tau + 2\eta\sqrt{\tau}) e^{-\eta^{2}} d\eta.$$

According to Lebesgue's dominated convergence theorem, since *limy*−→−∞*h*(−*y* + 2*aτ* + 2*η* <sup>√</sup>*τ*) = *<sup>u</sup>*0(0) for each fixed *<sup>η</sup>* and *<sup>τ</sup>*, one has *limy*−→−<sup>∞</sup> *<sup>I</sup>*2(*τ*, *<sup>y</sup>*) = *<sup>e</sup>τα*<sup>2</sup> *u*0(0). On the other hand:

 $|I\_3(\tau, y)| \le \frac{\text{const}}{2\sqrt{\pi \tau}} \int\_0^{+\infty} e^{a(y + \xi)} (-\frac{(\xi - y)^2}{4\tau}) d\xi = \text{const}.$  $e^{\pi a^2} \int\_{-\frac{y + 2a\tau}{2\sqrt{\tau}}}^{+\infty} \exp[-\mu^2 + 2ay - 4\pi a^2 + 2a\mu\sqrt{\tau}] d\mu = \text{const}.$  $e^{2ay - 2\tau a^2} \int\_{\frac{-\sqrt{\tau}}{2\sqrt{\tau}}}^{+\infty} e^{-\varepsilon^2} d\varepsilon.$ 

Thus, for fixed *τ*, 0 < *τ* < *<sup>σ</sup>*<sup>2</sup> <sup>2</sup> *T*, and *y* << −1, we have

<sup>|</sup>*I*3(*τ*, *<sup>y</sup>*)<sup>|</sup> <sup>≤</sup>const.*e*2*αy*−2*τα*<sup>2</sup> <sup>√</sup>*<sup>τ</sup>* <sup>−</sup>*<sup>y</sup> <sup>e</sup>*<sup>−</sup> *<sup>y</sup>*<sup>2</sup> <sup>4</sup>*<sup>τ</sup>* , which implies that *limy*−→−<sup>∞</sup> *I*3(*τ*, *y*) = 0. Finally, we observe that:

<sup>|</sup>*I*1(*τ*, *<sup>y</sup>*)<sup>|</sup> <sup>≤</sup> max|*g*<sup>|</sup> 2 <sup>√</sup>*<sup>π</sup>* <sup>|</sup>*y*|*eα<sup>y</sup>* � *<sup>τ</sup>* <sup>0</sup> *<sup>e</sup>*−*βτ* (*τ*−*γ*)3/2 exp(<sup>−</sup> *<sup>y</sup>*<sup>2</sup> <sup>4</sup>(*τ*−*γ*))*d<sup>γ</sup>* as *<sup>β</sup>* <sup>≤</sup> 0 implies 0 ≤ −*βγ* ≤ −*βτ*. The change *θ* = <sup>−</sup>*<sup>y</sup>* 2 <sup>√</sup>*τ*−*<sup>γ</sup>* yields

$$\left|I\_{1}(\tau, y)\right| \le \text{const.} e^{\alpha y} \int\_{\frac{-y}{2\sqrt{\tau}}}^{+\infty} e^{-\theta^{2}} d\theta \text{, that is }$$

10 Will-be-set-by-IN-TECH

**Example 1 (Single-asset inside barrier options)** The case of single-barrier zero-rebate down-and-out options was already priced in [18], while the case with rebate is found in [22]. A simple method for obtaining analytical formulas for barrier options is the reflection principle that has a long history in Physics and is commonly used in Finance. Here we write down the pricing formula for a general payoff and rebate and study its analytical properties. Let us

*Lu* = 0 in Ω = {(*t*, *S*); 0 < *t* < *T*, 0 < *S* < *S*∗}

*<sup>σ</sup>*<sup>2</sup> , *β* = −*r*

[ <sup>1</sup> �(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*)

Let us study the properties of *u*(*t*, *S*) analytically. Without loss of generality we can assume

<sup>2</sup>*σ*2(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*) ) <sup>−</sup> exp(<sup>−</sup> [ln(*S*/*S*∗) <sup>−</sup> *<sup>ξ</sup>*]

2

*<sup>S</sup>*<sup>∗</sup> <sup>=</sup> 1 and therefore *<sup>e</sup>*−*β*(*T*−*t*)*u*(*t*, *<sup>S</sup>*) = *<sup>u</sup>*�(*t*, *<sup>S</sup>*) is written in the form *<sup>I</sup>*<sup>1</sup> <sup>+</sup> *<sup>I</sup>*<sup>2</sup> <sup>+</sup> *<sup>I</sup>*<sup>3</sup> with:

<sup>4</sup>(*τ*−*γ*))*d<sup>γ</sup>*

2 <sup>4</sup>*<sup>τ</sup>* )*dξ*

2 <sup>4</sup>*<sup>τ</sup>* )*d<sup>ξ</sup>* <sup>=</sup> *<sup>e</sup>τα*<sup>2</sup>

<sup>√</sup>*τ*) = *<sup>u</sup>*0(0) for each fixed *<sup>η</sup>* and *<sup>τ</sup>*, one has *limy*−→−<sup>∞</sup> *<sup>I</sup>*2(*τ*, *<sup>y</sup>*) = *<sup>e</sup>τα*<sup>2</sup>

exp[−*μ*<sup>2</sup> <sup>+</sup> <sup>2</sup>*α<sup>y</sup>* <sup>−</sup> <sup>4</sup>*τα*<sup>2</sup> <sup>+</sup> <sup>2</sup>*αμ*√*τ*]*d<sup>μ</sup>* <sup>=</sup>

<sup>4</sup>*<sup>τ</sup>* )*dξ* =

(<sup>−</sup> (*ξ*−*y*)<sup>2</sup>

2 <sup>4</sup>*<sup>τ</sup>* )*dξ*

<sup>2</sup> *<sup>T</sup>* (i.e. 0 <sup>&</sup>lt; *<sup>t</sup>* <sup>&</sup>lt; *<sup>T</sup>*) fixed and for *<sup>y</sup>* −→ −<sup>∞</sup> (i.e. *<sup>S</sup>* <sup>→</sup> <sup>0</sup>+). Put *<sup>h</sup>*(*ξ*) = *<sup>u</sup>*0(*e*−*<sup>ξ</sup>* ), *<sup>ξ</sup>* <sup>≥</sup> 0.

<sup>√</sup>*<sup>π</sup>*

According to Lebesgue's dominated convergence theorem, since *limy*−→−∞*h*(−*y* + 2*aτ* +

� <sup>+</sup>∞*y*−2*ατ* 2 √*τ*

� 1 <sup>2</sup> <sup>+</sup> *<sup>r</sup> σ*2 � , *C* =

> � +∞ 0

<sup>2</sup> exp(<sup>−</sup> ln2(*S*/*S*∗)

<sup>2</sup> (*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*). We shall examine the asymptotics of *<sup>v</sup>*�(*τ*, *<sup>y</sup>*) = *<sup>u</sup>*�(*t*, *<sup>S</sup>*) for

*h*(−*y* + 2*aτ* + 2*η*

<sup>√</sup>*τ*)*e*−*η*<sup>2</sup>

*dη*.

*u*0(0). On the other

*<sup>S</sup>* − *r* , *u*<sup>0</sup> and *g* are continuous and *u*0(*S*∗) = *g*(*T*). Using the

√2

*u*0(*S*∗*e*−*<sup>ξ</sup>* )*e*

<sup>2</sup>*σ*2(*<sup>T</sup>* − *<sup>t</sup>* − *<sup>s</sup>*)

2 <sup>2</sup>*σ*2(*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>*) )]*dξ*<sup>+</sup>

*<sup>σ</sup>* we straightforwardly

*αξ*<sup>×</sup> (27)

<sup>2</sup>*<sup>λ</sup>* <sup>=</sup> ln *<sup>S</sup>*<sup>∗</sup> <sup>−</sup> *<sup>ξ</sup>*):

)*ds*]

*u*|*t*=*<sup>T</sup>* = *u*0(*S*), 0 ≤ *S* ≤ *S*<sup>∗</sup> *u*|*S*=*S*<sup>∗</sup> = *g*(*t*), 0 ≤ *t* ≤ *T*

obtain the following pricing formula (after changing to variables <sup>√</sup>*<sup>σ</sup>*

<sup>2</sup> <sup>−</sup> *<sup>r</sup>*

�*<sup>α</sup> eβ*(*T*−*t*) √ 2*πσ*

*g*(*T* − *s*) (*<sup>T</sup>* <sup>−</sup> *<sup>t</sup>* <sup>−</sup> *<sup>s</sup>*)3/2 *<sup>e</sup>*<sup>−</sup> *βσ*2*<sup>s</sup>*

exp(<sup>−</sup> *<sup>y</sup>*<sup>2</sup>

consider the following boundary value problem:

<sup>2</sup> *<sup>σ</sup>*2*S*2*∂*<sup>2</sup>

� *S S*∗

<sup>×</sup>[exp(<sup>−</sup> [ln(*S*/*S*∗) + *<sup>ξ</sup>*]

<sup>0</sup> *<sup>u</sup>*0(*e*−*<sup>ξ</sup>* )*eαξ* exp(<sup>−</sup> [*y*+*ξ*]

<sup>0</sup> *<sup>h</sup>*(*ξ*) exp(<sup>−</sup> [*y*+*ξ*−2*ατ*]

<sup>0</sup> *<sup>u</sup>*0(*e*−*<sup>ξ</sup>* )*eαξ* exp(<sup>−</sup> [*y*−*ξ*]

� *T*−*t* 0

⎧ ⎨ ⎩

notation of Section 2 and taking *α* = <sup>1</sup>

*u*(*t*, *S*) =

<sup>+</sup> ln *<sup>S</sup>*<sup>∗</sup> *S*

> *<sup>g</sup>*(*T*<sup>−</sup> <sup>2</sup>*<sup>γ</sup> σ*2 ) (*τ*−*γ*)3/2 *<sup>e</sup>*−*βγ*

� +<sup>∞</sup>

� +<sup>∞</sup>

� +<sup>∞</sup> <sup>0</sup> *<sup>e</sup>α*(*y*+*ξ*)

−*y* 2 √*τ e*−*ε*<sup>2</sup> *dε*.

� +<sup>∞</sup>

where *L* = *∂<sup>t</sup>* + *rS∂<sup>S</sup>* + <sup>1</sup>

*<sup>I</sup>*1(*τ*, *<sup>y</sup>*) = <sup>−</sup>*yeα<sup>y</sup>* 2 <sup>√</sup>*<sup>π</sup>* � *τ* 0

*<sup>I</sup>*2(*τ*, *<sup>y</sup>*) = *<sup>e</sup>α<sup>y</sup>* 2 <sup>√</sup>*πτ*

*<sup>I</sup>*3(*τ*, *<sup>y</sup>*) = <sup>−</sup> *<sup>e</sup>α<sup>y</sup>*

0 < *τ* < *<sup>σ</sup>*<sup>2</sup>

*<sup>I</sup>*2(*τ*, *<sup>y</sup>*) = *<sup>e</sup>τα*<sup>2</sup> 2 <sup>√</sup>*πτ*

<sup>|</sup>*I*3(*τ*, *<sup>y</sup>*)<sup>|</sup> <sup>≤</sup> *const* 2 <sup>√</sup>*πτ*

<sup>=</sup>const.*eτα*<sup>2</sup> � <sup>+</sup>∞−*y*+2*ατ*

=const.*e*2*αy*−2*τα*<sup>2</sup> � <sup>+</sup><sup>∞</sup>

2 √*τ*

Then:

2*η*

hand:

2 <sup>√</sup>*πτ*

where *y* = ln *S* and *τ* = *<sup>σ</sup>*<sup>2</sup>

<sup>|</sup>*I*1(*τ*, *<sup>y</sup>*)<sup>|</sup> <sup>≤</sup>const.*eα<sup>y</sup>* <sup>2</sup> √*τ* √*π*|*y*| *<sup>e</sup>*<sup>−</sup> *<sup>y</sup>*<sup>2</sup> <sup>4</sup>*<sup>τ</sup>* for *y* −→ −∞, *τ* fixed. Therefore we get:

$$\lim\_{t \to -0^+} \mu(t, \mathbf{S}) = \mu\_0(0)e^{-r(T-t)}, 0 < t < T.$$

**Remark 4.** Assume that *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω). Then, putting *<sup>S</sup>* <sup>=</sup> 0, *<sup>U</sup>*(*t*) = *<sup>u</sup>*(*t*, 0), we get *<sup>U</sup>*� (*t*) = *rU*, *U*(*T*) = *u*0(0). Evidently, *U*(*t*) = *u*0(0)*e*−*r*(*T*−*t*) is the only solution of that Cauchy problem. So *<sup>u</sup>* <sup>|</sup>Σ<sup>0</sup> , with <sup>Σ</sup><sup>0</sup> <sup>=</sup> � 0 < *t* < *T*, *S* = 0+� , is uniquely determined by *u*0(0).

For this example our CNN model is the following:

$$\frac{dS\_{ij}}{dt} + rS\_{ij}A\_1 \* S\_{ij} + \frac{1}{2}\sigma^2 S\_{ij}^2 A\_2 \* S\_{ij} - r = 0,\tag{28}$$

where ∗ is the convolution operator [24], *M* ≤ *i*, *j* ≤ *M*. We shall consider this model with free-boundary conditions:

$$u\_{ij}(\mathbf{x},t) = \mathbf{x} - k\_{\prime} \frac{\partial u\_{ij}(\mathbf{x},t)}{dt} = +1,$$

$$u\_{ij}(\mathbf{x},t) = k - \mathbf{x}\_{\prime} \frac{\partial u\_{ij}(\mathbf{x},t)}{dt} = -1.$$

These are classical first-order contact free-boundary conditions for obstacle problems.

Based on the above CNN model (28) we obtain the following simulations for different values of the parameters:

**Example 2. (Multi-asset option with single barrier)** Analytic valuation formulas for standard European options with single external barrier have been provided in Heynen-Kat (1994), Kwok-Wu-Yu (1998) and Buchen (2001). Here we give a slightly more general formula in that we allow for any payoff and for both an internal and an external barrier. We confine ourselves to the case of an upstream barrier and zero rebate for simplicity of exposition. Consider the following boundary value problem in Ω = {(*t*, *S*1, *S*2); 0 < *t* < *T*, 0 < *S*1, 0 < *S*<sup>2</sup> < *S*∗}:

$$\begin{cases} Lu = 0 \\ u|\_{t=T} = u\_0(\mathcal{S}\_1, \mathcal{S}\_2) & 0 \le \mathcal{S}\_2 \le \mathcal{S}^\* \\ u|\_{\mathcal{S}\_2=\mathcal{S}^\*} = 0 & 0 \le t \le T \end{cases}$$

where *L* = *∂<sup>t</sup>* + ∑<sup>2</sup> *i*=1 *σ*2 *<sup>i</sup> S*<sup>2</sup> *i* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *Si* <sup>+</sup> *ρσ*1*σ*2*S*1*S*2*∂*<sup>2</sup> *<sup>S</sup>*1*S*<sup>2</sup> <sup>+</sup> *<sup>r</sup>* <sup>∑</sup><sup>2</sup> *<sup>i</sup>*=<sup>1</sup> *Si∂Si* − *r* , *u*<sup>0</sup> is continuous and *u*0(*S*1, *S*∗) = 0. Assume that *σ*1, *σ*<sup>2</sup> > 0, *ρ*<sup>2</sup> < 1. Using the notation of Section 2 and

12 Will-be-set-by-IN-TECH 42 Risk Management – Current Issues and Challenges Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach <sup>13</sup>

*<sup>I</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup>*−*r<sup>τ</sup>* 2*π* <sup>√</sup>1−*ρ*<sup>2</sup>

*S*2*eμ*2*τ*+*σ*<sup>2</sup>

becomes:

*<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>e</sup>*−*r<sup>τ</sup>* 2*π*

*u*0(*S*1*S*

−ln( *<sup>S</sup>*<sup>2</sup>

where *d*

*ωe* <sup>−</sup><sup>2</sup> *<sup>μ</sup>*<sup>2</sup> *σ*2 2 ln( *<sup>S</sup>*<sup>2</sup> *<sup>S</sup>*<sup>∗</sup> ) [*e* <sup>−</sup>2*<sup>ρ</sup> <sup>σ</sup>*<sup>1</sup> *<sup>σ</sup>*<sup>2</sup> ln( *<sup>S</sup>*<sup>2</sup> *<sup>S</sup>*<sup>∗</sup> )

*<sup>S</sup>*<sup>∗</sup> )+*μ*2*τ σ*2

 +<sup>∞</sup> −∞

<sup>√</sup>*τX*<sup>2</sup> )*dX*1*dX*2.

