**Appendix**

One can find results concerning regularity of the generalized solutions of degenerate parabolic operators in cylindrical domains in [14] and [19]. For the sake of simplicity we shall consider only one example from Il'in as the conditions are simple and clear. Consider

$$N(\mu) = \frac{\partial \mu}{\partial t} + h(t, \mathbf{x}) \frac{\partial^2 \mu}{\partial \mathbf{x}^2} + \mathbf{g}(t, \mathbf{x}) \frac{\partial \mu}{\partial \mathbf{x}} + c(t, \mathbf{x}) \mu = F(t, \mathbf{x}) \tag{51}$$

**Remark 8.** If *a*<sup>2</sup> < *x* < *a*1, *a*<sup>2</sup> < 0, *a*<sup>1</sup> > 0, the Black-Scholes equation (44) is with *h*(*t*, *x*) =

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

53

possesses a unique classical solution. As we know, *u* |*x*=0= *U*(*t*) satisfies in classical sense the

*U*�(*t*) − *rU*(*t*) = *f*(*t*, 0), *U*(*T*) = 0. Therefore, we can consider the restrictions: *u* |*x*>0, *u* |*x*<<sup>0</sup> and conclude that they are classical solutions of the respective boundary value problems with

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[4] Boyarchenko, S. I., and S. Z. Levendorski˘ı (2002) Barrier options and touch-and-out options under regular Lévy processes of exponential type.Annals of Applied Probability.

[6] Budak, B., Samarskii, A., Tihonov, A. (1972) A Problem Book on the Equations of

[7] Carr, P. (1995) Two extensions to barrier option valuation. Applied Mathematical Finance.

[8] Ericsson, J., Reneby, J. (1998) A framework for valuing corporate securities. Applied

[9] Fichera, G. (1956) Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine. Atti dell'Accademia Nazionale dei Lincei. Cl. Sci. Fis. Mat. Natur. Rend. Lincei.

[10] Geman, H., Yor, M. (1996) Pricing and hedging double barrier options: a probabilistic

[11] Haug, E. G. (1997) The Complete Guide to Option Pricing Formulas. Mc-Graw Hill, N.Y.

[13] Heynen, R., Kat, H. (1996) Discrete partial barrier options with a moving barrier. Journal

[14] Il'in, A. (1960) Degenerate Elliptic and Parabolic Equations. Russian Math. Sbornik. vol.

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

forthcoming in Mathematical Methods in the Applied Sciences, 19 pp.

[5] Buchen, P. (2001) Image options and the road to barriers. Risk. (Sept.)

<sup>2</sup> *<sup>x</sup>*<sup>2</sup> <sup>&</sup>gt; 0 on *<sup>I</sup>* <sup>∪</sup> *I I*, i.e. <sup>Σ</sup><sup>3</sup> <sup>=</sup> *<sup>I</sup>* <sup>∪</sup> *I I* and the equation

<sup>3</sup> , respectively at <sup>Σ</sup>(2)

NY-Toronto-London, McGraw-Hill book Company.

provisions. Journal of Finance. 31: 351-367.

Mathematical Physics. Nauka, Moscow.

approach. Mathematical Finance. 6: 365-378.

of Financial Engineering. 5: 199-209.

[12] Heynen, R., Kat, H. (1994) Crossing the barrier. Risk. 7: 46-51.

(Eds.). Handbooks in OR & MS, Elsevier. vol. 15: 343-373.

Mathematical Finance. 5: 143-163.

*σ*2

ODE:

0 data at Σ(1)

**6. References**

12: 1261-1298.

2: 173-209.

5: 1-30.

50. 4: 443-498

Finance. 2: 275-298.

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

 *L*(*u*) = *f* in *Q u* |*I*∪*I I*∪*IV*= 0

in the rectangle *<sup>Q</sup>* <sup>=</sup> {(*t*, *<sup>x</sup>*) : 0 <sup>&</sup>lt; *<sup>t</sup>* <sup>&</sup>lt; *<sup>T</sup>*, *<sup>a</sup>*<sup>2</sup> <sup>&</sup>lt; *<sup>x</sup>* <sup>&</sup>lt; *<sup>a</sup>*1} and *<sup>h</sup>*, *<sup>g</sup>*, *<sup>c</sup>*, *<sup>F</sup>* <sup>∈</sup> *<sup>C</sup>*3(*Q*¯). Moreover, we assume that in some domain

*(i) Q*� <sup>⊃</sup> *<sup>Q</sup>*¯ the function *<sup>h</sup>* <sup>≥</sup> 0 and *<sup>h</sup>* <sup>∈</sup> *<sup>C</sup>*2(*Q*� ).

