**2. The Freidlin-Wentzell theory**

In this section, we briefly review the classical Freidlin-Wentzell theory of large deviations for the stochastic differential equations (SDEs) with small noise terms. In what follows, let us denote the solution of the SDEs in Eqs. (1)–(3) by a bold face letter **x***<sup>ε</sup> t* � � *<sup>t</sup>*≥<sup>0</sup> <sup>¼</sup> *<sup>x</sup>ε*,1 *<sup>t</sup>* , *xε*,2 *<sup>t</sup>* , *xε*,3 *t* � � *t*≥0 ≜ *c<sup>ε</sup>* ð Þ*<sup>t</sup>* ,*r<sup>ε</sup>* ð Þ*t* , *f <sup>ε</sup>* ð Þ ð Þ*<sup>t</sup> <sup>t</sup>*≥<sup>0</sup> as an <sup>3</sup> -valued diffusion process and rewrite the above equations as follows

$$d\mathbf{x}\_t^\varepsilon = \mathbf{f}\left(\mathbf{x}\_t^\varepsilon\right)dt + \sqrt{\varepsilon}\,\sigma\left(\mathbf{x}\_t^\varepsilon\right)dW\_t,\tag{4}$$

where  $\mathbf{f}\left(\mathbf{x}\_{t}^{\varepsilon}\right) = \left[f\_{1}\left(\mathbf{x}\_{t}^{\varepsilon}\right), f\_{2}\left(\mathbf{x}\_{t}^{\varepsilon}\right), f\_{3}\left(\mathbf{x}\_{t}^{\varepsilon}\right)\right]^{T}$  with 
$$f\_{1}\left(\mathbf{x}\_{t}^{\varepsilon,1}, \mathbf{x}\_{t}^{\varepsilon,2}, \mathbf{x}\_{t}^{\varepsilon,3}\right) = \left(-\mu + \beta k \left(1 - \mathbf{x}\_{t}^{\varepsilon,2} - \mathbf{x}\_{t}^{\varepsilon,1}\right)\right) \mathbf{x}\_{t}^{\varepsilon,1}$$
 
$$f\_{2}\left(\mathbf{x}\_{t}^{\varepsilon,1}, \mathbf{x}\_{t}^{\varepsilon,2}, \mathbf{x}\_{t}^{\varepsilon,3}\right) = \mu \mathbf{x}\_{t}^{\varepsilon,2} \tag{5}$$
 
$$f\_{3}\left(\mathbf{x}\_{t}^{\varepsilon,1}, \mathbf{x}\_{t}^{\varepsilon,2}, \mathbf{x}\_{t}^{\varepsilon,3}\right) = \left(-\beta k \left(1 - \mathbf{x}\_{t}^{\varepsilon,2} - \mathbf{x}\_{t}^{\varepsilon,1}\right)\right) \mathbf{x}\_{t}^{\varepsilon,1}$$

and *σ* **x***<sup>ε</sup> t* � � <sup>¼</sup> *<sup>σ</sup>*<sup>1</sup> **<sup>x</sup>***<sup>ε</sup> t* � �, *σ*<sup>2</sup> **x***<sup>ε</sup> t* � �, *σ*<sup>3</sup> **x***<sup>ε</sup> t* � � � � *<sup>T</sup>* with *σ*<sup>1</sup> *x<sup>ε</sup>*,1 *<sup>t</sup>* , *x<sup>ε</sup>*,2 *<sup>t</sup>* , *x<sup>ε</sup>*,3 *t* � � <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>μ</sup>* <sup>þ</sup> *<sup>β</sup><sup>k</sup>* <sup>1</sup> <sup>þ</sup> *<sup>x</sup><sup>ε</sup>*,2 *<sup>t</sup>* <sup>þ</sup> *<sup>x</sup><sup>ε</sup>*,1 *t* � � � � *x<sup>ε</sup>*,1 *t* q *σ*<sup>2</sup> *x<sup>ε</sup>*,1 *<sup>t</sup>* , *x<sup>ε</sup>*,2 *<sup>t</sup>* , *x<sup>ε</sup>*,3 *t* � � <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi *μx<sup>ε</sup>*,2 *t* q *σ*<sup>3</sup> *x<sup>ε</sup>*,1 *<sup>t</sup>* , *x<sup>ε</sup>*,2 *<sup>t</sup>* , *x<sup>ε</sup>*,3 *t* � � <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>β</sup><sup>k</sup>* <sup>1</sup> <sup>þ</sup> *<sup>x</sup><sup>ε</sup>*,2 *<sup>t</sup>* <sup>þ</sup> *<sup>x</sup><sup>ε</sup>*,1 *t* � � � � *x<sup>ε</sup>*,1 *t* q *:* (6)