<sup>√</sup>1−*ρ*<sup>2</sup> (*S*2)

<sup>−</sup>2*<sup>ρ</sup> <sup>σ</sup>*<sup>1</sup> *σ*2 <sup>2</sup> *<sup>e</sup>μ*1*τ*−*σ*<sup>1</sup>

Changing to the variables *<sup>X</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*<sup>1</sup>

<sup>−</sup> <sup>2</sup>*μ*<sup>2</sup> *σ*2 2 +<sup>∞</sup> −∞

Then *I*<sup>1</sup> can be written in the form:

<sup>√</sup>*<sup>τ</sup>* , *<sup>e</sup>*<sup>+</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>−</sup> *ρσ*<sup>1</sup>

<sup>±</sup> <sup>=</sup> *<sup>d</sup>*<sup>±</sup> <sup>−</sup> <sup>2</sup>*<sup>ρ</sup>*

days, *σ* = 1.5, *ρ* = 0.06 .

*σ*2

<sup>√</sup>*<sup>τ</sup>* ln( *<sup>S</sup>*<sup>2</sup>

figure with different values of the parameter set:

<sup>√</sup>*τX*<sup>1</sup> , *<sup>S</sup>*−<sup>1</sup>

*<sup>ω</sup>S*1*N*2(*ωd*+,*e*+; <sup>−</sup>*ρω*) <sup>−</sup> *<sup>ω</sup>Ke*−*rτN*2(*ωd*−,*e*−; <sup>−</sup>*ρω*)

 <sup>−</sup> ln(*S*2/*S*∗)+*μ*2*<sup>τ</sup> σ*2 √*τ* <sup>−</sup><sup>∞</sup> exp[<sup>−</sup> <sup>1</sup>

<sup>2</sup>(1−*ρ*<sup>2</sup>)(*X*<sup>2</sup>

<sup>√</sup>*<sup>τ</sup>* <sup>−</sup> <sup>2</sup>*<sup>ρ</sup>* ln *<sup>S</sup>*<sup>2</sup> *σ*2

<sup>2</sup>(1−*ρ*<sup>2</sup>)(*X*<sup>2</sup>

*σ*1

<sup>√</sup>*τX*<sup>2</sup> )*dX*1*dX*2.

where *<sup>N</sup>*<sup>2</sup> is the bivariate cumulative normal distribution function, *<sup>d</sup>*<sup>±</sup> <sup>=</sup> ln( *<sup>S</sup>*<sup>1</sup>

*<sup>S</sup>*<sup>∗</sup> ), *<sup>e</sup>*<sup>±</sup> <sup>=</sup> *<sup>e</sup>*<sup>±</sup> <sup>+</sup> <sup>2</sup>

In the special case of standard options one has: *u*0(*S*1, *S*2) = max(*ω*(*S*<sup>1</sup> − *K*), 0), *ω* = ±1.

<sup>√</sup>*τ*. Similarly *<sup>I</sup>*<sup>2</sup> is written in the form:

<sup>+</sup>, *<sup>e</sup>*+; <sup>−</sup>*ρω*) <sup>−</sup> *Ke*−*rτN*2(*ω<sup>d</sup>*

Simulating CNN for multi-asset option with single barrier model, we obtain the following

(a) (b)

**Example 3. (Two-asset barrier options with simultaneous barriers)** While single-asset barrier options have received substantial coverage in the literature, multi-asset options with several barriers have been discussed only in some special cases (e.g. sequential barriers, radial

**Figure 3.** CNN simulations for Example 2. (a) *r* = 1, *T* = 60 days, *σ* = 1, *ρ* = 0.05; (b) *r* = 0.5, *T* = 120

<sup>√</sup>*<sup>τ</sup>* ln( *<sup>S</sup>*<sup>2</sup> *<sup>S</sup>*<sup>∗</sup> ).

*σ*2

 ln(*S*2*S*∗)−*μ*2*<sup>τ</sup> σ*2 √*τ* <sup>−</sup><sup>∞</sup> exp[<sup>−</sup> <sup>1</sup>

<sup>2</sup> *<sup>e</sup>μ*2*τ*+*σ*<sup>2</sup>

*S*1*N*2(*ωd*

<sup>1</sup> <sup>+</sup> *<sup>X</sup>*<sup>2</sup>

<sup>2</sup> + <sup>2</sup>*ρX*1*X*2)]*u*0(*S*1*eμ*1*τ*−*σ*<sup>1</sup>

Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

*σ*2

<sup>2</sup> + 2*ρX*1*X*2)]

<sup>−</sup>, *<sup>e</sup>*−; <sup>−</sup>*ρω*)

<sup>√</sup>*<sup>τ</sup>* , *<sup>X</sup>*<sup>2</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>X</sup>*<sup>2</sup>

<sup>√</sup>*τX*<sup>1</sup> ,

43

<sup>√</sup>*τ*, the second integral

*<sup>K</sup>* )+(*r*±*<sup>σ</sup>*<sup>2</sup> 1 <sup>2</sup> )*τ*

<sup>√</sup>*<sup>τ</sup>* , *<sup>e</sup>*<sup>−</sup> =

*σ*1

**Figure 2.** CNN simulations for Example 1. (a) *r* = 1, 1 ≤ *t* ≤ 30, *σ* = 1; (b) *r* = 0.5, 1 ≤ *t* ≤ 30, *σ* = 1.5 . taking *<sup>μ</sup><sup>i</sup>* <sup>=</sup> *<sup>r</sup>* <sup>−</sup> *<sup>σ</sup>*<sup>2</sup> *i* <sup>2</sup> for *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, 2, we have *<sup>α</sup><sup>i</sup>* <sup>=</sup> <sup>−</sup>*μi*+*ρμjσi*/*σ<sup>j</sup> σ*2 *<sup>i</sup>* (1−*ρ*<sup>2</sup>) for *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, 2 and *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*, *β* = ∑*i*,*j*=1,2 *σiσ<sup>j</sup>* <sup>2</sup> *αiα<sup>j</sup>* + ∑*i*=1,2 *μiα<sup>i</sup>* − *r*. Then we have the following pricing formula:

$$\begin{cases} \mu(t, \mathcal{S}\_1, \mathcal{S}\_2) = S\_1^{\mathfrak{sl}\_1} S\_2^{\mathfrak{sl}\_2} \frac{e^{\mathfrak{gl}\tau}}{4\pi\tau} \int\_{\mathcal{R}^2} w\_0(\lambda\_1, \lambda\_2) \exp\left[ -\frac{\left(\frac{\sqrt{\mathfrak{T}\ln S\_1}}{\sigma\_1\sqrt{1-\rho^2}} - \rho \frac{\sqrt{\mathfrak{T}\ln S\_2}}{\sigma\_2\sqrt{1-\rho^2}} - \lambda\_1 \right)^2}{4\pi} \right] \\ \left\{ \exp[-\frac{(\sqrt{\mathfrak{T}\ln S\_2}/\sigma\_2 - \lambda\_2)^2}{4\tau}] - \exp[-\frac{(\sqrt{\mathfrak{T}\ln S\_2}/\sigma\_2 + \lambda\_2)^2}{4\tau}] \right\} d\lambda\_1 d\lambda\_2 \end{cases}$$

where

 $w\_{0}(\lambda\_{1},\lambda\_{2}) = \exp[-\frac{\hbar\_{1}\sigma\_{1}}{\sqrt{2}}(\lambda\_{1}\sqrt{1-\rho^{2}}+\rho\lambda\_{2})-\frac{\hbar\_{2}\sigma\_{2}}{\sqrt{2}}\lambda\_{2}]$  $\mu\_{0}(\frac{\sigma\_{1}}{\sqrt{2}}(\lambda\_{1}\sqrt{1-\rho^{2}}+\rho\lambda\_{2}),\frac{\sigma\_{2}\lambda\_{2}}{\sqrt{2}})$  $\lambda\_{3} < \frac{\sqrt{2}\ln S^{\*}}{\sigma\_{2}}.$ 

Splitting the integral into two integrals and changing to variables *<sup>η</sup>*<sup>1</sup> <sup>=</sup> *<sup>λ</sup>*<sup>1</sup> <sup>√</sup>1−*ρ*<sup>2</sup>+*ρλ*2<sup>−</sup> √ 2 ln *S*<sup>1</sup> *σ*1 <sup>√</sup>2*<sup>τ</sup>* , *η*<sup>2</sup> = *<sup>λ</sup>*2<sup>−</sup> <sup>√</sup>2 ln *<sup>S</sup>*<sup>2</sup> *σ*2 <sup>√</sup>2*<sup>τ</sup>* (*η*<sup>2</sup> <sup>=</sup> *<sup>λ</sup>*2<sup>+</sup> √ 2 ln *S*<sup>2</sup> *σ*2 <sup>√</sup>2*<sup>τ</sup>* ) in the first (second) integral, one gets:

$$\begin{split} &u(t, S\_1, S\_2) = I\_1 - I\_2 \\ &\text{where} \\ &I\_1 = \frac{e^{\theta\tau}}{2\pi\sqrt{1-\rho^2}} \int\_{-\infty}^{+\infty} \int\_{-\infty}^{\frac{\ln(\xi^s/\xi\_2)}{\rho\_2\sqrt{\tau}}} \exp[- (\alpha\_1\sigma\_1\eta\_1 + \alpha\_2\sigma\_2\eta\_2)\sqrt{\tau}] \mu\_0(S\_1e^{\sigma\_1\sqrt{\tau}\eta\_1}, S\_2e^{\sigma\_2\sqrt{\tau}\eta\_2}) \\ &\exp[- \frac{(\eta\_1-\rho\eta\_2)^2}{2(1-\rho^2)} - \frac{\eta\_2^2}{2}] d\eta\_1 d\eta\_2 \\ &I\_2 = \frac{S\_2e^{\theta\tau}}{2\pi\sqrt{1-\rho^2}} \int\_{-\infty}^{+\infty} \int\_{-\infty}^{\frac{\ln(\xi^s/\xi\_1)}{\rho\_2\sqrt{\tau}}} \exp[- (\alpha\_1\sigma\_1\eta\_1 + \alpha\_2\sigma\_2\eta\_2)\sqrt{\tau}] \mu\_0(S\_1e^{\sigma\_1\sqrt{\tau}\eta\_1}, S\_2^{-1}e^{\sigma\_2\sqrt{\tau}\eta\_2}) \\ &\exp[- \frac{(-\eta\_1+\rho\eta\_2 - 2\rho\ln S\_2/(\sigma\_2\sqrt{\tau}))^2}{2(1-\rho^2)} - \frac{\eta\_2^2}{2}] d\eta\_1 d\eta\_2. \end{split}$$

Note that (*<sup>β</sup>* <sup>+</sup> *<sup>r</sup>*)(<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*2) + *<sup>μ</sup>*<sup>2</sup> 1 2*σ*<sup>2</sup> 1 <sup>+</sup> *<sup>μ</sup>*<sup>2</sup> 2 2*σ*<sup>2</sup> 2 <sup>−</sup> *<sup>ρ</sup> <sup>μ</sup>*1*μ*<sup>2</sup> *<sup>σ</sup>*1*σ*<sup>2</sup> = 0. Then the first integral (after changing to variables *<sup>X</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*<sup>1</sup> *σ*1 <sup>√</sup>*τ*, *<sup>X</sup>*<sup>2</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup> *σ*2 <sup>√</sup>*τ*) is written in the form:

42 Risk Management – Current Issues and Challenges Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach <sup>13</sup> 43 Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

*<sup>I</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup>*−*r<sup>τ</sup>* 2*π* <sup>√</sup>1−*ρ*<sup>2</sup> +<sup>∞</sup> −∞ <sup>−</sup> ln(*S*2/*S*∗)+*μ*2*<sup>τ</sup> σ*2 √*τ* <sup>−</sup><sup>∞</sup> exp[<sup>−</sup> <sup>1</sup> <sup>2</sup>(1−*ρ*<sup>2</sup>)(*X*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>X</sup>*<sup>2</sup> <sup>2</sup> + <sup>2</sup>*ρX*1*X*2)]*u*0(*S*1*eμ*1*τ*−*σ*<sup>1</sup> <sup>√</sup>*τX*<sup>1</sup> ,

*S*2*eμ*2*τ*+*σ*<sup>2</sup> <sup>√</sup>*τX*<sup>2</sup> )*dX*1*dX*2.

12 Will-be-set-by-IN-TECH

(a) (b)

**Figure 2.** CNN simulations for Example 1. (a) *r* = 1, 1 ≤ *t* ≤ 30, *σ* = 1; (b) *r* = 0.5, 1 ≤ *t* ≤ 30, *σ* = 1.5 .

 <sup>√</sup>2 ln *<sup>S</sup>*<sup>1</sup> *σ*1 <sup>√</sup><sup>1</sup>−*ρ*<sup>2</sup> <sup>−</sup>*<sup>ρ</sup>*

<sup>2</sup> *αiα<sup>j</sup>* + ∑*i*=1,2 *μiα<sup>i</sup>* − *r*. Then we have the following pricing formula:

√2*σ*2 2

<sup>√</sup>2*<sup>τ</sup>* ) in the first (second) integral, one gets:

*dλ*1*dλ*<sup>2</sup>

*λ*2]*u*0( <sup>√</sup>*σ*<sup>1</sup> 2 (*λ*<sup>1</sup>

<sup>√</sup>*τ*]*u*0(*S*1*eσ*<sup>1</sup>

<sup>√</sup>*τ*]*u*0(*S*1*eσ*<sup>1</sup>

<sup>√</sup>*τ*) is written in the form:

*σ*2

<sup>√</sup>2 ln *<sup>S</sup>*<sup>2</sup> *σ*2

<sup>√</sup><sup>1</sup>−*ρ*<sup>2</sup> <sup>−</sup>*λ*<sup>1</sup>

<sup>4</sup>*<sup>τ</sup>* ]

 2

*<sup>i</sup>* (1−*ρ*<sup>2</sup>) for *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, 2 and *<sup>i</sup>* �<sup>=</sup> *<sup>j</sup>*,

<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*<sup>2</sup> <sup>+</sup> *ρλ*2), *<sup>σ</sup>*

<sup>√</sup>*τη*<sup>1</sup> , *<sup>S</sup>*2*eσ*<sup>2</sup>

<sup>√</sup>*τη*<sup>1</sup> , *<sup>S</sup>*−<sup>1</sup> <sup>2</sup> *<sup>e</sup>σ*<sup>2</sup>

*<sup>σ</sup>*1*σ*<sup>2</sup> = 0. Then the first integral (after changing to

√2*λ*2 <sup>2</sup> )1*λ*<sup>2</sup> <sup>&</sup>lt;

> √ 2 ln *S*<sup>1</sup>

<sup>√</sup>2*<sup>τ</sup>* ,

<sup>√</sup>1−*ρ*<sup>2</sup>+*ρλ*2<sup>−</sup>

*σ*1

<sup>√</sup>*τη*<sup>2</sup> )

<sup>√</sup>*τη*<sup>2</sup> )

<sup>2</sup> for *<sup>i</sup>*, *<sup>j</sup>* <sup>=</sup> 1, 2, we have *<sup>α</sup><sup>i</sup>* <sup>=</sup> <sup>−</sup>*μi*+*ρμjσi*/*σ<sup>j</sup>*

2 ln *S*2/*σ*2+*λ*2)<sup>2</sup> <sup>4</sup>*<sup>τ</sup>* ]

<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*<sup>2</sup> <sup>+</sup> *ρλ*2) <sup>−</sup> *<sup>α</sup>*

Splitting the integral into two integrals and changing to variables *<sup>η</sup>*<sup>1</sup> <sup>=</sup> *<sup>λ</sup>*<sup>1</sup>

<sup>−</sup><sup>∞</sup> exp[−(*α*1*σ*1*η*<sup>1</sup> <sup>+</sup> *<sup>α</sup>*2*σ*2*η*2)

<sup>−</sup><sup>∞</sup> exp[−(*α*1*σ*1*η*<sup>1</sup> <sup>+</sup> *<sup>α</sup>*2*σ*2*η*2)

<sup>−</sup> *<sup>ρ</sup> <sup>μ</sup>*1*μ*<sup>2</sup>

*σ*2

2 <sup>2</sup> ]*dη*1*dη*2.