*(ii)* Suppose that if *h*(*t*, *a*1) = 0 (*h*(*t*, *a*2) = 0), 0 ≤ *t* ≤ *T*, then *g*(*t*, *a*1) > 0 (*g*(*t*, *a*2) < 0).

Moreover, we assume that the following compatibility conditions hold:

$$(\text{iii})\ D\_{t,\mathbf{x}}^{\alpha}F(T,a\_1) = D\_{t,\mathbf{x}}^{\alpha}F(T,a\_2) = 0, |\alpha| \le 2.$$

Define now the following parts of the boundary *∂Q*:

$$I = \{(t, \mathbf{x}) : 0 < t < T, \mathbf{x} = a\_2\}, \\ II = \{(t, \mathbf{x}) : 0 < t < T, \mathbf{x} = a\_1\},$$

$$III = \{(t, \mathbf{x}) : a\_2 < \mathbf{x} < a\_1, t = 0\} \text{ and } IV = \{(t, \mathbf{x}) : a\_2 < \mathbf{x} < a\_1, t = T\}.$$

One can easily see that: <sup>Σ</sup><sup>3</sup> <sup>=</sup> {(*t*, *<sup>x</sup>*) <sup>∈</sup> *<sup>I</sup>* <sup>∪</sup> *I I* : *<sup>h</sup>*(*t*, *<sup>x</sup>*) <sup>&</sup>gt; <sup>0</sup>}, <sup>Σ</sup><sup>0</sup> <sup>=</sup> {(*t*, *<sup>x</sup>*) <sup>∈</sup> *<sup>I</sup>* <sup>∪</sup> *I I* : *<sup>h</sup>*(*t*, *<sup>x</sup>*) = <sup>0</sup>} <sup>∪</sup> {(*t*, *<sup>x</sup>*) <sup>∈</sup> *III* <sup>∪</sup> *IV*}, *<sup>β</sup>* <sup>=</sup> *gn*<sup>1</sup> <sup>+</sup> *<sup>n</sup>*<sup>2</sup> <sup>−</sup> *<sup>∂</sup><sup>h</sup> <sup>∂</sup><sup>x</sup> <sup>n</sup>*1, i.e. (*t*, *<sup>x</sup>*) <sup>∈</sup> <sup>Σ</sup>0, (*t*, *<sup>x</sup>*) <sup>∈</sup> *<sup>I</sup>* <sup>∪</sup> *I I* <sup>⇒</sup> *<sup>h</sup>*(*t*, *<sup>x</sup>*) = <sup>0</sup> <sup>⇒</sup> *∂h <sup>∂</sup><sup>x</sup>* <sup>=</sup> 0 and −→ *n* = (1, 0) on *I*, −→ *<sup>n</sup>* = (−1, 0) on *I I*. Thus *<sup>β</sup>* <sup>|</sup>*I*∩Σ<sup>0</sup><sup>=</sup> *gn*<sup>1</sup> <sup>=</sup> *<sup>g</sup>* <sup>&</sup>lt; 0, while *<sup>β</sup>* <sup>|</sup>*I I*∩Σ<sup>0</sup><sup>=</sup> <sup>−</sup>*<sup>g</sup>* <sup>&</sup>lt; 0. Therefore, *<sup>I</sup>* <sup>∩</sup> <sup>Σ</sup><sup>0</sup> <sup>⊂</sup> <sup>Σ</sup>2, *I I* <sup>∩</sup> <sup>Σ</sup><sup>0</sup> <sup>⊂</sup> <sup>Σ</sup>2. Evidently, *<sup>β</sup>* <sup>|</sup>*III*<sup>=</sup> *<sup>n</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>⇒</sup> *III* <sup>⊂</sup> <sup>Σ</sup>1, while *IV* ⊂ Σ2; Σ<sup>0</sup> = ∅.

In conclusion, *III* is free of data as it is of the type <sup>Σ</sup>1; (*<sup>I</sup>* <sup>∪</sup> *I I*) <sup>∩</sup> <sup>Σ</sup><sup>0</sup> and *IV* are of the type <sup>Σ</sup>2, while Σ<sup>3</sup> = (*I* ∪ *I I*) ∩ {*h* > 0}. Part of *I* ∪ *I I* is non-characteristic, part of *I* ∪ *I I* is of Σ<sup>2</sup> type. Data are prescribed on Σ<sup>2</sup> ∪ Σ3, i.e. on *I* ∪ *I I* ∪ *IV*.