Moreover, *Wt* is a standard three-dimensional Wiener process. Note that the corresponding backward operator for the diffusion process **x***<sup>ε</sup> <sup>t</sup>* , when applied to a certain function *υε* ð Þ *t*, **x** , is given by

$$
\partial\_t \boldsymbol{\upsilon}^{\varepsilon} + \mathcal{L}^{\varepsilon} \boldsymbol{\upsilon} \triangleq \frac{\partial \boldsymbol{\upsilon}^{\varepsilon}(t, \mathbf{x})}{\partial t} + \frac{\varepsilon}{2} \sum\_{i, j = 1}^{3} a\_{ij}(\mathbf{x}) \frac{\partial^2 \boldsymbol{\upsilon}^{\varepsilon}(t, \mathbf{x})}{\partial \mathbf{x}^{i} \partial \mathbf{x}^{j}} + \mathbf{f}(\mathbf{x}) \cdot \nabla\_{\mathbf{x}} \boldsymbol{\upsilon}^{\varepsilon}(t, \mathbf{x}), \tag{7}
$$

where *<sup>a</sup>*ð Þ¼ **<sup>x</sup>** *<sup>σ</sup>*ð Þ **<sup>x</sup>** *<sup>σ</sup>T*ð Þ **<sup>x</sup>** .

Let Ω ∈ <sup>3</sup> be bounded open domains with smooth boundary (i.e., ∂Ω is a manifold of class *C*<sup>2</sup> ) and let Ω*<sup>T</sup>* be an open set defined by

$$
\boldsymbol{\Omega}^T = (\mathbf{0}, T) \times \boldsymbol{\Omega}. \tag{8}
$$

Furthermore, let us denote by *C*<sup>∞</sup> Ω*<sup>T</sup>* � � the spaces of infinitely differentiable functions on Ω*<sup>T</sup>* and by *C*<sup>∞</sup> <sup>0</sup> <sup>Ω</sup>*<sup>T</sup>* � � the space of the functions *<sup>ϕ</sup>* <sup>∈</sup>*C*<sup>∞</sup> <sup>Ω</sup>*<sup>T</sup>* � � with compact support in Ω*T*. A locally square integrable function *υε* ð Þ *<sup>t</sup>*, **<sup>x</sup>** on <sup>Ω</sup>*<sup>T</sup>* is said to be a distribution solution to the following equation

$$
\partial\_t v^\varepsilon + \mathcal{L}^\varepsilon v^\varepsilon = \mathbf{0}, \tag{9}
$$

if, for any test function *ϕ*∈*C*<sup>∞</sup> <sup>0</sup> <sup>Ω</sup>*<sup>T</sup>* � �, the following holds true

$$\int\_{\Omega^{T}} (-\partial\_{t}\phi + \mathcal{L}^{\varepsilon \*}\phi) v^{\varepsilon} d\Omega^{T} = 0,\tag{10}$$

where *<sup>d</sup>*Ω*<sup>T</sup>* denotes the Lebesgue measure on <sup>3</sup> � <sup>þ</sup> and <sup>L</sup>*<sup>ε</sup>* <sup>∗</sup> is an adjoint operator corresponding to the infinitesimal generator <sup>L</sup>*<sup>ε</sup>* of the process **<sup>x</sup>***<sup>ε</sup> t* .