<sup>√</sup>*τ*, *<sup>X</sup>*<sup>2</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup>

*<sup>R</sup>*<sup>2</sup> *w*0(*λ*1, *λ*2) exp[−

√

taking *<sup>μ</sup><sup>i</sup>* <sup>=</sup> *<sup>r</sup>* <sup>−</sup> *<sup>σ</sup>*<sup>2</sup>

*σiσ<sup>j</sup>*

*β* = ∑*i*,*j*=1,2

 exp[<sup>−</sup> (

where

<sup>√</sup>2 ln *<sup>S</sup>*<sup>∗</sup> *<sup>σ</sup>*<sup>2</sup> .

*η*<sup>2</sup> = *<sup>λ</sup>*2<sup>−</sup>

where *<sup>I</sup>*<sup>1</sup> <sup>=</sup> *<sup>e</sup>βτ* 2*π* <sup>√</sup>1−*ρ*<sup>2</sup>

*<sup>u</sup>*(*t*, *<sup>S</sup>*1, *<sup>S</sup>*2) = *<sup>S</sup>α*<sup>1</sup>

√

*<sup>w</sup>*0(*λ*1, *<sup>λ</sup>*2) = exp[<sup>−</sup> *<sup>α</sup>*

<sup>√</sup>2 ln *<sup>S</sup>*<sup>2</sup> *σ*2

*u*(*t*, *S*1, *S*2) = *I*<sup>1</sup> − *I*<sup>2</sup>

exp[<sup>−</sup> (*η*1−*ρη*<sup>2</sup> )<sup>2</sup>

*<sup>I</sup>*<sup>2</sup> <sup>=</sup> *<sup>S</sup>*<sup>2</sup> *<sup>e</sup>βτ* 2*π* <sup>√</sup>1−*ρ*<sup>2</sup> *i*

<sup>1</sup> *<sup>S</sup>α*<sup>2</sup> <sup>2</sup> *<sup>e</sup>βτ* 4*πτ* 

<sup>√</sup>2*<sup>τ</sup>* (*η*<sup>2</sup> <sup>=</sup> *<sup>λ</sup>*2<sup>+</sup>

 +<sup>∞</sup> −∞

 +<sup>∞</sup> −∞

Note that (*<sup>β</sup>* <sup>+</sup> *<sup>r</sup>*)(<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*2) + *<sup>μ</sup>*<sup>2</sup>

2 <sup>2</sup> ]*dη*1*dη*<sup>2</sup>

<sup>2</sup>(1−*ρ*<sup>2</sup>) <sup>−</sup> *<sup>η</sup>*<sup>2</sup>

exp[<sup>−</sup> (−*η*1+*ρη*2−2*<sup>ρ</sup>* ln *<sup>S</sup>*2/(*σ*<sup>2</sup>

variables *<sup>X</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*<sup>1</sup>

<sup>4</sup>*<sup>τ</sup>* ] <sup>−</sup> exp[<sup>−</sup> (

√1*σ*1 <sup>2</sup> (*λ*<sup>1</sup>

 ln(*S*∗/*S*2) *σ*2 √*τ*

 ln(*S*∗/*S*2) *σ*2 √*τ*

*σ*1

<sup>√</sup>*τ*))<sup>2</sup> <sup>2</sup>(1−*ρ*<sup>2</sup>) <sup>−</sup> *<sup>η</sup>*<sup>2</sup>

> 1 2*σ*<sup>2</sup> 1 <sup>+</sup> *<sup>μ</sup>*<sup>2</sup> 2 2*σ*<sup>2</sup> 2

√ 2 ln *S*<sup>2</sup> *σ*2

2 ln *<sup>S</sup>*2/*σ*2−*λ*2)<sup>2</sup>

Changing to the variables *<sup>X</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>*η*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*<sup>1</sup> *σ*1 <sup>√</sup>*<sup>τ</sup>* <sup>−</sup> <sup>2</sup>*<sup>ρ</sup>* ln *<sup>S</sup>*<sup>2</sup> *σ*2 <sup>√</sup>*<sup>τ</sup>* , *<sup>X</sup>*<sup>2</sup> <sup>=</sup> *<sup>η</sup>*<sup>2</sup> <sup>−</sup> *<sup>μ</sup>*<sup>2</sup> *σ*2 <sup>√</sup>*τ*, the second integral becomes:

$$\begin{split} I\_{2} &= \frac{e^{-r\tau}}{2\pi\sqrt{1-\rho^{2}}} (\mathcal{S}\_{2})^{-\frac{2\mu\_{2}}{\sigma\_{2}^{2}}} \int\_{-\infty}^{+\infty} \int\_{-\infty}^{\frac{\ln(\mathcal{S}\_{2}\mathcal{S}^{+}) - \mu\_{2}\tau}{\sigma\_{2}\sqrt{\tau}}} \exp[-\frac{1}{2(1-\rho^{2})} (X\_{1}^{2} + X\_{2}^{2} + 2\rho X\_{1}X\_{2})] \, d\tau \\ &\mu\_{0} (\mathcal{S}\_{1}\mathcal{S}\_{2}^{-2\rho\frac{\sigma\_{1}}{\sigma\_{2}}}e^{\mu\_{1}\tau - \sigma\_{1}\sqrt{\tau}X\_{1}} \mathcal{S}\_{2}^{-1} e^{\mu\_{2}\tau + \sigma\_{2}\sqrt{\tau}X\_{2}}) dX\_{1} dX\_{2} . \end{split}$$

In the special case of standard options one has: *u*0(*S*1, *S*2) = max(*ω*(*S*<sup>1</sup> − *K*), 0), *ω* = ±1. Then *I*<sup>1</sup> can be written in the form:

$$
\omega \text{S}\_1 \text{N}\_2(\omega d^+, e^+; -\rho \omega) - \omega \text{K} e^{-r \text{T}} \text{N}\_2(\omega d^-, e^-; -\rho \omega),
$$

where *<sup>N</sup>*<sup>2</sup> is the bivariate cumulative normal distribution function, *<sup>d</sup>*<sup>±</sup> <sup>=</sup> ln( *<sup>S</sup>*<sup>1</sup> *<sup>K</sup>* )+(*r*±*<sup>σ</sup>*<sup>2</sup> 1 <sup>2</sup> )*τ σ*1 <sup>√</sup>*<sup>τ</sup>* , *<sup>e</sup>*<sup>−</sup> = −ln( *<sup>S</sup>*<sup>2</sup> *<sup>S</sup>*<sup>∗</sup> )+*μ*2*τ σ*2 <sup>√</sup>*<sup>τ</sup>* , *<sup>e</sup>*<sup>+</sup> <sup>=</sup> *<sup>e</sup>*<sup>−</sup> <sup>−</sup> *ρσ*<sup>1</sup> <sup>√</sup>*τ*. Similarly *<sup>I</sup>*<sup>2</sup> is written in the form: *ωe* <sup>−</sup><sup>2</sup> *<sup>μ</sup>*<sup>2</sup> *σ*2 2 ln( *<sup>S</sup>*<sup>2</sup> *<sup>S</sup>*<sup>∗</sup> ) [*e* <sup>−</sup>2*<sup>ρ</sup> <sup>σ</sup>*<sup>1</sup> *<sup>σ</sup>*<sup>2</sup> ln( *<sup>S</sup>*<sup>2</sup> *<sup>S</sup>*<sup>∗</sup> ) *S*1*N*2(*ωd* <sup>+</sup>, *<sup>e</sup>*+; <sup>−</sup>*ρω*) <sup>−</sup> *Ke*−*rτN*2(*ω<sup>d</sup>* <sup>−</sup>, *<sup>e</sup>*−; <sup>−</sup>*ρω*)

$$\text{where } \hat{d}^{\pm} = d^{\pm} - \frac{2\rho}{\sigma\_2 \sqrt{\tau}} \ln(\frac{S\_2}{S^+}), \hat{e}^{\pm} = e^{\pm} + \frac{2}{\sigma\_2 \sqrt{\tau}} \ln(\frac{S\_2}{S^+}).$$

Simulating CNN for multi-asset option with single barrier model, we obtain the following figure with different values of the parameter set:

**Figure 3.** CNN simulations for Example 2. (a) *r* = 1, *T* = 60 days, *σ* = 1, *ρ* = 0.05; (b) *r* = 0.5, *T* = 120 days, *σ* = 1.5, *ρ* = 0.06 .

**Example 3. (Two-asset barrier options with simultaneous barriers)** While single-asset barrier options have received substantial coverage in the literature, multi-asset options with several barriers have been discussed only in some special cases (e.g. sequential barriers, radial

#### 14 Will-be-set-by-IN-TECH 44 Risk Management – Current Issues and Challenges Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach <sup>15</sup>

options, etc.). Here we show how the case of two simultaneous barriers can be valued straightforwardly from the arguments in Section 2. Let us confine ourselves to zero-rebate options for simplicity's sake, although Section 2 deals with the general case too. Then the boundary value problem takes the form:

$$\begin{cases} Lu = 0 \quad \text{in} \quad \Omega\\ u|\_{t=T} = u\_0(\mathcal{S}\_1, \mathcal{S}\_2) \\\ u|\_{\mathcal{S}\_1 = \mathcal{S}\_1^\*} = 0 \quad \text{and} \quad u|\_{\mathcal{S}\_2 = \mathcal{S}\_2^\*} = 0 \quad 0 \le t \le T \end{cases}$$

where *L* = *∂<sup>t</sup>* + ∑<sup>2</sup> *i*=1 *σ*2 *<sup>i</sup> S*<sup>2</sup> *i* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup> *Si* <sup>+</sup> *ρσ*1*σ*2*S*1*S*2*∂*<sup>2</sup> *<sup>S</sup>*1*S*<sup>2</sup> <sup>+</sup> *<sup>r</sup>* <sup>∑</sup><sup>2</sup> *<sup>i</sup>*=<sup>1</sup> *Si∂Si* − *r*, Ω = {(*t*, *S*1, *S*2); 0 < *t* < *T*, 0 < *S*<sup>1</sup> < *S*<sup>∗</sup> <sup>1</sup>, 0 < *S*<sup>2</sup> < *S*<sup>∗</sup> <sup>2</sup> }. Arguing as in the last part of Section 2 and taking

$$D = \begin{pmatrix} \sigma\_1 & \rho \sigma\_2 \\ 0 & \sqrt{1 - \rho^2} \sigma\_2 \end{pmatrix}, \rho^2 < 1, \sigma\_1 > 0, \sigma\_2 > 0$$

and *<sup>ϕ</sup>*<sup>0</sup> as the opening of the angle between � *<sup>x</sup>* <sup>≤</sup> <sup>0</sup> *<sup>y</sup>* <sup>=</sup> <sup>0</sup> and � *x* = *ρσ*2*η*, *η* ≥ 0 *<sup>y</sup>* <sup>=</sup> �<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*2*σ*2*<sup>η</sup>* , from (21) we have

$$w(\pi, r, \varphi) = \int\_0^{\varphi\_0} \int\_0^{\infty} w\_0(\xi, \eta) G(r, \varphi, \xi, \eta, \tau) \xi d\xi d\eta,\tag{29}$$

**4. Comparison principle for multi-asset Black-Scholes equations**

*aijxixjuxixj* +

*III* = {*x*<sup>2</sup> = 0, 0 < *x*<sup>1</sup> < *a*1, 0 < *t* < *T*}. The Dirichlet data are prescribed on Γ:

2 ∑ *i*=1

where (*aij*)<sup>∗</sup> = (*aij*), (*aij*) > 0, *aij*, *bi*, *c* are real constants and *c* < 0 in the domain

into two parts: Parabolic Γ = {*x*<sup>1</sup> = *a*1, 0 < *x*<sup>2</sup> < *a*2, 0 < *t* < *T*}∪{*x*<sup>2</sup> = *a*2, 0 < *x*<sup>1</sup> < *a*1, 0 < *t* < *T*}∪{*t* = *T*, 0 < *xj* < *aj*, *j* = 1, 2} and free of boundary data part Γ<sup>1</sup> = *I* ∪ *I I* ∪ *III*, where *I* = {0 < *xj* < *aj*, *j* = 1, 2; *t* = 0}, *I I* = {*x*<sup>1</sup> = 0, 0 < *x*<sup>2</sup> < *a*2, 0 < *t* < *T*},

*Assume that u is a classical solution of (30), (31), i.e. u* <sup>∈</sup> *<sup>C</sup>*2(*<sup>D</sup>* <sup>∪</sup> <sup>Γ</sup>¯ <sup>1</sup>) <sup>∩</sup> *<sup>C</sup>*0(*D*¯ )*. Let v be another solution of (30), (31) belonging to C*2(*<sup>D</sup>* <sup>∪</sup> <sup>Γ</sup>¯ <sup>1</sup>) <sup>∩</sup> *<sup>C</sup>*0(*D*¯ )*. Suppose that u*|<sup>Γ</sup> <sup>≤</sup> *<sup>v</sup>*|Γ*. Then u* <sup>≤</sup> *<sup>v</sup>*

**Proof.** Put *<sup>w</sup>* <sup>=</sup> *<sup>u</sup>* <sup>−</sup> *<sup>v</sup>*. Assume that *max w* <sup>=</sup> *<sup>w</sup>*(*t*0, *<sup>x</sup>*0) = *<sup>M</sup>* <sup>&</sup>gt; 0, *<sup>P</sup>*<sup>0</sup> = (*t*0, *<sup>x</sup>*0) <sup>∈</sup> *<sup>D</sup>*¯ . Evidently,

Case a). (*t*0, *x*0) ∈ *D*. Having in mind that ∑ *aijxixjwxixj* is a strictly elliptic operator in the open rectangle {0 < *xj* < *aj*, *j* = 1, 2} we shall apply the interior parabolic maximum principle ( see A.Friedman, Partial Differential equations of parabolic type, Prentice Hall, Inc.

such that *x*<sup>0</sup> ∈ Π = (*ε*1, *a*1) × (*ε*2, *a*2), 0 < *t*<sup>0</sup> < *T*. Then Th1 from Chapter II of the above mentioned book gives: *<sup>w</sup>* <sup>≡</sup> *<sup>M</sup>* <sup>&</sup>gt; 0 for *<sup>T</sup>* <sup>≥</sup> *<sup>t</sup>* <sup>≥</sup> *<sup>t</sup>*0, *<sup>x</sup>* <sup>∈</sup> <sup>Π</sup>¯ and this is a contradiction with

b). (2) Again *wt*(*P*0) ≤ 0 and *wx*<sup>1</sup> (*P*0) = 0, *wx*<sup>1</sup> *<sup>x</sup>*<sup>1</sup> (*P*0) ≤ 0 as *P*<sup>0</sup> is interior point for the interval

*∂w ∂x*<sup>1</sup>

(*P*0) + *b*<sup>1</sup>

, (2)

(*P*0) ≤ 0 as it is shown in Friedman book. Obviously, *wt*(*P*0) ≤ 0, as

 0 < *x*<sup>10</sup> < *a*<sup>1</sup> 0 < *x*<sup>20</sup> < *a*<sup>2</sup> *bixiuxi* + *cu* = *f* , (30)

Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

*u*|<sup>Γ</sup> = *g* (31)

 0 < *x*<sup>10</sup> < *a*<sup>1</sup> *<sup>x</sup>*<sup>20</sup> <sup>=</sup> <sup>0</sup> , (3)

*∂xj*

<sup>1</sup> *aijxi*0*xj*<sup>0</sup>

(*P*0) + *cw*(*P*0) + *wt*(*P*0) = 0 → contradiction.