#### **Theorem 9.** (see [14]).

*There exists a unique classical solution u of (51), u* |*I*∪*I I*∪*IV*= 0 *under the conditions (i), (ii), (iii). More specifically, there exists Lipschitz continuous derivatives: u, <sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>t</sup> , <sup>∂</sup><sup>u</sup> <sup>∂</sup><sup>x</sup> , <sup>∂</sup>*2*<sup>u</sup> <sup>∂</sup>x*<sup>2</sup> <sup>∈</sup> *<sup>C</sup>*0,*α*(*Q*)*,* <sup>0</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>&</sup>lt; <sup>1</sup>*.*

In [19] it is mentioned that under several restrictions on the coefficients the boundary value problem

$$\begin{cases} \begin{array}{c} \mathcal{N}(\boldsymbol{\mu}) = 0 \\ \boldsymbol{\mu} \mid\_{I \cup II \cup IV} = \mathbf{0} \end{array} \tag{52}$$

possesses a unique generalized bounded solution which is Lipschitz continuous in *Q*. The proof relies on the method of elliptic regularization.

**Remark 8.** If *a*<sup>2</sup> < *x* < *a*1, *a*<sup>2</sup> < 0, *a*<sup>1</sup> > 0, the Black-Scholes equation (44) is with *h*(*t*, *x*) = *σ*2 <sup>2</sup> *<sup>x</sup>*<sup>2</sup> <sup>&</sup>gt; 0 on *<sup>I</sup>* <sup>∪</sup> *I I*, i.e. <sup>Σ</sup><sup>3</sup> <sup>=</sup> *<sup>I</sup>* <sup>∪</sup> *I I* and the equation

$$\begin{cases} L(\mu) = f \quad \text{in} \quad Q \\ \cdot \end{cases}$$

22 Will-be-set-by-IN-TECH

One can find results concerning regularity of the generalized solutions of degenerate parabolic operators in cylindrical domains in [14] and [19]. For the sake of simplicity we shall consider

*<sup>∂</sup>x*<sup>2</sup> <sup>+</sup> *<sup>g</sup>*(*t*, *<sup>x</sup>*)

in the rectangle *<sup>Q</sup>* <sup>=</sup> {(*t*, *<sup>x</sup>*) : 0 <sup>&</sup>lt; *<sup>t</sup>* <sup>&</sup>lt; *<sup>T</sup>*, *<sup>a</sup>*<sup>2</sup> <sup>&</sup>lt; *<sup>x</sup>* <sup>&</sup>lt; *<sup>a</sup>*1} and *<sup>h</sup>*, *<sup>g</sup>*, *<sup>c</sup>*, *<sup>F</sup>* <sup>∈</sup> *<sup>C</sup>*3(*Q*¯). Moreover, we

One can easily see that: <sup>Σ</sup><sup>3</sup> <sup>=</sup> {(*t*, *<sup>x</sup>*) <sup>∈</sup> *<sup>I</sup>* <sup>∪</sup> *I I* : *<sup>h</sup>*(*t*, *<sup>x</sup>*) <sup>&</sup>gt; <sup>0</sup>}, <sup>Σ</sup><sup>0</sup> <sup>=</sup> {(*t*, *<sup>x</sup>*) <sup>∈</sup> *<sup>I</sup>* <sup>∪</sup> *I I* : *<sup>h</sup>*(*t*, *<sup>x</sup>*) = <sup>0</sup>}

<sup>−</sup>*<sup>g</sup>* <sup>&</sup>lt; 0. Therefore, *<sup>I</sup>* <sup>∩</sup> <sup>Σ</sup><sup>0</sup> <sup>⊂</sup> <sup>Σ</sup>2, *I I* <sup>∩</sup> <sup>Σ</sup><sup>0</sup> <sup>⊂</sup> <sup>Σ</sup>2. Evidently, *<sup>β</sup>* <sup>|</sup>*III*<sup>=</sup> *<sup>n</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>⇒</sup> *III* <sup>⊂</sup> <sup>Σ</sup>1, while

In conclusion, *III* is free of data as it is of the type <sup>Σ</sup>1; (*<sup>I</sup>* <sup>∪</sup> *I I*) <sup>∩</sup> <sup>Σ</sup><sup>0</sup> and *IV* are of the type <sup>Σ</sup>2, while Σ<sup>3</sup> = (*I* ∪ *I I*) ∩ {*h* > 0}. Part of *I* ∪ *I I* is non-characteristic, part of *I* ∪ *I I* is of Σ<sup>2</sup> type.

*There exists a unique classical solution u of (51), u* |*I*∪*I I*∪*IV*= 0 *under the conditions (i), (ii), (iii).*

In [19] it is mentioned that under several restrictions on the coefficients the boundary value

*N*(*u*) = 0

possesses a unique generalized bounded solution which is Lipschitz continuous in *Q*. The

).