Moreover, we also assume that the following statements hold for the SDE in (4). Assumption 1


$$
\langle \mathbf{f}(t, \mathbf{x}), n(\mathbf{x}) \rangle \tag{11}
$$

is positive and zero, respectively. **Remark 1** *Note that*

$$\mathbb{P}\_{\mathfrak{s},\mathbf{x}\_{\varepsilon}^{\varepsilon}}^{\varepsilon}\left\{ \left(\boldsymbol{\tau}^{\varepsilon},\mathbf{x}\_{\tau^{\varepsilon}}^{\varepsilon}\right) \in \Gamma^{+} \bigcup \Gamma^{0}, \ \boldsymbol{\tau}^{\varepsilon} < \infty \right\} = \mathbbm{1}, \quad \forall \left(\boldsymbol{\varepsilon},\mathbf{x}\_{\varepsilon}^{\varepsilon}\right) \in \mathfrak{Q}\_{0}^{\infty}.\tag{12}$$

where *<sup>τ</sup><sup>ε</sup>* <sup>¼</sup> inf *<sup>t</sup>*>*s*j**x***<sup>ε</sup> <sup>t</sup>* ∈ ∂<sup>Ω</sup> � �. Moreover, if

$$\mathbb{P}^{\varepsilon}\_{\mathfrak{s}, \mathbf{x}\_{\varepsilon}^{\varepsilon}} \left\{ (t, \mathbf{x}\_{t}^{\varepsilon}) \in \Gamma^{0} \text{ for some } t \in [\mathfrak{s}, T] \right\} = \mathbf{0}, \quad \forall (\mathfrak{s}, \mathbf{x}\_{\varepsilon}^{\varepsilon}) \in \mathfrak{Q}\_{0}^{\infty}, \tag{13}$$

and *τ<sup>ε</sup>* ≤*T*, then we have *τ<sup>ε</sup>* , **x***<sup>ε</sup> τε* � �∈ Γþ, almost surely (see [24], Section 7).

In what follows, let **x***<sup>ε</sup> <sup>t</sup>* , for 0 ≤*t*≤ *T*, be the diffusion process associated with (4) (or Eqs. (1)–(3)) and consider the following boundary value problem

$$\begin{aligned} \partial\_{\boldsymbol{v}} \boldsymbol{v}^{\boldsymbol{\epsilon}} + \mathcal{L}^{\boldsymbol{\epsilon}} \boldsymbol{v}^{\boldsymbol{\epsilon}} &= \mathbf{0} \quad \text{in} \quad \Omega^{T} \\ \boldsymbol{v}^{\boldsymbol{\epsilon}}(\boldsymbol{s}, \mathbf{x}) &= \mathbf{1} \quad \text{on} \quad \Gamma^{+}\_{T} \\ \boldsymbol{v}^{\boldsymbol{\epsilon}}(\boldsymbol{s}, \mathbf{x}) &= \mathbf{0} \quad \text{on} \quad \{T\} \times \Omega \end{aligned} \tag{14}$$

*Rare Event Simulation in a Dynamical Model Describing the Spread of Traffic Congestions… DOI: http://dx.doi.org/10.5772/intechopen.95789*

where <sup>L</sup>*<sup>ε</sup>* is the backward operator in (7) and

$$
\Gamma\_T^+ = \{ (\mathfrak{s}, \mathfrak{x}) \in \Gamma^+ \, | \, \mathbf{0} < \mathfrak{s} \le T \}. \tag{15}
$$

Further, let <sup>Ω</sup>0*<sup>T</sup>* be the set consisting of <sup>Ω</sup>*<sup>T</sup>* <sup>∪</sup> f g *<sup>T</sup>* � <sup>Ω</sup>, together with the boundary points ð Þ *s*, **x** ∈Γþ, with 0< *s*<*T*. Then, the following proposition, whose proof is given in [25], provides a solution to the exit probability *<sup>ε</sup> s*,**x***<sup>ε</sup> <sup>s</sup> <sup>τ</sup><sup>ε</sup>* f g <sup>≤</sup>*<sup>T</sup>* with which the diffusion process **x***<sup>ε</sup> <sup>t</sup>* exits from the domain Ω.