0 < *t* < *T*

<sup>0</sup> <sup>&</sup>lt; *<sup>ε</sup><sup>j</sup>* <sup>&</sup>lt; *xj* <sup>&</sup>lt; *aj*, *<sup>j</sup>* <sup>=</sup> 1, 2

(*P*0) = 0, *j* = 1, 2, while

*∂*2*w ∂xi∂xj*  *x*<sup>10</sup> = 0 *<sup>x</sup>*<sup>20</sup> <sup>=</sup> <sup>0</sup> and

(*P*0) + *cw*(*P*0) +

,

45

, *aj* = *const* > 0. The boundary of the parallelepiped *D* is split

For the sake of simplicity consider

**Theorem 1.** (Comparison principle)

(*t*0, *x*0) ∈ *D* ∪ Γ<sup>1</sup> as *w*|<sup>Γ</sup> ≤ 0.

Case b). (*t*0, *x*0) ∈ *I* ⇒ *t*<sup>0</sup> = 0, (1)

*∂*2*w ∂xi∂xj*

(0, *a*1). According to (30) : *a*11*x*<sup>2</sup>

0 < *t* < *T* 0 < *xj* < *aj*,

*everywhere in D.* ¯

*w* ≤ 0 on *t* = *T*.

*<sup>i</sup>*,*<sup>j</sup> aijxi*0*xj*<sup>0</sup>

∑2

*D* :  *ut* +

*<sup>j</sup>* <sup>=</sup> 1, 2

2 ∑ *i*,*j*=1

(1964), Chapter II). To do this we shall work in the domain *D*<sup>1</sup> :

a similar case with respect to *x*<sup>20</sup> ∈ [0, *a*2), *x*<sup>10</sup> = 0. Thus,

*wt*(*P*0) = 0 -contradiction with *c* < 0, *w*(*P*0) > 0.

b). (1) *<sup>x</sup>*<sup>0</sup> is interior point of (0, *<sup>a</sup>*1) <sup>×</sup> (0, *<sup>a</sup>*2) and therefore *<sup>∂</sup><sup>w</sup>*

*<sup>w</sup>*(0, *<sup>x</sup>*0) = *<sup>M</sup>* <sup>=</sup> *maxD*¯ *<sup>w</sup>*. As we know, (30) is satisfied on I <sup>⇒</sup> <sup>∑</sup><sup>2</sup>

10 *∂*2*w ∂x*<sup>2</sup> 1

where *G*(*r*, *ϕ*, *ξ*, *η*, *τ*) = <sup>1</sup> *<sup>ϕ</sup>*0*<sup>τ</sup> <sup>e</sup>*<sup>−</sup> (*r*2+*ξ*2) <sup>4</sup>*<sup>τ</sup>* ∑<sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> *I <sup>n</sup><sup>π</sup> ϕ*0 ( *rξ* <sup>2</sup>*<sup>τ</sup>* )*sin <sup>n</sup><sup>π</sup> <sup>ϕ</sup>*<sup>0</sup> *<sup>ϕ</sup>sin <sup>n</sup><sup>π</sup> <sup>ϕ</sup>*<sup>0</sup> *η* and *Iv* is the modified Bessel function satisfying (22). Here *<sup>w</sup>*0(*r*, *<sup>ϕ</sup>*) = *<sup>v</sup>*�0(*D*∗*z*) <sup>|</sup>*z*1=*<sup>r</sup>* cos *<sup>ϕ</sup>*,*z*2=*<sup>r</sup>* sin *<sup>ϕ</sup>* where *<sup>v</sup>*�0(*λ*) = *u*0(*S*∗ <sup>1</sup> *<sup>e</sup>*−*λ*<sup>1</sup> , *<sup>S</sup>*<sup>∗</sup> <sup>2</sup> *<sup>e</sup>*−*λ*<sup>2</sup> )*e*−Σ*αi*(ln *<sup>S</sup>*<sup>∗</sup> *<sup>i</sup>* <sup>−</sup>*λi*). Changing back the variables one obtains *u*(*t*, *S*1, *S*2).

Simulating CNN for two-asset barrier options with simultaneous barriers model, we obtain the following figure with different values of the parameter set:

**Figure 4.** CNN simulations for Example 3. (a) *r* = 1, *T* = 120 days, *σ* = 1, *ρ* = 0.05; (b) *r* = 0.5, *T* = 180 days, *σ* = 1.5, *ρ* = 0.06 .

#### **4. Comparison principle for multi-asset Black-Scholes equations**

For the sake of simplicity consider

14 Will-be-set-by-IN-TECH

options, etc.). Here we show how the case of two simultaneous barriers can be valued straightforwardly from the arguments in Section 2. Let us confine ourselves to zero-rebate options for simplicity's sake, although Section 2 deals with the general case too. Then the

<sup>1</sup> = 0 and, *u*|*S*2=*S*<sup>∗</sup>

*<sup>S</sup>*1*S*<sup>2</sup> <sup>+</sup> *<sup>r</sup>* <sup>∑</sup><sup>2</sup>

� *<sup>x</sup>* <sup>≤</sup> <sup>0</sup> *<sup>y</sup>* <sup>=</sup> <sup>0</sup> and

<sup>2</sup> }. Arguing as in the last part of Section 2 and taking

<sup>2</sup> = 0 0 ≤ *t* ≤ *T*

�

*<sup>ϕ</sup>*<sup>0</sup> *<sup>ϕ</sup>sin <sup>n</sup><sup>π</sup>*

*<sup>i</sup>* <sup>−</sup>*λi*). Changing back the variables one obtains *u*(*t*, *S*1, *S*2).

*<sup>i</sup>*=<sup>1</sup> *Si∂Si* − *r*, Ω = {(*t*, *S*1, *S*2); 0 < *t* <

*x* = *ρσ*2*η*, *η* ≥ 0

*w*0(*ξ*, *η*)*G*(*r*, *ϕ*, *ξ*, *η*, *τ*)*ξdξdη*, (29)

*<sup>y</sup>* <sup>=</sup> �<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*2*σ*2*<sup>η</sup>* , from (21) we

*<sup>ϕ</sup>*<sup>0</sup> *η* and *Iv* is the modified Bessel

boundary value problem takes the form:

*i*=1 *σ*2 *<sup>i</sup> S*<sup>2</sup> *i* <sup>2</sup> *<sup>∂</sup>*<sup>2</sup>

<sup>1</sup>, 0 < *S*<sup>2</sup> < *S*<sup>∗</sup>

�

and *ϕ*<sup>0</sup> as the opening of the angle between

<sup>2</sup> *<sup>e</sup>*−*λ*<sup>2</sup> )*e*−Σ*αi*(ln *<sup>S</sup>*<sup>∗</sup>

*w*(*τ*,*r*, *ϕ*) =

*<sup>ϕ</sup>*0*<sup>τ</sup> <sup>e</sup>*<sup>−</sup> (*r*2+*ξ*2)

the following figure with different values of the parameter set:

where *L* = *∂<sup>t</sup>* + ∑<sup>2</sup>

*σ*<sup>1</sup> *ρσ*<sup>2</sup> <sup>0</sup> �<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>*2*σ*<sup>2</sup>

where *G*(*r*, *ϕ*, *ξ*, *η*, *τ*) = <sup>1</sup>

<sup>1</sup> *<sup>e</sup>*−*λ*<sup>1</sup> , *<sup>S</sup>*<sup>∗</sup>

days, *σ* = 1.5, *ρ* = 0.06 .

*T*, 0 < *S*<sup>1</sup> < *S*<sup>∗</sup>

*D* = �

have

*u*0(*S*∗

⎧ ⎪⎨

*Lu* = 0 in Ω *u*|*t*=*<sup>T</sup>* = *u*0(*S*1, *S*2)

*Si* <sup>+</sup> *ρσ*1*σ*2*S*1*S*2*∂*<sup>2</sup>

, *ρ*<sup>2</sup> < 1, *σ*<sup>1</sup> > 0, *σ*<sup>2</sup> > 0

� *ϕ*<sup>0</sup> 0

<sup>4</sup>*<sup>τ</sup>* ∑<sup>∞</sup>

� ∞ 0

*<sup>n</sup>*=<sup>1</sup> *I <sup>n</sup><sup>π</sup> ϕ*0 ( *rξ* <sup>2</sup>*<sup>τ</sup>* )*sin <sup>n</sup><sup>π</sup>*

function satisfying (22). Here *<sup>w</sup>*0(*r*, *<sup>ϕ</sup>*) = *<sup>v</sup>*�0(*D*∗*z*) <sup>|</sup>*z*1=*<sup>r</sup>* cos *<sup>ϕ</sup>*,*z*2=*<sup>r</sup>* sin *<sup>ϕ</sup>* where *<sup>v</sup>*�0(*λ*) =

Simulating CNN for two-asset barrier options with simultaneous barriers model, we obtain

(a) (b)

**Figure 4.** CNN simulations for Example 3. (a) *r* = 1, *T* = 120 days, *σ* = 1, *ρ* = 0.05; (b) *r* = 0.5, *T* = 180

⎪⎩

*u*|*S*1=*S*<sup>∗</sup>

$$u\_t + \sum\_{i,j=1}^{2} a\_{ij} \mathbf{x}\_i \mathbf{x}\_j u\_{\mathbf{x}\_i \mathbf{x}\_j} + \sum\_{i=1}^{2} b\_i \mathbf{x}\_i u\_{\mathbf{x}\_i} + cu = f\_\prime \tag{30}$$

where (*aij*)<sup>∗</sup> = (*aij*), (*aij*) > 0, *aij*, *bi*, *c* are real constants and *c* < 0 in the domain *D* : 0 < *t* < *T* 0 < *xj* < *aj*, *<sup>j</sup>* <sup>=</sup> 1, 2 , *aj* = *const* > 0. The boundary of the parallelepiped *D* is split into two parts: Parabolic Γ = {*x*<sup>1</sup> = *a*1, 0 < *x*<sup>2</sup> < *a*2, 0 < *t* < *T*}∪{*x*<sup>2</sup> = *a*2, 0 < *x*<sup>1</sup> < *a*1, 0 < *t* < *T*}∪{*t* = *T*, 0 < *xj* < *aj*, *j* = 1, 2} and free of boundary data part Γ<sup>1</sup> = *I* ∪ *I I* ∪ *III*, where *I* = {0 < *xj* < *aj*, *j* = 1, 2; *t* = 0}, *I I* = {*x*<sup>1</sup> = 0, 0 < *x*<sup>2</sup> < *a*2, 0 < *t* < *T*}, *III* = {*x*<sup>2</sup> = 0, 0 < *x*<sup>1</sup> < *a*1, 0 < *t* < *T*}. The Dirichlet data are prescribed on Γ:

$$\mathfrak{u}|\_{\Gamma} = \mathfrak{g} \tag{31}$$

#### **Theorem 1.** (Comparison principle)

*Assume that u is a classical solution of (30), (31), i.e. u* <sup>∈</sup> *<sup>C</sup>*2(*<sup>D</sup>* <sup>∪</sup> <sup>Γ</sup>¯ <sup>1</sup>) <sup>∩</sup> *<sup>C</sup>*0(*D*¯ )*. Let v be another solution of (30), (31) belonging to C*2(*<sup>D</sup>* <sup>∪</sup> <sup>Γ</sup>¯ <sup>1</sup>) <sup>∩</sup> *<sup>C</sup>*0(*D*¯ )*. Suppose that u*|<sup>Γ</sup> <sup>≤</sup> *<sup>v</sup>*|Γ*. Then u* <sup>≤</sup> *<sup>v</sup> everywhere in D.* ¯

**Proof.** Put *<sup>w</sup>* <sup>=</sup> *<sup>u</sup>* <sup>−</sup> *<sup>v</sup>*. Assume that *max w* <sup>=</sup> *<sup>w</sup>*(*t*0, *<sup>x</sup>*0) = *<sup>M</sup>* <sup>&</sup>gt; 0, *<sup>P</sup>*<sup>0</sup> = (*t*0, *<sup>x</sup>*0) <sup>∈</sup> *<sup>D</sup>*¯ . Evidently, (*t*0, *x*0) ∈ *D* ∪ Γ<sup>1</sup> as *w*|<sup>Γ</sup> ≤ 0.

Case a). (*t*0, *x*0) ∈ *D*. Having in mind that ∑ *aijxixjwxixj* is a strictly elliptic operator in the open rectangle {0 < *xj* < *aj*, *j* = 1, 2} we shall apply the interior parabolic maximum principle ( see A.Friedman, Partial Differential equations of parabolic type, Prentice Hall, Inc.

(1964), Chapter II). To do this we shall work in the domain *D*<sup>1</sup> : 0 < *t* < *T* <sup>0</sup> <sup>&</sup>lt; *<sup>ε</sup><sup>j</sup>* <sup>&</sup>lt; *xj* <sup>&</sup>lt; *aj*, *<sup>j</sup>* <sup>=</sup> 1, 2 ,

such that *x*<sup>0</sup> ∈ Π = (*ε*1, *a*1) × (*ε*2, *a*2), 0 < *t*<sup>0</sup> < *T*. Then Th1 from Chapter II of the above mentioned book gives: *<sup>w</sup>* <sup>≡</sup> *<sup>M</sup>* <sup>&</sup>gt; 0 for *<sup>T</sup>* <sup>≥</sup> *<sup>t</sup>* <sup>≥</sup> *<sup>t</sup>*0, *<sup>x</sup>* <sup>∈</sup> <sup>Π</sup>¯ and this is a contradiction with *w* ≤ 0 on *t* = *T*.

Case b). (*t*0, *x*0) ∈ *I* ⇒ *t*<sup>0</sup> = 0, (1) 0 < *x*<sup>10</sup> < *a*<sup>1</sup> 0 < *x*<sup>20</sup> < *a*<sup>2</sup> , (2) 0 < *x*<sup>10</sup> < *a*<sup>1</sup> *<sup>x</sup>*<sup>20</sup> <sup>=</sup> <sup>0</sup> , (3) *x*<sup>10</sup> = 0 *<sup>x</sup>*<sup>20</sup> <sup>=</sup> <sup>0</sup> and a similar case with respect to *x*<sup>20</sup> ∈ [0, *a*2), *x*<sup>10</sup> = 0. Thus,

b). (1) *<sup>x</sup>*<sup>0</sup> is interior point of (0, *<sup>a</sup>*1) <sup>×</sup> (0, *<sup>a</sup>*2) and therefore *<sup>∂</sup><sup>w</sup> ∂xj* (*P*0) = 0, *j* = 1, 2, while ∑2 *<sup>i</sup>*,*<sup>j</sup> aijxi*0*xj*<sup>0</sup> *∂*2*w ∂xi∂xj* (*P*0) ≤ 0 as it is shown in Friedman book. Obviously, *wt*(*P*0) ≤ 0, as *<sup>w</sup>*(0, *<sup>x</sup>*0) = *<sup>M</sup>* <sup>=</sup> *maxD*¯ *<sup>w</sup>*. As we know, (30) is satisfied on I <sup>⇒</sup> <sup>∑</sup><sup>2</sup> <sup>1</sup> *aijxi*0*xj*<sup>0</sup> *∂*2*w ∂xi∂xj* (*P*0) + *cw*(*P*0) + *wt*(*P*0) = 0 -contradiction with *c* < 0, *w*(*P*0) > 0.

b). (2) Again *wt*(*P*0) ≤ 0 and *wx*<sup>1</sup> (*P*0) = 0, *wx*<sup>1</sup> *<sup>x</sup>*<sup>1</sup> (*P*0) ≤ 0 as *P*<sup>0</sup> is interior point for the interval (0, *a*1). According to (30) : *a*11*x*<sup>2</sup> 10 *∂*2*w ∂x*<sup>2</sup> 1 (*P*0) + *b*<sup>1</sup> *∂w ∂x*<sup>1</sup> (*P*0) + *cw*(*P*0) + *wt*(*P*0) = 0 → contradiction. b). (3) Then (30) takes the form: *cw*(*P*0) + *wt*(*P*0) = 0 - contradiction.