*(ii)* Suppose that if *h*(*t*, *a*1) = 0 (*h*(*t*, *a*2) = 0), 0 ≤ *t* ≤ *T*, then *g*(*t*, *a*1) > 0 (*g*(*t*, *a*2) < 0).

*∂u ∂x*

+ *c*(*t*, *x*)*u* = *F*(*t*, *x*) (51)

*<sup>∂</sup><sup>x</sup> <sup>n</sup>*1, i.e. (*t*, *<sup>x</sup>*) <sup>∈</sup> <sup>Σ</sup>0, (*t*, *<sup>x</sup>*) <sup>∈</sup> *<sup>I</sup>* <sup>∪</sup> *I I* <sup>⇒</sup> *<sup>h</sup>*(*t*, *<sup>x</sup>*) = <sup>0</sup> <sup>⇒</sup>

*<sup>n</sup>* = (−1, 0) on *I I*. Thus *<sup>β</sup>* <sup>|</sup>*I*∩Σ<sup>0</sup><sup>=</sup> *gn*<sup>1</sup> <sup>=</sup> *<sup>g</sup>* <sup>&</sup>lt; 0, while *<sup>β</sup>* <sup>|</sup>*I I*∩Σ<sup>0</sup><sup>=</sup>

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

*<sup>u</sup>* <sup>|</sup>*I*∪*I I*∪*IV*<sup>=</sup> <sup>0</sup> (52)

*<sup>∂</sup>x*<sup>2</sup> <sup>∈</sup> *<sup>C</sup>*0,*α*(*Q*)*,* <sup>0</sup> <sup>&</sup>lt; *<sup>α</sup>* <sup>&</sup>lt; <sup>1</sup>*.*

only one example from Il'in as the conditions are simple and clear. Consider

*∂*2*u*

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>h</sup>*(*t*, *<sup>x</sup>*)

Moreover, we assume that the following compatibility conditions hold:

*III* = {(*t*, *x*) : *a*<sup>2</sup> < *x* < *a*1, *t* = 0} and *IV* = {(*t*, *x*) : *a*<sup>2</sup> < *x* < *a*1, *t* = *T*}.

*<sup>t</sup>*,*xF*(*T*, *a*2) = 0, |*α*| ≤ 2.

*I* = {(*t*, *x*) : 0 < *t* < *T*, *x* = *a*2}, *I I* = {(*t*, *x*) : 0 < *t* < *T*, *x* = *a*1),

Define now the following parts of the boundary *∂Q*:

Petar Popivanov and Angela Slavova

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

*(i) Q*� <sup>⊃</sup> *<sup>Q</sup>*¯ the function *<sup>h</sup>* <sup>≥</sup> 0 and *<sup>h</sup>* <sup>∈</sup> *<sup>C</sup>*2(*Q*�

<sup>∪</sup> {(*t*, *<sup>x</sup>*) <sup>∈</sup> *III* <sup>∪</sup> *IV*}, *<sup>β</sup>* <sup>=</sup> *gn*<sup>1</sup> <sup>+</sup> *<sup>n</sup>*<sup>2</sup> <sup>−</sup> *<sup>∂</sup><sup>h</sup>*

*n* = (1, 0) on *I*, −→

Data are prescribed on Σ<sup>2</sup> ∪ Σ3, i.e. on *I* ∪ *I I* ∪ *IV*.

proof relies on the method of elliptic regularization.

*More specifically, there exists Lipschitz continuous derivatives: u, <sup>∂</sup><sup>u</sup>*

assume that in some domain

*<sup>t</sup>*,*xF*(*T*, *<sup>a</sup>*1) = *<sup>D</sup><sup>α</sup>*

**Appendix**

*(iii) D<sup>α</sup>*

*∂h*

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> 0 and −→

*IV* ⊂ Σ2; Σ<sup>0</sup> = ∅.

**Theorem 9.** (see [14]).

problem

*Institute of Mathematics, Bulgarian Academy of Sciences, Bulgaria*

*u* |*I*∪*I I*∪*IV*= 0 possesses a unique classical solution. As we know, *u* |*x*=0= *U*(*t*) satisfies in classical sense the ODE:

*U*�(*t*) − *rU*(*t*) = *f*(*t*, 0), *U*(*T*) = 0. Therefore, we can consider the restrictions: *u* |*x*>0, *u* |*x*<<sup>0</sup> and conclude that they are classical solutions of the respective boundary value problems with 0 data at Σ(1) <sup>2</sup> <sup>∪</sup> <sup>Σ</sup>(1) <sup>3</sup> , respectively at <sup>Σ</sup>(2) <sup>3</sup> <sup>∪</sup> <sup>Σ</sup>(2) <sup>2</sup> .