**Proposition 1** Suppose that the statements in Assumption 1 hold true. Then, the exit probability *<sup>q</sup><sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ¼ *<sup>ε</sup> s*,**x***<sup>ε</sup> <sup>s</sup> <sup>τ</sup><sup>ε</sup>* f g <sup>≤</sup>*<sup>T</sup>* is a smooth solution to the boundary value problem in (14) and, moreover, it is a continuous function on Ω<sup>0</sup>*T.*

Note that, from Proposition 1, the exit probability *<sup>q</sup><sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ is a smooth solution to the boundary value problem in (14). Further, if we introduce the following logarithmic transformation (e.g., see [22, 26] or [23])

$$I^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}) = -\varepsilon \log q^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}).\tag{16}$$

Then, using ideas from stochastic control theory (see [22] for similar arguments), we present results useful for proving the following asymptotic property

$$I^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}) \to I^{0}(\mathfrak{s}, \mathbf{x}^{\varepsilon}) \quad \text{as} \quad \mathfrak{e} \to \mathbf{0}. \tag{17}$$

The starting point for such an analysis is to introduce a family of related stochastic control problems whose dynamic programming equation, for *ε*>0, is given below by (21). Then, this also allows us to reinterpret the exit probability function as a value function for a family of stochastic control problems associated with the underlying urban traffic network dynamics with small random perturbation. Moreover, as discussed later in Section 5, such a connection provides a computational paradigm – based on an exponentially-tilted biasing distribution – for constructing an efficient importance sampling estimators for rare-event simulations that further improves the efficiency of Monte Carlo simulations.

Then, we consider the following boundary value problem

$$\begin{aligned} \partial\_t \mathbf{g}^{\varepsilon} + \frac{\varepsilon}{2} \mathcal{L}^{\varepsilon} &= \mathbf{0} \quad \text{in} \quad \mathfrak{Q}^T\\ \mathbf{g}^{\varepsilon} &= \mathbb{E}\_{\mathbf{s}, \mathbf{x}}^{\varepsilon} \left\{ \exp \left( -\frac{1}{\varepsilon} \boldsymbol{\Phi}^{\varepsilon} \right) \right\} \quad \text{on} \quad \boldsymbol{\partial}^\* \boldsymbol{\Omega}^T \end{aligned} \tag{18}$$

where <sup>Φ</sup>*<sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ is a bounded, nonnegative Lipschitz function such that

$$\Phi^{\mathfrak{e}}(\mathfrak{s}, \mathbf{x}^{\mathfrak{e}}) = \mathbf{0}, \quad \forall (\mathfrak{s}, \mathbf{x}^{\mathfrak{e}}) \in \Gamma\_T^+. \tag{19}$$

Observe that the function *<sup>g</sup><sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ is a smooth solution in <sup>Ω</sup>*<sup>T</sup>* to the backward operator in (9); and it is also continuous on *∂* <sup>∗</sup> Ω*<sup>T</sup>*. Moreover, if we introduce the following logarithm transformation

$$J^{\varepsilon}(\boldsymbol{\varepsilon}, \mathbf{x}^{\varepsilon}) = -\varepsilon \log \mathbf{g}^{\varepsilon}(\boldsymbol{\varepsilon}, \mathbf{x}^{\varepsilon}). \tag{20}$$

Then, *J <sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ satisfies the following dynamic programming equation (i.e., the Hamilton-Jacobi-Bellman equation)

$$\partial\_i f^\epsilon + \frac{\varepsilon}{2} \sum\_{i,j=1}^3 a\_{ij} \frac{\partial^2 f^\epsilon}{\partial \mathbf{x}^i \partial \mathbf{x}^j} + H^\epsilon = \mathbf{0}, \quad \text{in} \quad \Omega^T,\tag{21}$$

where *<sup>H</sup><sup>ε</sup>* <sup>¼</sup> *<sup>H</sup><sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* , ∇**x***J <sup>ε</sup>* ð Þ is given by

$$H^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}, \nabla\_{\mathbf{x}} J^{\varepsilon}) = \mathbf{f}(\mathbf{x}^{\varepsilon}) \cdot \nabla\_{\mathbf{x}} J^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}) - \frac{1}{2} (\nabla\_{\mathbf{x}} J^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}))^{T} a(\mathbf{x}^{\varepsilon}) \nabla\_{\mathbf{x}} J^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}). \tag{22}$$