$$\begin{array}{l}\text{(Case c). } (t\_0, \mathbf{x}\_0) \in II \Rightarrow 0 \le t\_0 < T, \ \mathbf{x}\_{10} = \mathbf{0}; \ \text{(1)} \quad \begin{cases} 0 < t\_0 < T \\ 0 < \mathbf{x}\_{20} < a\_2 \end{cases} \text{(2)} \quad \begin{cases} t\_0 = 0 \\ 0 < \mathbf{x}\_2^0 < a\_2 \end{cases} \text{(3)} \\\ \text{(3)} \quad \begin{cases} t\_0 = 0 \\ \mathbf{x}\_2^0 = \mathbf{0} \end{cases} \text{(4)} \quad \begin{cases} T > t\_0 > 0 \\ \mathbf{x}\_2^0 = \mathbf{0} \end{cases} \text{(4)} \quad \begin{cases} t\_0 = 0 \\ \mathbf{x}\_2^0 = \mathbf{0} \end{cases} \text{(5)} \quad \begin{cases} t\_0 = 0 \\ \mathbf{x}\_2^0 = \mathbf{0} \end{cases} \text{(6)} \quad \begin{cases} t\_0 = 0 \\ \mathbf{x}\_2^0 = \mathbf{0} \end{cases} \text{(7)}$$

Certainly, *wt*(*P*0) ≤ 0 in each case (1) -(4).

c). (1) As *P*<sup>0</sup> is interior point in the rectangle {0 < *t* < *T*}×{0 < *x*<sup>2</sup> < *a*2} ⇒ *wt*(*P*0) = 0, *wx*<sup>2</sup> (*P*0) = 0, *wx*<sup>2</sup> *<sup>x</sup>*<sup>2</sup> (*P*0) <sup>≤</sup> 0. According to (30) *<sup>a</sup>*2*x*<sup>2</sup> <sup>20</sup>*wx*<sup>2</sup> *<sup>x</sup>*<sup>2</sup> (*P*0) + *b*2*x*20*wx*<sup>2</sup> (*P*0) + *cw*(*P*0) + *wt*(*P*0) = 0 - contradiction.

c). (2) As *x*<sup>0</sup> <sup>2</sup> ∈ (0, *a*2) ⇒ *wx*<sup>2</sup> (*P*0) = 0, *wx*<sup>2</sup> *<sup>x</sup>*<sup>2</sup> (*P*0) ≤ 0. The contradiction is obvious.

c). (3) The equation (30) takes the form:

$$
\sigma w(P\_0) + w\_t(P\_0) = 0 \tag{32}
$$

**5. The approach of Fichera-Oleinik-Radkeviˇc**

*k*,*j*=1,...,*m*

*<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup>; <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kj*(*x*)*nknj* <sup>&</sup>gt; <sup>0</sup>

*β*(*x*) = ∑

= *β*.*A* where *A* > 0 and *A* is continuous.

*k*,*j*=1,...,*m*

Σ. Following Fichera (1956) we introduce on Σ<sup>0</sup> the Fichera function:

*k*=1,...,*m*

*L*(*u*) = ∑

*<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup>; <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kj*(*x*)*nknj* <sup>=</sup> <sup>0</sup>

Then we split Σ<sup>0</sup> into three parts, namely

 ,

 ,

 .

Black-Scholes type.

smooth boundary Σ:

*<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup>0; *<sup>β</sup>*(*x*) <sup>&</sup>gt; <sup>0</sup>

*<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup>0; *<sup>β</sup>*(*x*) <sup>&</sup>lt; <sup>0</sup>

*<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup>0; *<sup>β</sup>*(*x*) = <sup>0</sup>

change *y* = *F*(*x*) it takes the form

Assume now that *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω) and *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup>

<sup>Ω</sup> *uL*∗(*v*)*dx*,

*L*∗(*v*) = ∑

*kj*

*<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω); *<sup>v</sup>* <sup>=</sup> 0 at <sup>Σ</sup><sup>1</sup> <sup>∪</sup> <sup>Σ</sup><sup>3</sup>

 Ω

and let Σ<sup>3</sup> =

Σ<sup>0</sup> = 

<sup>Σ</sup><sup>1</sup> =

Σ<sup>2</sup> =

Σ<sup>0</sup> =

*β* . Then *β*

where

<sup>V</sup> <sup>=</sup>

<sup>Ω</sup> *<sup>L</sup>*(*u*)*vdx* <sup>=</sup>

and *b*∗*<sup>k</sup>* = 2 ∑*j*=1,...,*<sup>m</sup> a*

In this section we revise the results of [9] and [20] for the Dirichlet problem for PDEs of second order having non-negative characteristic form; then the method is applied to some PDEs of

To begin with consider the following equation in a bounded domain <sup>Ω</sup> <sup>⊂</sup> **<sup>R</sup>***<sup>m</sup>* with piecewise

where <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kj*(*x*)*ξkξ<sup>j</sup>* <sup>≥</sup> 0 , <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>, <sup>∀</sup>*<sup>ξ</sup>* <sup>∈</sup> **<sup>R</sup>***m*; *<sup>a</sup>kj*(*x*) = *<sup>a</sup>jk*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>. Moreover, *<sup>a</sup>kj* <sup>∈</sup> *<sup>C</sup>*2(Ω), *<sup>b</sup><sup>k</sup>* <sup>∈</sup> *<sup>C</sup>*1(Ω), *<sup>c</sup>* <sup>∈</sup> *<sup>C</sup>*0(Ω). Denote the unit inner normal to <sup>Σ</sup> by −→*<sup>n</sup>* = (*n*1, ..., *nm*)

(*bk*(*x*) <sup>−</sup> ∑

As it is proved in Oleinik and Radkeviˇc (1971) the sets Σ0, Σ1, Σ2, Σ<sup>3</sup> are invariant under smooth non-degenerate changes of the variables. More precisely, let *L*(*u*) = *f* in Ω; after the

<sup>0</sup> (Ω). Then

*<sup>a</sup>kj*(*x*)*vxkxj* <sup>+</sup> ∑

if we denote the Fichera function for *L*∗(*v*) by *β*∗, then *β*<sup>∗</sup> = −*β* and *β* is defined by (34).

Assume now that *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω), *<sup>u</sup>* <sup>=</sup> 0 at <sup>Σ</sup><sup>2</sup> <sup>∪</sup> <sup>Σ</sup>3, and define the following set of test functions:

 Ω

*k*=1,...,*m*

*b*∗*<sup>k</sup>*

. In view of the Green formula for *L* we get:

 Ω

*L*(*u*)*vdx* =

*kj xkxj* <sup>−</sup> *<sup>b</sup><sup>k</sup> xk*

*<sup>L</sup>*(*u*) = *<sup>f</sup>*

*xj* <sup>−</sup> *<sup>b</sup>k*, *<sup>c</sup>*<sup>∗</sup> <sup>=</sup> <sup>∑</sup>*k*=1,...,*m*(∑*j*=1,...,*<sup>m</sup> <sup>a</sup>*

(*L*(*u*)*v* − *L*∗(*v*)*u*)*dx* = 0 ⇔

*j*=1,...,*m*

*a kj xj*

*k*=1,...,*m*

*<sup>b</sup>k*(*x*)*uxk* + *<sup>c</sup>*(*x*)*<sup>u</sup>* = *<sup>f</sup>*(*x*) (33)

Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

be the non-characteristic part of Σ. Define

(*x*))*nk*, *<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup><sup>0</sup> (34)

(*x*)*vxk* + *c*∗(*x*)*v* (35)

) + *c*. One can easily see that

*uL*∗(*v*)*dx* (36)

*<sup>L</sup>*(*u*) = *<sup>f</sup>*

by

47

, i.e. <sup>Σ</sup> <sup>=</sup> <sup>Σ</sup><sup>0</sup> <sup>∪</sup> <sup>Σ</sup><sup>3</sup> and <sup>Σ</sup><sup>0</sup> is the characteristic part of

in Ω . Denote the Fichera function for

*<sup>a</sup>kj*(*x*)*uxkxj* <sup>+</sup> ∑

and again a contradiction.

c). (4). Then *wt*(*P*0) = 0 and according to (30) *cw*(*P*0) + *wt*(*P*0) = 0 - contradiction.

We conclude that *<sup>M</sup>* <sup>=</sup> *supD*¯ *<sup>w</sup>* <sup>≤</sup> <sup>0</sup> <sup>⇒</sup> *<sup>u</sup>* <sup>−</sup> *<sup>v</sup>* <sup>≤</sup> 0 in *<sup>D</sup>*¯ <sup>⇒</sup> *<sup>u</sup>* <sup>≤</sup> *<sup>v</sup>* in *<sup>D</sup>*¯ .

The comparison principle is proved.

**Remark 5.** The operator

$$Lu = u\_t + \sum\_{i,j=1}^n a\_{ij} \mathbf{x}\_i \mathbf{x}\_j u\_{\mathcal{X}\_i \mathbf{x}\_j} + \sum\_{i=1}^n b\_i \mathbf{x}\_i u\_{\mathcal{X}\_i} + cu\_t$$

is non-hypoelliptic. The constants *aij*, *bi*, *c* are arbitrary. To verify this we recall that the function *s<sup>a</sup>* <sup>+</sup> = *sa*, *s* > 0 0, *<sup>s</sup>* <sup>≤</sup> <sup>0</sup> considered as a Schwartz distribution in *<sup>D</sup>*� (**R**1) satisfies for *Re a* > 1 the following identities:

$$\mathbf{s} \mathbf{s}\_+^a = \mathbf{s}\_+^{a+1} \text{ / } \frac{d}{ds} \mathbf{s}\_+^a = a \mathbf{s}\_+^{a-1} \text{ / } \frac{d^2}{ds^2} \mathbf{s}\_+^a = a(a-1) \mathbf{s}\_+^{a-2} \text{ .} $$

Consider now the distribution *u* = *eλ<sup>t</sup> u*1(*x*1) ... *un*(*xn*), where *λ* = *const*, *uj*(*xj*) = *x dj j* ∈ *D*� (**R**<sup>1</sup> *xj* ), *Redj* <sup>&</sup>gt; 1. Then *<sup>u</sup>* <sup>∈</sup> *<sup>D</sup>*� (**R***n*+1) satisfies in distribution sense *Lu* = 0 if

$$\lambda + \sum\_{i \neq j}^{n} a\_{ij} d\_i d\_j + \sum\_{i=j}^{n} a\_{ii} d\_i (d\_i - 1) + \sum\_{i=1}^{n} b\_i d\_i + c = 0$$

Of course, *sing supp u* <sup>=</sup> *<sup>∂</sup>*{*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* : *xj* <sup>≥</sup> 0, 1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>*}, i.e. *sing supp u* is the boundary of the first octant of **R***<sup>n</sup> <sup>x</sup>* multiplied by **R**<sup>1</sup> *<sup>t</sup>* . The nonhypoellipticity is proved. Evidently, under (4) *L* is hypoelliptic in the open domain {*xj* > 0, 1 ≤ *j* ≤ *n*} as it is strictly parabolic there.

### **5. The approach of Fichera-Oleinik-Radkeviˇc**

In this section we revise the results of [9] and [20] for the Dirichlet problem for PDEs of second order having non-negative characteristic form; then the method is applied to some PDEs of Black-Scholes type.

To begin with consider the following equation in a bounded domain <sup>Ω</sup> <sup>⊂</sup> **<sup>R</sup>***<sup>m</sup>* with piecewise smooth boundary Σ:

$$L(u) = \sum\_{k,j=1,\ldots,m} a^{kj}(\mathbf{x}) u\_{\mathbf{x}\_k \mathbf{x}\_j} + \sum\_{k=1,\ldots,m} b^k(\mathbf{x}) u\_{\mathbf{x}\_k} + c(\mathbf{x}) u = f(\mathbf{x}) \tag{33}$$

where <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kj*(*x*)*ξkξ<sup>j</sup>* <sup>≥</sup> 0 , <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>, <sup>∀</sup>*<sup>ξ</sup>* <sup>∈</sup> **<sup>R</sup>***m*; *<sup>a</sup>kj*(*x*) = *<sup>a</sup>jk*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>. Moreover, *<sup>a</sup>kj* <sup>∈</sup> *<sup>C</sup>*2(Ω), *<sup>b</sup><sup>k</sup>* <sup>∈</sup> *<sup>C</sup>*1(Ω), *<sup>c</sup>* <sup>∈</sup> *<sup>C</sup>*0(Ω). Denote the unit inner normal to <sup>Σ</sup> by −→*<sup>n</sup>* = (*n*1, ..., *nm*) and let Σ<sup>3</sup> = *<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup>; <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kj*(*x*)*nknj* <sup>&</sup>gt; <sup>0</sup> be the non-characteristic part of Σ. Define Σ<sup>0</sup> = *<sup>x</sup>* <sup>∈</sup> <sup>Σ</sup>; <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kj*(*x*)*nknj* <sup>=</sup> <sup>0</sup> , i.e. <sup>Σ</sup> <sup>=</sup> <sup>Σ</sup><sup>0</sup> <sup>∪</sup> <sup>Σ</sup><sup>3</sup> and <sup>Σ</sup><sup>0</sup> is the characteristic part of Σ. Following Fichera (1956) we introduce on Σ<sup>0</sup> the Fichera function:

$$\beta(\mathbf{x}) = \sum\_{k=1,\ldots,m} (b^k(\mathbf{x}) - \sum\_{j=1,\ldots,m} a^{kj}\_{\mathbf{x}\_j}(\mathbf{x})) n\_{\mathbf{k}\prime} \mathbf{x} \in \boldsymbol{\Sigma}^0 \tag{34}$$

Then we split Σ<sup>0</sup> into three parts, namely

$$\Sigma\_1 = \{ \mathbf{x} \in \Sigma^0; \mathcal{B}(\mathbf{x}) > 0 \},$$

$$\Sigma\_2 = \{ \mathbf{x} \in \Sigma^0; \mathcal{B}(\mathbf{x}) < 0 \},$$

$$\Sigma\_0 = \{ \mathbf{x} \in \Sigma^0; \mathcal{B}(\mathbf{x}) = 0 \}.$$

16 Will-be-set-by-IN-TECH

c). (1) As *P*<sup>0</sup> is interior point in the rectangle {0 < *t* < *T*}×{0 < *x*<sup>2</sup> < *a*2} ⇒ *wt*(*P*0) = 0,

<sup>2</sup> ∈ (0, *a*2) ⇒ *wx*<sup>2</sup> (*P*0) = 0, *wx*<sup>2</sup> *<sup>x</sup>*<sup>2</sup> (*P*0) ≤ 0. The contradiction is obvious.

*aijxixjuxixj* +

is non-hypoelliptic. The constants *aij*, *bi*, *c* are arbitrary. To verify this we recall that the

<sup>+</sup> , *<sup>d</sup>*<sup>2</sup> *ds*<sup>2</sup> *<sup>s</sup><sup>a</sup>*

*aiidi*(*di* − 1) +

Of course, *sing supp u* <sup>=</sup> *<sup>∂</sup>*{*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* : *xj* <sup>≥</sup> 0, 1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>*}, i.e. *sing supp u* is the boundary of

*L* is hypoelliptic in the open domain {*xj* > 0, 1 ≤ *j* ≤ *n*} as it is strictly parabolic there.

0, *<sup>s</sup>* <sup>≤</sup> <sup>0</sup> considered as a Schwartz distribution in *<sup>D</sup>*�

<sup>+</sup> = *asa*−<sup>1</sup>

*u*1(*x*1)

*n* ∑ *i*=*j*

*n* ∑ *i*=1

(**R***n*+1) satisfies in distribution sense *Lu* = 0 if

*n* ∑ *i*=1

*bixiuxi* + *cu*

<sup>+</sup> <sup>=</sup> *<sup>a</sup>*(*<sup>a</sup>* <sup>−</sup> <sup>1</sup>)*sa*−<sup>2</sup>

<sup>+</sup> .

... *un*(*xn*), where *λ* = *const*, *uj*(*xj*) = *x*

*bidi* + *c* = 0

*<sup>t</sup>* . The nonhypoellipticity is proved. Evidently, under (4)

c). (4). Then *wt*(*P*0) = 0 and according to (30) *cw*(*P*0) + *wt*(*P*0) = 0 - contradiction.

We conclude that *<sup>M</sup>* <sup>=</sup> *supD*¯ *<sup>w</sup>* <sup>≤</sup> <sup>0</sup> <sup>⇒</sup> *<sup>u</sup>* <sup>−</sup> *<sup>v</sup>* <sup>≤</sup> 0 in *<sup>D</sup>*¯ <sup>⇒</sup> *<sup>u</sup>* <sup>≤</sup> *<sup>v</sup>* in *<sup>D</sup>*¯ .

*n* ∑ *i*,*j*=1

*Lu* = *ut* +

 0 < *t*<sup>0</sup> < *T* 0 < *x*<sup>20</sup> < *a*<sup>2</sup>

*cw*(*P*0) + *wt*(*P*0) = 0 (32)

, (2)

<sup>20</sup>*wx*<sup>2</sup> *<sup>x</sup>*<sup>2</sup> (*P*0) + *b*2*x*20*wx*<sup>2</sup> (*P*0) + *cw*(*P*0) +

 *t*<sup>0</sup> = 0 0 < *x*<sup>0</sup>

(**R**1) satisfies for *Re a* >

*dj j* ∈

<sup>2</sup> < *a*<sup>2</sup> ,

b). (3) Then (30) takes the form: *cw*(*P*0) + *wt*(*P*0) = 0 - contradiction.

Case c). (*t*0, *x*0) ∈ *I I* ⇒ 0 ≤ *t*<sup>0</sup> < *T*, *x*<sup>10</sup> = 0; (1)

Certainly, *wt*(*P*0) ≤ 0 in each case (1) -(4).

c). (3) The equation (30) takes the form:

The comparison principle is proved.

*sa*, *s* > 0

Consider now the distribution *u* = *eλ<sup>t</sup>*

), *Redj* <sup>&</sup>gt; 1. Then *<sup>u</sup>* <sup>∈</sup> *<sup>D</sup>*�

*ssa*

*λ* + *n* ∑ *i*�=*j*

<sup>+</sup> = *<sup>s</sup>a*+<sup>1</sup> <sup>+</sup> , *<sup>d</sup> dssa*

*<sup>x</sup>* multiplied by **R**<sup>1</sup>

*aijdidj* +

 *T* > *t*<sup>0</sup> > 0 *x*0

*wx*<sup>2</sup> (*P*0) = 0, *wx*<sup>2</sup> *<sup>x</sup>*<sup>2</sup> (*P*0) <sup>≤</sup> 0. According to (30) *<sup>a</sup>*2*x*<sup>2</sup>

<sup>2</sup> <sup>=</sup> <sup>0</sup> .

(3)

 *t*<sup>0</sup> = 0 *x*0

c). (2) As *x*<sup>0</sup>

<sup>2</sup> <sup>=</sup> <sup>0</sup> , (4)

*wt*(*P*0) = 0 - contradiction.

and again a contradiction.

**Remark 5.** The operator

<sup>+</sup> =

1 the following identities:

the first octant of **R***<sup>n</sup>*

function *s<sup>a</sup>*

*D*� (**R**<sup>1</sup> *xj* As it is proved in Oleinik and Radkeviˇc (1971) the sets Σ0, Σ1, Σ2, Σ<sup>3</sup> are invariant under smooth non-degenerate changes of the variables. More precisely, let *L*(*u*) = *f* in Ω; after the change *y* = *F*(*x*) it takes the form *<sup>L</sup>*(*u*) = *<sup>f</sup>* in Ω . Denote the Fichera function for *<sup>L</sup>*(*u*) = *<sup>f</sup>* by *β* . Then *β* = *β*.*A* where *A* > 0 and *A* is continuous.

Assume now that *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω) and *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> (Ω). Then

 $\int\_{\Omega} L(u)v dx = \int\_{\Omega} \mu L^\*(v) dx,$  where

*L*∗(*v*) = ∑ *k*,*j*=1,...,*m <sup>a</sup>kj*(*x*)*vxkxj* <sup>+</sup> ∑ *k*=1,...,*m b*∗*<sup>k</sup>* (*x*)*vxk* + *c*∗(*x*)*v* (35)

and *b*∗*<sup>k</sup>* = 2 ∑*j*=1,...,*<sup>m</sup> a kj xj* <sup>−</sup> *<sup>b</sup>k*, *<sup>c</sup>*<sup>∗</sup> <sup>=</sup> <sup>∑</sup>*k*=1,...,*m*(∑*j*=1,...,*<sup>m</sup> <sup>a</sup> kj xkxj* <sup>−</sup> *<sup>b</sup><sup>k</sup> xk* ) + *c*. One can easily see that if we denote the Fichera function for *L*∗(*v*) by *β*∗, then *β*<sup>∗</sup> = −*β* and *β* is defined by (34).

Assume now that *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω), *<sup>u</sup>* <sup>=</sup> 0 at <sup>Σ</sup><sup>2</sup> <sup>∪</sup> <sup>Σ</sup>3, and define the following set of test functions: <sup>V</sup> <sup>=</sup> *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω); *<sup>v</sup>* <sup>=</sup> 0 at <sup>Σ</sup><sup>1</sup> <sup>∪</sup> <sup>Σ</sup><sup>3</sup> . In view of the Green formula for *L* we get:

$$\int\_{\Omega} (L(\mathfrak{u})\mathfrak{v} - L^\*(\mathfrak{v})\mathfrak{u})d\mathfrak{x} = 0 \Leftrightarrow \int\_{\Omega} L(\mathfrak{u})\mathfrak{v}d\mathfrak{x} = \int\_{\Omega} \mathfrak{u}L^\*(\mathfrak{v})d\mathfrak{x} \tag{36}$$

for any *u* and *v* ∈ V. Let us now recall the definitions of generalized solution.

**Definition 3.** *The function u* <sup>∈</sup> *<sup>L</sup>p*(Ω)*, p* <sup>≥</sup> <sup>1</sup>*, is called a generalized solution of the boundary value problem*

$$\begin{cases} L(u) = f & \text{in } \Omega \\ u = 0 & \text{at } \Sigma\_2 \cup \Sigma\_3 \end{cases} \tag{37}$$

 Ω

*xk* − ∑*<sup>j</sup> a*

*kj xkxj*

Finally we propose the existence of a generalized solution of (37) in the space *L*∞(Ω). To fix the ideas we assume that the coefficients of *L* and *L*<sup>∗</sup> belong to *C*1(Ω) and Σ is thrice piecewise smooth (i.e. Σ can be split into several parts and each of them is *C*<sup>3</sup> smooth). Consider the

*L*(*u*) = *f* in Ω

If *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω) is a classical solution of (41) and *<sup>v</sup>* ∈ V then according to the Green formula

*f vdx* −

*<sup>∂</sup>*−→*<sup>ν</sup>* = <sup>∑</sup>*<sup>k</sup> <sup>ν</sup><sup>k</sup>*

**Definition 5.** *We shall say that the function u* <sup>∈</sup> *<sup>L</sup>*∞(Ω) *is a generalized solution of (41) if for each*

*Assume that the coefficient c*(*x*) *of L is such that c*(*x*) ≤ −*c*<sup>0</sup> <sup>&</sup>lt; <sup>0</sup> *in* <sup>Ω</sup>*, f* <sup>∈</sup> *<sup>L</sup>*∞(Ω)*, g* <sup>∈</sup> *<sup>L</sup>*∞(Σ<sup>2</sup> <sup>∪</sup> Σ3) *and β*(*x*) ≤ 0 *in the interior points of* Σ<sup>2</sup> ∪ Σ0*. Then there exists a generalized solution of (41) in*

**Remark 6.** In Th.6 it is assumed that <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kj*(*x*)*ξkξ<sup>j</sup>* <sup>≥</sup> 0 in an *<sup>m</sup>*−dimensional

*Suppose that g is continuous in the interior points of* Σ<sup>2</sup> ∪ Σ3*. Then the generalized solution u of (41)*

As we shall deal with (degenerate) parabolic PDEs we shall have to work in cylindrical domains (rectangles in **R**2). Therefore Σ = *∂*Ω will be piecewise smooth. Consider now the bounded domain Ω having piecewise *C*<sup>3</sup> smooth boundary Σ. The corresponding boundary

*u* = 0 on Σ<sup>2</sup> ∪ Σ<sup>3</sup>

We shall say that the point *P* ∈ Σ is regular if locally near to *P* the surface Σ can be written in the form *xk* = *<sup>ϕ</sup>k*(*x*1, ..., *xk*−1, *xk*+1, ..., *xm*), (*x*1,..., *xk*−1, *xk*+1,..., *xm*) describing some

*L*(*u*) = *f* in Ω,

*constructed in Th. 6 is continuous at those points and, moreover, u* = *g there.*

*u* = *g* on Σ<sup>2</sup> ∪ Σ<sup>3</sup>

 Σ3 *g ∂v ∂*−→*ν*

> *∂ ∂xk* .

*dσ* + Σ2

*<sup>c</sup>*<sup>0</sup> , sup |*g*|)*.*

*problem (37) possesses a generalized solution u* ∈ H *(i.e. a weak solution) in the sense of (40).*

<sup>2</sup> <sup>∑</sup>*k*(*b<sup>k</sup>*

**Theorem 5.** (See [20], Th. 1.4.1).

*Assume that f* <sup>∈</sup> *<sup>L</sup>*2(Ω) *and* <sup>1</sup>

boundary value problem:

 Ω

where −→*ν* = (*ν*1,..., *νm*), *ν<sup>k</sup>* = ∑*<sup>j</sup> akjnj*, *<sup>∂</sup>*

*test function v* ∈ V *the identity (42) is fulfilled.*

**Theorem 6.** (See [20], Th. 1.5.1).

neighbourhood of <sup>Σ</sup>0, <sup>∀</sup>*<sup>ξ</sup>* <sup>∈</sup> **<sup>R</sup>***m*. **Theorem 7.** (See [20], Th. 1.5.2).

value problem is:

We point out that *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω) and *<sup>g</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Σ<sup>2</sup> <sup>∪</sup> <sup>Σ</sup>3).

*the sense of Definition 5. Moreover,* <sup>|</sup>*u*(*x*)<sup>|</sup> <sup>≤</sup> max(sup <sup>|</sup> *<sup>f</sup>* <sup>|</sup>

*L*∗(*v*)*udx* =

 Ω *vf dx* = *B*(*u*, *v*). (40)

Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

) − *c* ≥ *c*<sup>0</sup> > 0 *in* Ω*. Then the boundary value*

(41)

49

(43)

*βgvdσ*, (42)

*if for each test function v* ∈ V *the following integral identity holds:*

$$\int\_{\Omega} fv d\mathbf{x} = \int\_{\Omega} \mu \mathcal{L}^\*(v) d\mathbf{x}.\tag{38}$$

#### **Theorem 2.** (See [20],Th.1.3.1).

*Suppose that c* <sup>&</sup>lt; <sup>0</sup>*, c*<sup>∗</sup> <sup>&</sup>lt; <sup>0</sup> *in* <sup>Ω</sup> *and p* <sup>&</sup>gt; <sup>1</sup>*. Then for each f* <sup>∈</sup> *<sup>L</sup>p*(Ω) *there exists a generalized solution u* <sup>∈</sup> *<sup>L</sup>p*(Ω) *of (37) in the sense of (38) and such that*

$$\inf\_{\mu\_0 \in \mathcal{Z}} \|\mu + \mu\_0\|\_{L^p(\Omega)} \le K \|f\|\_{L^p(\Omega)}\tag{39}$$

*K* = *const* > 0*. The set Z* = *<sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>p*(Ω) : <sup>Ω</sup> *<sup>u</sup>*0*L*∗(*v*)*dx* <sup>=</sup> 0, <sup>∀</sup>*<sup>v</sup>* ∈ V .

**Theorem 3.** (See [20], Th. 1.3.2).

*Let c* < 0 *in* Ω*,* <sup>1</sup> *<sup>p</sup>* <sup>+</sup> <sup>1</sup> *<sup>q</sup>* <sup>=</sup> <sup>1</sup> *and* <sup>−</sup>*<sup>c</sup>* + (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*)*c*<sup>∗</sup> <sup>&</sup>gt; <sup>0</sup> *in* <sup>Ω</sup>*. Then for each f* <sup>∈</sup> *<sup>L</sup>p*(Ω) *there exists a generalized solution u of (37) satisfying the a-priori estimate (39).*

**Theorem 4.** (See [20], Th. 1.3.3).

*Let c*<sup>∗</sup> <sup>&</sup>lt; <sup>0</sup> *in* <sup>Ω</sup> *and* <sup>−</sup>*<sup>c</sup>* + (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*)*c*<sup>∗</sup> <sup>&</sup>gt; <sup>0</sup> *in* <sup>Ω</sup>*,* <sup>1</sup> *<sup>p</sup>* <sup>+</sup> <sup>1</sup> *<sup>q</sup>* <sup>=</sup> <sup>1</sup>*. Then for each f* <sup>∈</sup> *<sup>L</sup>p*(Ω) *there exists a generalized solution u of (37) satisfying the estimate (39).*

**Conclusion**. Assume that *c* < 0. Then (37) is solvable in the sense of Definition 1 for *p* >> 1 as *p* → +∞ ⇒ *q* → 1. On the other hand, *c*<sup>∗</sup> < 0 implies the solvability of (40) for *p* ≥ 1, *p* ≈ 1 as *p* → 1 ⇒ *q* → +∞.

We shall now discuss the problem for existence of a generalized solution of (37) in the Sobolev space *H*1(Ω) with an appropriate weight. Define the following set of test functions:

$$\mathcal{W} = \{ v \in \mathbb{C}^1(\overline{\Omega}); v \mid\_{\Sigma\_3} = 0 \}$$

and equip <sup>W</sup> with the scalar product: (*u*, *<sup>v</sup>*)<sup>H</sup> <sup>=</sup> <sup>Ω</sup>(∑*k*,*<sup>j</sup> <sup>a</sup>kjuxj vxk* + *uv*)*dx* +

 <sup>Σ</sup>1∪Σ<sup>3</sup> *uv* <sup>|</sup>*β*<sup>|</sup> *<sup>d</sup>σ*. The completion of <sup>W</sup> with respect to the norm �*u*�H is a real Hilbert space denoted by H. For each two functions *u*, *v* ∈ W we consider the bilinear form *B*(*u*, *v*) = − <sup>Ω</sup>[∑*k*,*<sup>j</sup> <sup>a</sup>kjuxj vxk* + ∑*k*(*u*�*kvxk* + (�*<sup>k</sup> xk* <sup>−</sup> *<sup>c</sup>*)*uv*)]*dx* <sup>−</sup> <sup>Σ</sup><sup>1</sup> *uvβdσ*, where *l <sup>k</sup>* <sup>=</sup> *<sup>b</sup><sup>k</sup>* <sup>−</sup> <sup>∑</sup>*<sup>j</sup> <sup>a</sup> kj xj* . According to the Cauchy-Schwartz inequality <sup>|</sup>*B*(*u*, *<sup>v</sup>*)<sup>|</sup> <sup>≤</sup> *const*[�*v*�*H*<sup>1</sup> (Ω) <sup>+</sup> �*v*�*L*<sup>2</sup> (Σ1)] �*u*�H . Therefore, *B*(*u*, *v*) is well defined for *v* ∈ W and *u* ∈ H.