Note that the duality relation between *<sup>H</sup><sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ , � and *<sup>L</sup><sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ , � , i.e.,

$$H^{\varepsilon}(\mathbf{s}, \mathbf{x}^{\varepsilon}, \nabla\_{\mathbf{x}} J^{\varepsilon}) = \inf\_{\hat{u}} \left\{ L^{\varepsilon}(\mathbf{s}, \mathbf{x}^{\varepsilon}, \hat{u}) + \| \nabla\_{\mathbf{x}} J^{\varepsilon} \cdot \hat{u} \| \right\}, \tag{23}$$

with

$$L^{\varepsilon}(\mathbf{s}, \mathbf{x}^{\varepsilon}, \hat{u}) = \frac{1}{2} \left\| \mathbf{f}(\mathbf{x}^{\varepsilon}) - \hat{u} \right\|\_{[\mathbf{a}(\mathbf{x}^{\varepsilon})]^{-1}}^{2},\tag{24}$$

where <sup>∥</sup> � <sup>∥</sup><sup>2</sup> *<sup>a</sup>* **<sup>x</sup>***<sup>ε</sup>* ½ � ð Þ �<sup>1</sup> denotes the Riemannian norm of a tangent vector.

Then, it is easy to see that *J <sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ is a solution in <sup>Ω</sup>*<sup>T</sup>*, with *<sup>J</sup> <sup>ε</sup>* <sup>¼</sup> <sup>Φ</sup>*<sup>ε</sup>* on *<sup>∂</sup>* <sup>∗</sup> <sup>Ω</sup>*<sup>T</sup>*, to the dynamic programming in (21), where the latter is associated with the following stochastic control problem

$$J^{\varepsilon}(\boldsymbol{\varsigma}, \mathbf{x}^{\varepsilon}) = \inf\_{\boldsymbol{\hat{u}} \in \mathcal{U}(\boldsymbol{\varsigma}, \mathbf{x}^{\varepsilon}\_{\cdot})} \mathbb{E}\_{\boldsymbol{\varsigma}, \mathbf{x}^{\varepsilon}\_{\cdot}} \left\{ \int\_{\boldsymbol{\varsigma}}^{\theta} L^{\varepsilon}(\boldsymbol{\varsigma}, \mathbf{x}^{\varepsilon}, \boldsymbol{\hat{u}}) dt + \Phi^{\varepsilon}(\theta, \mathbf{x}^{\varepsilon}) \right\} \tag{25}$$

that corresponds to the following system of SDEs

$$d\mathbf{x}\_t^\varepsilon = \hat{u}(t)dt + \sqrt{\varepsilon}\sigma\left(\mathbf{x}\_t^\varepsilon\right)dW\_t,\tag{26}$$

with an initial condition **x***<sup>ε</sup> <sup>s</sup>* <sup>¼</sup> **<sup>x</sup>***<sup>ε</sup>* and *U s* ^ , **<sup>x</sup>***<sup>ε</sup>* ð Þ is a class of continuous functions for which *θ* ≤*T* and *θ*, *x<sup>ε</sup> θ* � �∈Γ<sup>þ</sup> *T* .

Next, we provide bounds, i.e., the asymptotic lower and upper bounds, on the exit probability *<sup>q</sup><sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ.

Define

$$\begin{aligned} I\_{\Omega}^{\varepsilon}((s, \mathbf{x}^{\varepsilon}); \partial \Omega) &= -\lim\_{\varepsilon \to 0} \varepsilon \log \mathbb{P}\_{s, \mathbf{x}\_{\varepsilon}^{\varepsilon}}^{\varepsilon} \{ \mathbf{x}\_{\theta}^{\varepsilon} \in \partial \Omega \}, \\ &\triangleq -\lim\_{\varepsilon \to 0} \varepsilon \log q^{\varepsilon}(s, \mathbf{x}^{\varepsilon}), \end{aligned} \tag{27}$$

where *<sup>θ</sup>* (or *<sup>θ</sup>* <sup>¼</sup> *<sup>τ</sup><sup>ε</sup>*∧*T*) is the first exit-time of **<sup>x</sup>***<sup>ε</sup> <sup>t</sup>* from the domain Ω. Furthermore, let us introduce the following supplementary minimization problem

$$\hat{I}^{\varepsilon}\_{\Omega}(\mathfrak{s},\mathfrak{\boldsymbol{\rho}},\mathfrak{\boldsymbol{\theta}}) = \inf\_{\substack{\boldsymbol{\rho} \in \mathrm{C}\_{\mathcal{T}}\left([\boldsymbol{\varepsilon},T],\mathbb{R}^{\boldsymbol{\beta}}\right),\boldsymbol{\theta} \geq \mathfrak{s}}} \int\_{\mathfrak{s}}^{\boldsymbol{\theta}} L^{\varepsilon}(\mathfrak{t},\boldsymbol{\rho}(\mathfrak{t}),\dot{\boldsymbol{\rho}}(\mathfrak{t}))d\mathfrak{t},\tag{28}$$