**Definition 4.** *Let f* <sup>∈</sup> *<sup>L</sup>*2(Ω)*. We shall say that the function u* ∈ H *is a generalized solution of (37) if for each v* ∈ W *the following identity is satisfied:*

48 Risk Management – Current Issues and Challenges Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach <sup>19</sup> 49 Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

$$\int\_{\Omega} vf d\mathbf{x} = B(\boldsymbol{u}, \boldsymbol{v}).\tag{40}$$

#### **Theorem 5.** (See [20], Th. 1.4.1).

18 Will-be-set-by-IN-TECH

**Definition 3.** *The function u* <sup>∈</sup> *<sup>L</sup>p*(Ω)*, p* <sup>≥</sup> <sup>1</sup>*, is called a generalized solution of the boundary value*

*L*(*u*) = *f* in Ω

*f vdx* =

*u* = 0 at Σ<sup>2</sup> ∪ Σ<sup>3</sup>

 Ω

*Suppose that c* <sup>&</sup>lt; <sup>0</sup>*, c*<sup>∗</sup> <sup>&</sup>lt; <sup>0</sup> *in* <sup>Ω</sup> *and p* <sup>&</sup>gt; <sup>1</sup>*. Then for each f* <sup>∈</sup> *<sup>L</sup>p*(Ω) *there exists a generalized*

*<sup>p</sup>* <sup>+</sup> <sup>1</sup>

**Conclusion**. Assume that *c* < 0. Then (37) is solvable in the sense of Definition 1 for *p* >> 1 as *p* → +∞ ⇒ *q* → 1. On the other hand, *c*<sup>∗</sup> < 0 implies the solvability of (40) for *p* ≥ 1, *p* ≈ 1

We shall now discuss the problem for existence of a generalized solution of (37) in the Sobolev

<sup>W</sup> <sup>=</sup> {*<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*1(Ω); *<sup>v</sup>* <sup>|</sup>Σ3<sup>=</sup> <sup>0</sup>}

<sup>Σ</sup>1∪Σ<sup>3</sup> *uv* <sup>|</sup>*β*<sup>|</sup> *<sup>d</sup>σ*. The completion of <sup>W</sup> with respect to the norm �*u*�H is a real Hilbert space denoted by H. For each two functions *u*, *v* ∈ W we consider the bilinear form *B*(*u*, *v*) =

*xk* <sup>−</sup> *<sup>c</sup>*)*uv*)]*dx* <sup>−</sup>

According to the Cauchy-Schwartz inequality <sup>|</sup>*B*(*u*, *<sup>v</sup>*)<sup>|</sup> <sup>≤</sup> *const*[�*v*�*H*<sup>1</sup> (Ω) <sup>+</sup> �*v*�*L*<sup>2</sup> (Σ1)] �*u*�H .

**Definition 4.** *Let f* <sup>∈</sup> *<sup>L</sup>*2(Ω)*. We shall say that the function u* ∈ H *is a generalized solution of (37)*

space *H*1(Ω) with an appropriate weight. Define the following set of test functions:

(37)

*uL*∗(*v*)*dx*. (38)

.

*<sup>q</sup>* <sup>=</sup> <sup>1</sup>*. Then for each f* <sup>∈</sup> *<sup>L</sup>p*(Ω) *there exists a*

*vxk* + *uv*)*dx* +

*<sup>k</sup>* <sup>=</sup> *<sup>b</sup><sup>k</sup>* <sup>−</sup> <sup>∑</sup>*<sup>j</sup> <sup>a</sup>*

*kj xj* .

<sup>Σ</sup><sup>1</sup> *uvβdσ*, where *l*

�*u* + *u*0�*Lp*(Ω) ≤ *K* � *f* �*Lp*(Ω) (39)

<sup>Ω</sup> *<sup>u</sup>*0*L*∗(*v*)*dx* <sup>=</sup> 0, <sup>∀</sup>*<sup>v</sup>* ∈ V

*<sup>q</sup>* <sup>=</sup> <sup>1</sup> *and* <sup>−</sup>*<sup>c</sup>* + (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*)*c*<sup>∗</sup> <sup>&</sup>gt; <sup>0</sup> *in* <sup>Ω</sup>*. Then for each f* <sup>∈</sup> *<sup>L</sup>p*(Ω) *there exists a*

<sup>Ω</sup>(∑*k*,*<sup>j</sup> <sup>a</sup>kjuxj*

for any *u* and *v* ∈ V. Let us now recall the definitions of generalized solution.

*if for each test function v* ∈ V *the following integral identity holds:*

*solution u* <sup>∈</sup> *<sup>L</sup>p*(Ω) *of (37) in the sense of (38) and such that*

inf *u*0∈*Z*

*generalized solution u of (37) satisfying the a-priori estimate (39).*

*<sup>u</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>p*(Ω) :

 Ω

*problem*

**Theorem 2.** (See [20],Th.1.3.1).

*K* = *const* > 0*. The set Z* =

*Let c* < 0 *in* Ω*,* <sup>1</sup>

as *p* → 1 ⇒ *q* → +∞.

<sup>Ω</sup>[∑*k*,*<sup>j</sup> <sup>a</sup>kjuxj*

−

**Theorem 3.** (See [20], Th. 1.3.2).

**Theorem 4.** (See [20], Th. 1.3.3).

*<sup>p</sup>* <sup>+</sup> <sup>1</sup>

*Let c*<sup>∗</sup> <sup>&</sup>lt; <sup>0</sup> *in* <sup>Ω</sup> *and* <sup>−</sup>*<sup>c</sup>* + (<sup>1</sup> <sup>−</sup> *<sup>q</sup>*)*c*<sup>∗</sup> <sup>&</sup>gt; <sup>0</sup> *in* <sup>Ω</sup>*,* <sup>1</sup>

*generalized solution u of (37) satisfying the estimate (39).*

and equip <sup>W</sup> with the scalar product: (*u*, *<sup>v</sup>*)<sup>H</sup> <sup>=</sup>

*vxk* + ∑*k*(*u*�*kvxk* + (�*<sup>k</sup>*

Therefore, *B*(*u*, *v*) is well defined for *v* ∈ W and *u* ∈ H.

*if for each v* ∈ W *the following identity is satisfied:*

*Assume that f* <sup>∈</sup> *<sup>L</sup>*2(Ω) *and* <sup>1</sup> <sup>2</sup> <sup>∑</sup>*k*(*b<sup>k</sup> xk* − ∑*<sup>j</sup> a kj xkxj* ) − *c* ≥ *c*<sup>0</sup> > 0 *in* Ω*. Then the boundary value problem (37) possesses a generalized solution u* ∈ H *(i.e. a weak solution) in the sense of (40).*

Finally we propose the existence of a generalized solution of (37) in the space *L*∞(Ω). To fix the ideas we assume that the coefficients of *L* and *L*<sup>∗</sup> belong to *C*1(Ω) and Σ is thrice piecewise smooth (i.e. Σ can be split into several parts and each of them is *C*<sup>3</sup> smooth). Consider the boundary value problem:

$$\begin{cases} L(u) = f & \text{in} \quad \Omega\\ u = g & \text{on} \quad \Sigma\_2 \cup \Sigma\_3 \end{cases} \tag{41}$$

If *<sup>u</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω) is a classical solution of (41) and *<sup>v</sup>* ∈ V then according to the Green formula

$$\int\_{\Omega} L^\*(v)ud\mathbf{x} = \int\_{\Omega} fv d\mathbf{x} - \int\_{\Sigma\_3} \mathbf{g} \frac{\partial v}{\partial \overline{\mathcal{V}}} d\sigma + \int\_{\Sigma\_2} \beta gv d\sigma,\tag{42}$$

where −→*ν* = (*ν*1,..., *νm*), *ν<sup>k</sup>* = ∑*<sup>j</sup> akjnj*, *<sup>∂</sup> <sup>∂</sup>*−→*<sup>ν</sup>* = <sup>∑</sup>*<sup>k</sup> <sup>ν</sup><sup>k</sup> ∂ ∂xk* .

**Definition 5.** *We shall say that the function u* <sup>∈</sup> *<sup>L</sup>*∞(Ω) *is a generalized solution of (41) if for each test function v* ∈ V *the identity (42) is fulfilled.*

We point out that *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω) and *<sup>g</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Σ<sup>2</sup> <sup>∪</sup> <sup>Σ</sup>3).

**Theorem 6.** (See [20], Th. 1.5.1).

*Assume that the coefficient c*(*x*) *of L is such that c*(*x*) ≤ −*c*<sup>0</sup> <sup>&</sup>lt; <sup>0</sup> *in* <sup>Ω</sup>*, f* <sup>∈</sup> *<sup>L</sup>*∞(Ω)*, g* <sup>∈</sup> *<sup>L</sup>*∞(Σ<sup>2</sup> <sup>∪</sup> Σ3) *and β*(*x*) ≤ 0 *in the interior points of* Σ<sup>2</sup> ∪ Σ0*. Then there exists a generalized solution of (41) in the sense of Definition 5. Moreover,* <sup>|</sup>*u*(*x*)<sup>|</sup> <sup>≤</sup> max(sup <sup>|</sup> *<sup>f</sup>* <sup>|</sup> *<sup>c</sup>*<sup>0</sup> , sup |*g*|)*.*

**Remark 6.** In Th.6 it is assumed that <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kj*(*x*)*ξkξ<sup>j</sup>* <sup>≥</sup> 0 in an *<sup>m</sup>*−dimensional neighbourhood of <sup>Σ</sup>0, <sup>∀</sup>*<sup>ξ</sup>* <sup>∈</sup> **<sup>R</sup>***m*.

**Theorem 7.** (See [20], Th. 1.5.2).

*Suppose that g is continuous in the interior points of* Σ<sup>2</sup> ∪ Σ3*. Then the generalized solution u of (41) constructed in Th. 6 is continuous at those points and, moreover, u* = *g there.*

As we shall deal with (degenerate) parabolic PDEs we shall have to work in cylindrical domains (rectangles in **R**2). Therefore Σ = *∂*Ω will be piecewise smooth. Consider now the bounded domain Ω having piecewise *C*<sup>3</sup> smooth boundary Σ. The corresponding boundary value problem is:

$$\begin{cases} L(u) = f \text{ in } \Omega, \\ u = 0 \quad \text{on } \Sigma\_2 \cup \Sigma\_3 \end{cases} \tag{43}$$

We shall say that the point *P* ∈ Σ is regular if locally near to *P* the surface Σ can be written in the form *xk* = *<sup>ϕ</sup>k*(*x*1, ..., *xk*−1, *xk*+1, ..., *xm*), (*x*1,..., *xk*−1, *xk*+1,..., *xm*) describing some

neighborhood of the projection of *P* onto the plane *xk* = 0. The set of the boundary points which do not possess such a representation will be denoted by *B*.

As we know from [20] there exists an *Lp*(Ω1) solution of the boundary value problem

According to the Definition 3:

In a similar way there exists *<sup>u</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>p*(Ω2) such that

<sup>Ω</sup><sup>2</sup> *<sup>u</sup>*2*L*∗(*v*2)*dx* <sup>=</sup>

We conclude as follows: *(a)* If *ui* satisfies

The set Σ<sup>0</sup> is called interior boundary of Ω.

*i* = 1, 2, then (48) satisfies (49).

<sup>Ω</sup> *f vdx* <sup>=</sup>

**Acknowledgments**

**Author details**

Rossella Agliardi

Certainly, there exists *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>p*(Ω) such that

Σ(1) <sup>1</sup> <sup>∪</sup>Σ(1) 3 = 0.

*i* = 1, 2. Consequently,

satisfies the identity

*Lp*(Ω) solution of

*<sup>C</sup>*2(Ω1), *<sup>v</sup>*<sup>1</sup> <sup>|</sup>

Therefore:

the function

identity

 *<sup>L</sup>*(*u*1) = *<sup>f</sup>* <sup>∈</sup> <sup>Ω</sup><sup>1</sup> *<sup>u</sup>*<sup>1</sup> <sup>=</sup> 0 on <sup>Σ</sup>(1)

<sup>Ω</sup><sup>1</sup> *<sup>u</sup>*1*L*∗(*v*1)*dx* <sup>=</sup>

 *<sup>L</sup>*(*u*2) = *<sup>f</sup>* in <sup>Ω</sup><sup>2</sup> *<sup>u</sup>*<sup>2</sup> <sup>=</sup> 0 on <sup>Σ</sup>(2)

*<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω), *<sup>v</sup>* <sup>|</sup>Σ1∪Σ3<sup>=</sup> 0. Evidently, *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω), *<sup>v</sup>* <sup>|</sup>Σ1∪Σ3<sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω*i*), *<sup>v</sup>* <sup>|</sup>

<sup>Ω</sup><sup>1</sup> *f vdx* <sup>+</sup>

 *u*<sup>1</sup> in Ω<sup>1</sup> *u*<sup>2</sup> in Ω<sup>2</sup>

*L*(*W*) = *f* in Ω

 *<sup>L</sup>*(*ui*) = *<sup>f</sup>* in <sup>Ω</sup>*<sup>i</sup> ui* <sup>=</sup> 0 on <sup>Σ</sup>(*i*)

*(b)* In the special case when *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω), *ui* <sup>∈</sup> *<sup>L</sup>*∞(Ω*i*), *<sup>i</sup>* <sup>=</sup> 1, 2, *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω), *<sup>u</sup>* satisfies the

*W* = 0 on Σ<sup>2</sup> ∪ Σ<sup>3</sup>

<sup>Ω</sup><sup>1</sup> *<sup>u</sup>*1*L*∗(*v*)*dx* <sup>=</sup>

*W* =

The authors gratefully acknowledge financial support from CNR/BAS.

*Department of Mathematics for Economic and Social Sciences, University of Bologna, Italy*

<sup>Ω</sup> *f vdx* <sup>=</sup>

<sup>2</sup> <sup>∪</sup> <sup>Σ</sup>(1) 3

<sup>2</sup> <sup>∪</sup> <sup>Σ</sup>(2) 3

<sup>Ω</sup><sup>2</sup> *f v*2*dx* for each test function *<sup>v</sup>*<sup>2</sup> <sup>∈</sup> *<sup>C</sup>*2(Ω2), *<sup>v</sup>*2<sup>|</sup>

<sup>Ω</sup> *uL*∗(*v*)*dx* <sup>=</sup>

<sup>Ω</sup><sup>2</sup> *<sup>u</sup>*2*L*∗(*v*)*dx* <sup>=</sup>

<sup>∈</sup> *<sup>L</sup>p*(Ω) (48)

<sup>Ω</sup> *WL*∗(*v*)*dx*, i.e. *W* is a generalized

<sup>Ω</sup><sup>1</sup> *f vdx* and

<sup>Ω</sup><sup>2</sup> *f vdx* <sup>=</sup>

<sup>2</sup> <sup>∪</sup> <sup>Σ</sup>(*i*) 3

<sup>Ω</sup> *uL*∗(*v*)*dx*, we have a uniqueness theorem and therefore *u* = *W*.

<sup>Ω</sup><sup>1</sup> *f v*1*dx* for each test function *v*<sup>1</sup> ∈

Boundary-Value Problems for Second Order PDEs Arising in Risk Management and Cellular Neural Networks Approach

> Σ(2) <sup>1</sup> <sup>∪</sup>Σ(2) 3 = 0.

> > Σ(*i*) <sup>1</sup> <sup>∪</sup>Σ(*i*) 3 = 0,

<sup>Ω</sup><sup>2</sup> *f vdx*, and thus

<sup>Ω</sup> *f vdx* for each test function

(46)

51

(47)

(49)

(50)

**Definition 6.** *The function u* <sup>∈</sup> *<sup>L</sup>*∞(Ω) *is called a generalized solution of (43) for f* <sup>∈</sup> *<sup>L</sup>*∞(Ω) *if for each function v* <sup>∈</sup> *<sup>C</sup>*2(Ω)*, v* <sup>=</sup> <sup>0</sup> *at* <sup>Σ</sup><sup>1</sup> <sup>∪</sup> <sup>Σ</sup><sup>3</sup> <sup>∪</sup> *B the following identity holds:* <sup>Ω</sup> *uL*∗(*v*)*dx* <sup>=</sup> <sup>Ω</sup> *f vdx*.

**Theorem 8.** (See [20], Th. 1.5.5).

*Suppose that the boundary* <sup>Σ</sup> *of the bounded domain* <sup>Ω</sup> *is C*<sup>3</sup> *piecewise smooth, f* <sup>∈</sup> *<sup>L</sup>*∞(Ω)*, g* <sup>=</sup> <sup>0</sup>*, c*(*x*) ≤ −*c*<sup>0</sup> < 0 *in* Ω *and β* ≤ 0 *in the interior points of* Σ<sup>0</sup> ∪ Σ2*. Then there exists a generalized solution u of (43) in the sense of Definition 6 and such that* <sup>|</sup>*u*<sup>|</sup> <sup>≤</sup> sup <sup>|</sup> *<sup>f</sup>* <sup>|</sup> *c*0 *.*

We shall not discuss here in details the problems of uniqueness and regularity of the generalized solutions. Unicity results are given by Theorems 1.6.1.-1.6.2. in [20]. For domains with *C*<sup>3</sup> smooth boundary under several restrictions on the coefficients, including *c*(*x*) ≤ −*c*<sup>0</sup> < 0, *c*<sup>∗</sup> < 0 in Ω, *β* ≤ 0 in the interior points of Σ<sup>0</sup> ∪ Σ2, *β*<sup>∗</sup> = −*β* < 0 at Σ1, the maximum principle is valid for each generalized solution *u* in the sense of Definition 5: <sup>|</sup>*u*<sup>|</sup> <sup>≤</sup> max sup<sup>Ω</sup> | *f* | *<sup>c</sup>*<sup>0</sup> , supΣ3∪Σ<sup>2</sup> <sup>|</sup>*g*<sup>|</sup> .

In Th. 1.6.9. uniqueness result is proved for the boundary value problem (43) in the class *L*∞(Ω). The existence result is given Th. 8. Regularity result is given in the Appendix.

**Remark 7.** Backward parabolic and parabolic operators satisfy the conditions: *akm* = 0, *k* = 1, ..., *<sup>m</sup>*, and *<sup>b</sup><sup>m</sup>* <sup>=</sup> <sup>±</sup>1 if *<sup>x</sup>* = (*x*1, ..., *xm*−1, *<sup>t</sup>*), i.e. *<sup>t</sup>* <sup>=</sup> *xm*. Put now *<sup>u</sup>* <sup>=</sup> *veα<sup>t</sup>* in (33). Then *L*1(*v*) = ∑*k*,*j*=1,...,*<sup>m</sup> akjvxkxj* + ∑*k*=1,...,*<sup>m</sup> bkvxk* + (*c* + *α*)*v* = *f e*−*α<sup>t</sup>* and *L*∗ <sup>1</sup> (*w*) = <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kjwxkxj* <sup>+</sup> <sup>∑</sup>*k*=1,...,*<sup>m</sup> <sup>b</sup>*∗*kwxk* <sup>+</sup> *<sup>c</sup>*<sup>∗</sup> 1*w* where *c*<sup>1</sup> = *c* + *α*, *b*∗*<sup>k</sup>* = 2 ∑*j*=1,...,*<sup>m</sup> a kj xj* <sup>−</sup> *<sup>b</sup>k*, *<sup>c</sup>*<sup>∗</sup> <sup>1</sup> = ∑*k*,*j*=1,...,*<sup>m</sup> a kj xkxj* <sup>−</sup> <sup>∑</sup>*k*=1,...,*<sup>m</sup> <sup>b</sup><sup>k</sup> xk* + *c* + *α*.

Having in mind that <sup>|</sup>*c*<sup>|</sup> <sup>≤</sup> *<sup>c</sup>* <sup>=</sup> *const* we conclude that for *<sup>b</sup><sup>m</sup>* <sup>=</sup> <sup>±</sup>1 and *<sup>α</sup>* → ∓<sup>∞</sup> then *c*<sup>1</sup> → −∞, *c*<sup>∗</sup> <sup>1</sup> → −<sup>∞</sup> uniformly in (*x*1, ..., *xm*−1, *<sup>t</sup>*) ∈ <sup>Ω</sup>. So for parabolic (backward parabolic) equations the conditions of Theorems 2, 5 are fulfilled.

We shall illustrate the previous results by the backward parabolic equations:

$$L(u) = \frac{\partial u}{\partial t} + \frac{1}{2}\sigma^2 x^2 \frac{\partial^2 u}{\partial x^2} + rx \frac{\partial u}{\partial x} - ru = f(t, x) \tag{44}$$

which is the famous Black-Scholes equation, and

$$M(\mu) = \frac{\partial \mu}{\partial t} + \mathfrak{x}^2 \frac{\partial^2 \mu}{\partial \mathfrak{x}^2} + b(\mathfrak{x}) \frac{\partial \mu}{\partial \mathfrak{x}} + c(\mathfrak{x}) \mu = f(t, \mathfrak{x}) \tag{45}$$

We shall work in the following rectangles: Ω<sup>1</sup> = {(*t*, *x*) : 0 < *t* < *T*, 0 < *x* < *a*1}, Ω<sup>2</sup> = {(*t*, *x*) : 0 < *t* < *T*, *a*<sup>2</sup> < *x* < 0}, Ω = {(*t*, *x*) : 0 < *t* < *T*, *a*<sup>2</sup> < *x* < *a*1}. Under the previous notation for Ω we have: Σ<sup>1</sup> = {*t* = 0}, Σ<sup>2</sup> = {*t* = *T*}, Σ<sup>3</sup> = {*x* = *a*1} ∪ {*x* = *a*2}. Certainly, for Ω1, Ω<sup>2</sup> another part of the boundary appears, Σ<sup>0</sup> = {*x* = 0}.

As we know from [20] there exists an *Lp*(Ω1) solution of the boundary value problem

$$\begin{cases} L(\mu\_1) = f & \in \, \Omega\_1\\ \mu\_1 = 0 \quad \text{on} \quad \Sigma\_2^{(1)} \cup \Sigma\_3^{(1)} \end{cases} \tag{46}$$

According to the Definition 3: <sup>Ω</sup><sup>1</sup> *<sup>u</sup>*1*L*∗(*v*1)*dx* <sup>=</sup> <sup>Ω</sup><sup>1</sup> *f v*1*dx* for each test function *v*<sup>1</sup> ∈ *<sup>C</sup>*2(Ω1), *<sup>v</sup>*<sup>1</sup> <sup>|</sup> Σ(1) <sup>1</sup> <sup>∪</sup>Σ(1) 3 = 0.

In a similar way there exists *<sup>u</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>p*(Ω2) such that

20 Will-be-set-by-IN-TECH

neighborhood of the projection of *P* onto the plane *xk* = 0. The set of the boundary points

**Definition 6.** *The function u* <sup>∈</sup> *<sup>L</sup>*∞(Ω) *is called a generalized solution of (43) for f* <sup>∈</sup> *<sup>L</sup>*∞(Ω) *if for*

*Suppose that the boundary* <sup>Σ</sup> *of the bounded domain* <sup>Ω</sup> *is C*<sup>3</sup> *piecewise smooth, f* <sup>∈</sup> *<sup>L</sup>*∞(Ω)*, g* <sup>=</sup> <sup>0</sup>*, c*(*x*) ≤ −*c*<sup>0</sup> < 0 *in* Ω *and β* ≤ 0 *in the interior points of* Σ<sup>0</sup> ∪ Σ2*. Then there exists a generalized*

We shall not discuss here in details the problems of uniqueness and regularity of the generalized solutions. Unicity results are given by Theorems 1.6.1.-1.6.2. in [20]. For domains with *C*<sup>3</sup> smooth boundary under several restrictions on the coefficients, including *c*(*x*) ≤ −*c*<sup>0</sup> < 0, *c*<sup>∗</sup> < 0 in Ω, *β* ≤ 0 in the interior points of Σ<sup>0</sup> ∪ Σ2, *β*<sup>∗</sup> = −*β* < 0 at Σ1, the maximum principle is valid for each generalized solution *u* in the sense of Definition 5:

In Th. 1.6.9. uniqueness result is proved for the boundary value problem (43) in the class *L*∞(Ω). The existence result is given Th. 8. Regularity result is given in the Appendix.

**Remark 7.** Backward parabolic and parabolic operators satisfy the conditions: *akm* = 0, *k* =

Having in mind that <sup>|</sup>*c*<sup>|</sup> <sup>≤</sup> *<sup>c</sup>* <sup>=</sup> *const* we conclude that for *<sup>b</sup><sup>m</sup>* <sup>=</sup> <sup>±</sup>1 and *<sup>α</sup>* → ∓<sup>∞</sup> then

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *rx*

1*w*

<sup>1</sup> = ∑*k*,*j*=1,...,*<sup>m</sup> a*

<sup>1</sup> → −<sup>∞</sup> uniformly in (*x*1, ..., *xm*−1, *<sup>t</sup>*) ∈ <sup>Ω</sup>. So for parabolic (backward parabolic)

*∂u*

*∂u ∂x* *kj*

*xkxj* <sup>−</sup> <sup>∑</sup>*k*=1,...,*<sup>m</sup> <sup>b</sup><sup>k</sup>*

*<sup>∂</sup><sup>x</sup>* <sup>−</sup> *ru* <sup>=</sup> *<sup>f</sup>*(*t*, *<sup>x</sup>*) (44)

+ *c*(*x*)*u* = *f*(*t*, *x*) (45)

*xk* + *c* + *α*.

1, ..., *<sup>m</sup>*, and *<sup>b</sup><sup>m</sup>* <sup>=</sup> <sup>±</sup>1 if *<sup>x</sup>* = (*x*1, ..., *xm*−1, *<sup>t</sup>*), i.e. *<sup>t</sup>* <sup>=</sup> *xm*. Put now *<sup>u</sup>* <sup>=</sup> *veα<sup>t</sup>* in (33). Then

*c*0 *.*

which do not possess such a representation will be denoted by *B*.

<sup>Ω</sup> *uL*∗(*v*)*dx* <sup>=</sup>


and *L*∗

*c*<sup>1</sup> → −∞, *c*<sup>∗</sup>

 sup<sup>Ω</sup> | *f* |

<sup>Ω</sup> *f vdx*.

**Theorem 8.** (See [20], Th. 1.5.5).

*each function v* <sup>∈</sup> *<sup>C</sup>*2(Ω)*, v* <sup>=</sup> <sup>0</sup> *at* <sup>Σ</sup><sup>1</sup> <sup>∪</sup> <sup>Σ</sup><sup>3</sup> <sup>∪</sup> *B the following identity holds:*

*solution u of (43) in the sense of Definition 6 and such that* <sup>|</sup>*u*<sup>|</sup> <sup>≤</sup> sup <sup>|</sup> *<sup>f</sup>* <sup>|</sup>

 .

*L*1(*v*) = ∑*k*,*j*=1,...,*<sup>m</sup> akjvxkxj* + ∑*k*=1,...,*<sup>m</sup> bkvxk* + (*c* + *α*)*v* = *f e*−*α<sup>t</sup>*

*kj xj* <sup>−</sup> *<sup>b</sup>k*, *<sup>c</sup>*<sup>∗</sup>

We shall illustrate the previous results by the backward parabolic equations:

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>x</sup>*<sup>2</sup> *<sup>∂</sup>*2*<sup>u</sup>*

*<sup>σ</sup>*2*x*<sup>2</sup> *<sup>∂</sup>*2*<sup>u</sup>*

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*(*x*)

We shall work in the following rectangles: Ω<sup>1</sup> = {(*t*, *x*) : 0 < *t* < *T*, 0 < *x* < *a*1}, Ω<sup>2</sup> = {(*t*, *x*) : 0 < *t* < *T*, *a*<sup>2</sup> < *x* < 0}, Ω = {(*t*, *x*) : 0 < *t* < *T*, *a*<sup>2</sup> < *x* < *a*1}. Under the previous notation for Ω we have: Σ<sup>1</sup> = {*t* = 0}, Σ<sup>2</sup> = {*t* = *T*}, Σ<sup>3</sup> = {*x* = *a*1} ∪ {*x* = *a*2}. Certainly,

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> 1 2

*<sup>c</sup>*<sup>0</sup> , supΣ3∪Σ<sup>2</sup> <sup>|</sup>*g*<sup>|</sup>

<sup>1</sup> (*w*) = <sup>∑</sup>*k*,*j*=1,...,*<sup>m</sup> <sup>a</sup>kjwxkxj* <sup>+</sup> <sup>∑</sup>*k*=1,...,*<sup>m</sup> <sup>b</sup>*∗*kwxk* <sup>+</sup> *<sup>c</sup>*<sup>∗</sup>

equations the conditions of Theorems 2, 5 are fulfilled.

*<sup>L</sup>*(*u*) = *<sup>∂</sup><sup>u</sup>*

*<sup>M</sup>*(*u*) = *<sup>∂</sup><sup>u</sup>*

for Ω1, Ω<sup>2</sup> another part of the boundary appears, Σ<sup>0</sup> = {*x* = 0}.

which is the famous Black-Scholes equation, and

where *c*<sup>1</sup> = *c* + *α*, *b*∗*<sup>k</sup>* = 2 ∑*j*=1,...,*<sup>m</sup> a*

$$\begin{cases} L(\mu\_2) = f & \text{in } \Omega\_2 \\\ \mu\_2 = 0 & \text{on } \Sigma\_2^{(2)} \cup \Sigma\_3^{(2)} \end{cases} \tag{47}$$

Therefore: <sup>Ω</sup><sup>2</sup> *<sup>u</sup>*2*L*∗(*v*2)*dx* <sup>=</sup> <sup>Ω</sup><sup>2</sup> *f v*2*dx* for each test function *<sup>v</sup>*<sup>2</sup> <sup>∈</sup> *<sup>C</sup>*2(Ω2), *<sup>v</sup>*2<sup>|</sup> Σ(2) <sup>1</sup> <sup>∪</sup>Σ(2) 3 = 0.

Certainly, there exists *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>p*(Ω) such that <sup>Ω</sup> *uL*∗(*v*)*dx* <sup>=</sup> <sup>Ω</sup> *f vdx* for each test function *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω), *<sup>v</sup>* <sup>|</sup>Σ1∪Σ3<sup>=</sup> 0. Evidently, *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω), *<sup>v</sup>* <sup>|</sup>Σ1∪Σ3<sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*2(Ω*i*), *<sup>v</sup>* <sup>|</sup> Σ(*i*) <sup>1</sup> <sup>∪</sup>Σ(*i*) 3 = 0, *i* = 1, 2. Consequently, <sup>Ω</sup><sup>1</sup> *<sup>u</sup>*1*L*∗(*v*)*dx* <sup>=</sup> <sup>Ω</sup><sup>1</sup> *f vdx* and <sup>Ω</sup><sup>2</sup> *<sup>u</sup>*2*L*∗(*v*)*dx* <sup>=</sup> <sup>Ω</sup><sup>2</sup> *f vdx*, and thus the function

$$\mathcal{W} = \begin{cases} \mu\_1 \text{ in } \,\,\Omega\_1\\ \mu\_2 \text{ in } \,\,\Omega\_2 \end{cases} \in L^p(\Omega) \tag{48}$$

satisfies the identity <sup>Ω</sup> *f vdx* <sup>=</sup> <sup>Ω</sup><sup>1</sup> *f vdx* <sup>+</sup> <sup>Ω</sup><sup>2</sup> *f vdx* <sup>=</sup> <sup>Ω</sup> *WL*∗(*v*)*dx*, i.e. *W* is a generalized *Lp*(Ω) solution of

$$\begin{cases} L(W) = f & \text{in } \Omega \\ W = 0 & \text{on } \Sigma\_2 \cup \Sigma\_3 \end{cases} \tag{49}$$

We conclude as follows: *(a)* If *ui* satisfies

$$\begin{cases} L(\mu\_i) = f & \text{in } \Omega\_i \\\ u\_i = 0 & \text{on } \Sigma\_2^{(i)} \cup \Sigma\_3^{(i)} \end{cases} \tag{50}$$

*i* = 1, 2, then (48) satisfies (49).

*(b)* In the special case when *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω), *ui* <sup>∈</sup> *<sup>L</sup>*∞(Ω*i*), *<sup>i</sup>* <sup>=</sup> 1, 2, *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω), *<sup>u</sup>* satisfies the identity <sup>Ω</sup> *f vdx* <sup>=</sup> <sup>Ω</sup> *uL*∗(*v*)*dx*, we have a uniqueness theorem and therefore *u* = *W*.

The set Σ<sup>0</sup> is called interior boundary of Ω.

#### **Acknowledgments**

The authors gratefully acknowledge financial support from CNR/BAS.

#### **Author details**

Rossella Agliardi *Department of Mathematics for Economic and Social Sciences, University of Bologna, Italy* Petar Popivanov and Angela Slavova *Institute of Mathematics, Bulgarian Academy of Sciences, Bulgaria*