where the infimum is taken among all *<sup>φ</sup>*ð Þ� <sup>∈</sup>*CsT* ½ � *<sup>s</sup>*, *<sup>T</sup>* , <sup>3</sup> � � (i.e., from the space of *<sup>d</sup>*-valued locally absolutely continuous functions, with Ð *<sup>T</sup> <sup>s</sup>* j j *<sup>φ</sup>*\_ð Þ*<sup>t</sup>* <sup>2</sup> *dt* < ∞ for each *<sup>T</sup>* <sup>&</sup>gt;*s*) and *<sup>θ</sup>* <sup>≥</sup>*s*<sup>&</sup>gt; 0 such that *<sup>φ</sup>*ð Þ*<sup>s</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>T</sup>*, for all *<sup>t</sup>*<sup>∈</sup> ½ Þ *<sup>s</sup>*, *<sup>θ</sup>* , and ð Þ *<sup>θ</sup>*, *φ θ*ð Þ <sup>∈</sup>Γ<sup>þ</sup> *<sup>T</sup>* . Then, it is easy to see that

$$
\tilde{I}\_{\Omega}^{\varepsilon}(\mathfrak{s}, \mathfrak{q}, \theta) = I\_{\Omega}^{\varepsilon}((\mathfrak{s}, \mathbf{x}^{\varepsilon}); \partial \Omega). \tag{29}
$$

Next, we state the following lemma that will be useful for proving Proposition 2 (cf. [22], Lemma 3.1).

*Rare Event Simulation in a Dynamical Model Describing the Spread of Traffic Congestions… DOI: http://dx.doi.org/10.5772/intechopen.95789*

**Lemma 1** If *<sup>φ</sup>*∈*CsT* ½ � *<sup>s</sup>*, *<sup>T</sup>* , <sup>3</sup> � �, for *<sup>s</sup>*>0, and *<sup>φ</sup>*ðÞ¼ *<sup>s</sup>* **<sup>x</sup>***<sup>ε</sup> <sup>s</sup>* , ð Þ *<sup>t</sup>*, *<sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>∈</sup> <sup>Ω</sup>*T*, for all *t*∈½ Þ *s*, *T* , then lim *<sup>T</sup>*!<sup>∞</sup> Ð *T <sup>s</sup> <sup>L</sup><sup>ε</sup>* ð Þ *t*, *φ*ð Þ*t* , *φ*\_ð Þ*t dt* ¼ þ∞*.*

Consider again the stochastic control problem in (25) together with (26). Suppose that Φ*<sup>ε</sup> <sup>M</sup>* (with Φ*<sup>ε</sup> <sup>M</sup>* ≥0) is class *C*<sup>2</sup> such that Φ*<sup>ε</sup> <sup>M</sup>* ! þ∞ as *M* ! ∞ uniformly on any compact subset of <sup>Ω</sup>*T*n<sup>Γ</sup> þ *<sup>T</sup>* and Φ*<sup>ε</sup> <sup>M</sup>* on Γ<sup>þ</sup> *<sup>T</sup>* . Further, if we let *J <sup>ε</sup>* <sup>¼</sup> *<sup>J</sup> ε* <sup>Φ</sup>*<sup>M</sup>* , when <sup>Φ</sup>*<sup>ε</sup>* <sup>¼</sup> <sup>Φ</sup>*<sup>ε</sup> <sup>M</sup>*, then we have the following lemma.

**Lemma 2** Suppose that Lemma 1 holds, then we have

$$\liminf\_{M \to \infty} \quad J^{\epsilon}\_{\Phi\_M}((s, \mathbf{x}^{\epsilon})) \ge I^{\epsilon}(s, \mathbf{x}^{\epsilon}).\tag{30}$$
  $(\iota, \mathbf{x}^{\epsilon}\_{\iota}) \rightharpoonup (s, \mathbf{x}^{\epsilon}\_{\iota})$ 

Then, we have the following result.

**Proposition 2** [25, Proposition 2.8] Suppose that Lemma 1 holds, then we have

$$I^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}) \to I^{0}(\mathfrak{s}, \mathbf{x}^{\varepsilon}) \quad \text{as} \quad \mathfrak{e} \to \mathbf{0},\tag{31}$$

uniformly for all *s*, **x***<sup>ε</sup> s* � � in any compact subset Ω*<sup>T</sup>* . *Proof:* It is suffices to show the following conditions

$$\lim\_{\varepsilon \to 0} \sup e \, \log \mathbb{P}\_{\varepsilon, \mathbf{x}\_{\varepsilon}^{\varepsilon}}^{\varepsilon} \left\{ \mathbf{x}\_{\theta}^{\varepsilon} \in \partial \Omega \right\} \leq -I\_{\Omega}^{\varepsilon}((\varepsilon, \mathbf{x}^{\varepsilon}); \partial \Omega) \tag{32}$$

and

$$\lim\_{\varepsilon \to 0} \inf \, e \, \log \mathbb{P}\_{\boldsymbol{s}, \mathbf{x}\_{\boldsymbol{\tau}}^{\boldsymbol{\varepsilon}}}^{\boldsymbol{\varepsilon}} \left\{ \mathbf{x}\_{\boldsymbol{\theta}}^{\boldsymbol{\varepsilon}} \in \partial \Omega \right\} \geq -I\_{\Omega}^{\boldsymbol{\varepsilon}}((\boldsymbol{\varepsilon}, \mathbf{x}^{\boldsymbol{\varepsilon}}); \partial \Omega), \tag{33}$$

uniformly for all *s*, **x***<sup>ε</sup> s* � � in any compact subset Ω*<sup>T</sup>* . Note that *I ε* <sup>Ω</sup> *<sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ¼ ð Þ; <sup>∂</sup><sup>Ω</sup> *I <sup>ε</sup> <sup>s</sup>*, **<sup>x</sup>***<sup>ε</sup>* ð Þ (cf. Eq. (29)), then the upper bound in (32) can be verified using the Freidlin-Wentzell asymptotic estimates (e.g., see [27], pp. 332–334, [20] or [28]).

On the other hand, to prove the lower bound in (33), we introduce a penalty function Φ*<sup>ε</sup> <sup>M</sup>* (with Φ*<sup>ε</sup> <sup>M</sup> <sup>t</sup>*, **<sup>y</sup>** � � <sup>¼</sup> 0 for *<sup>t</sup>*, **<sup>y</sup>** � �∈Γ<sup>þ</sup> *<sup>T</sup>* ); and write *<sup>g</sup><sup>ε</sup>* <sup>¼</sup> *<sup>g</sup><sup>ε</sup> M* (� *<sup>ε</sup> s*,**x***<sup>ε</sup> <sup>s</sup>* exp � <sup>1</sup> *<sup>ε</sup>* Φ*<sup>ε</sup> M* � � � � ) and *J <sup>ε</sup>* <sup>¼</sup> *<sup>J</sup> ε* <sup>Φ</sup>*<sup>M</sup>* , with <sup>Φ</sup>*<sup>ε</sup>* <sup>¼</sup> <sup>Φ</sup>*<sup>ε</sup> <sup>M</sup>*. From the boundary condition in (18), then, for each *M*, we have

$$\mathbf{g}^{\boldsymbol{\epsilon}}(\boldsymbol{\varsigma}, \mathbf{x}^{\boldsymbol{\epsilon}}) \leq \mathbf{g}\_{M}^{\boldsymbol{\epsilon}}(\boldsymbol{\varsigma}, \mathbf{x}^{\boldsymbol{\epsilon}}).\tag{34}$$

Using Lemma 2 and noting further the following

$$J\_{\Phi\_M}^{\varepsilon}(\mathfrak{s}, \mathbf{x}^{\varepsilon}) \ge I\_{\Omega}^{\varepsilon}((\mathfrak{s}, \mathbf{x}^{\varepsilon}); \partial \Omega). \tag{35}$$

Then, the lower bound in (33) holds uniformly for all *s*, **x***<sup>ε</sup> s* � � in any compact subset Ω*<sup>T</sup>* . This completes the proof of Proposition 2. □
