**3. Importance sampling**

In this paper, we are mainly concerned with estimating the following quantity

$$\mathbb{E}\_{s, \mathbf{x}\_t^\varepsilon}^\varepsilon \left[ \exp \left( -\frac{1}{\varepsilon} \Phi^\varepsilon(\mathbf{x}^\varepsilon) \right) \right],\tag{36}$$

where <sup>Φ</sup>*<sup>ε</sup>* is an appropriate functional on *<sup>C</sup>* ½ � 0, *<sup>T</sup>* ; <sup>3</sup> � � and **<sup>x</sup>***<sup>ε</sup>* is a solution of the SDE in (4) and our analysis is in the situation where the level of the random perturbation is small, i.e., *ε* ≪ 1, and the functional *<sup>ε</sup> s*,**x***<sup>ε</sup> <sup>s</sup>* exp � <sup>1</sup> *<sup>ε</sup>* <sup>Φ</sup>*<sup>ε</sup>* **<sup>x</sup>***<sup>ε</sup>* ð Þ � � � � is rapidly varying in **x***<sup>ε</sup>* . Note that the challenge presented by such an analysis of rare event probabilities is well documented (see [12, 18, 29] for additional discussions). In the following (and see also Section 4), we specifically consider the case when the functional <sup>Φ</sup>*<sup>ε</sup>* is bounded and nonnegative Lipschitz, with <sup>Φ</sup>*<sup>ε</sup>* <sup>¼</sup> 0, if **x***ε <sup>t</sup>* <sup>∈</sup> <sup>Ω</sup>*<sup>T</sup>* <sup>⊂</sup>*<sup>C</sup>* ½ � 0, *<sup>T</sup>* : <sup>3</sup> � � and <sup>Φ</sup>*<sup>ε</sup>* <sup>¼</sup> <sup>∞</sup> otherwise; and we further consider analysis on the asymptotic estimates for exit probabilities from a given bounded open domain in the small noise limit case.

Consider the following simple estimator for the quantity of interest in (36)

$$\rho(\varepsilon) = \frac{1}{N} \sum\_{j=1}^{N} \exp\left(-\frac{1}{\varepsilon} \Phi^{\varepsilon} \left(\mathbf{x}^{\varepsilon(j)}\right)\right),\tag{37}$$

where **x***ε*ð Þ*<sup>j</sup>* � �*<sup>N</sup> <sup>j</sup>*¼<sup>1</sup> are *<sup>N</sup>*-copies of independent samples of **<sup>x</sup>***<sup>ε</sup>* . Here we remark that such an estimator is unbiased in the sense that

$$\mathbb{E}\_{s, \mathbf{x}\_{\varepsilon}^{\varepsilon}}^{\varepsilon}[\rho(\varepsilon)] = \mathbb{E}\_{s, \mathbf{x}\_{\varepsilon}^{\varepsilon}}^{\varepsilon} \left[ \exp \left( -\frac{1}{\varepsilon} \Phi^{\varepsilon}(\mathbf{x}^{\varepsilon}) \right) \right], \tag{38}$$

Moreover, its variance is given by

$$\text{Var}(\rho(\varepsilon)) = \frac{1}{N} \left( \mathbb{E}\_{\varepsilon, \mathbf{x}\_i'}^{\varepsilon} \left[ \exp \left( -\frac{2}{\varepsilon} \Phi^{\varepsilon}(\mathbf{x}^{\varepsilon}) \right) \right] - \mathbb{E}\_{\varepsilon, \mathbf{x}\_i'}^{\varepsilon} \left[ \exp \left( -\frac{1}{\varepsilon} \Phi^{\varepsilon}(\mathbf{x}^{\varepsilon}) \right) \right]^2 \right). \tag{39}$$

Then, we have the following for the relative estimation error

$$R\_{\text{err}}(\rho(\varepsilon)) = \frac{\sqrt{\text{Var}(\rho(\varepsilon))}}{\mathbb{E}\_{\varepsilon, \mathbf{x}\_i^{\varepsilon}}^{\varepsilon}[\rho(\varepsilon)]} \tag{40}$$

which can be further rewritten as follows

$$R\_{\text{err}}(\rho(\varepsilon)) = \left(\mathbf{1}/\sqrt{N}\right)\sqrt{\Delta(\rho(\varepsilon)) - 1},\tag{41}$$

where

$$\Delta(\rho(\varepsilon)) = \frac{\mathbb{E}\_{\varepsilon, \mathbf{x}\_{\varepsilon}^{\varepsilon}}^{\varepsilon} \left[ \exp \left( -\frac{2}{\varepsilon} \Phi^{\varepsilon} (\mathbf{x}^{\varepsilon}) \right) \right]}{\mathbb{E}\_{\varepsilon, \mathbf{x}\_{\varepsilon}^{\varepsilon}}^{\varepsilon} \left[ \exp \left( -\frac{1}{\varepsilon} \Phi^{\varepsilon} (\mathbf{x}^{\varepsilon}) \right) \right]^2} . \tag{42}$$

Note that, as we might expect, the relative estimation error may decrease with increasing the number of the sample size *N*. However, from Varahhan's lemma (e.g., see [30]; see also [20, 28]), under suitable assumptions, we also have the following conditions

$$\lim\_{\varepsilon \to 0} \sup e \, \log \mathbb{E}^{\varepsilon}\_{s, \mathbf{x}^{\varepsilon}\_{\cdot}} \left[ \exp \left( - \frac{1}{\varepsilon} \Phi^{\varepsilon} (\mathbf{x}^{\varepsilon}) \right) \right] = - \inf\_{\substack{\varrho \in \mathcal{C}\_{\mathcal{I}} \left( [\boldsymbol{\vartheta}, \boldsymbol{T}], \mathbb{R}^{nd} \right) \\ \varrho(\boldsymbol{\iota}) = \mathbf{x}\_{\iota}}} \{ I(\boldsymbol{\varrho}) + \Phi^{\varepsilon}(\boldsymbol{\varrho}) \} \tag{43}$$

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

and

$$\lim\_{\varepsilon \to 0} \sup e \, \log \mathbb{E}^{\varepsilon}\_{\boldsymbol{s}, \mathbf{x}^{\varepsilon}\_{\boldsymbol{t}}} \left[ \exp \left( - \frac{2}{\varepsilon} \Phi^{\varepsilon} (\mathbf{x}^{\varepsilon}) \right) \right] = - \inf\_{\substack{\boldsymbol{\varrho} \in \mathcal{C}\_{\mathcal{I}} \left[ \boldsymbol{s}, \mathcal{T} \right], \mathbb{R}^{\mathbf{x}^{\varepsilon}} \\ \boldsymbol{\varrho}(\boldsymbol{s}) = \mathbf{x}\_{\boldsymbol{t}}}} \left\{ I(\boldsymbol{\varrho}) + \mathcal{Q} \Phi^{\varepsilon} (\boldsymbol{\varrho}) \right\} \tag{44}$$

where *CsT* ½ � *<sup>s</sup>*, *<sup>T</sup>* , <sup>3</sup> � � is the set of absolutely continuous functions from ½ � *<sup>s</sup>*, *<sup>T</sup>* into 3 , with 0 <sup>≤</sup>*s*≤*t*<sup>≤</sup> *<sup>T</sup>*, and *<sup>I</sup>*ð Þ *<sup>φ</sup>* is the rate functional for the diffusion process **<sup>x</sup>***<sup>ε</sup> t* . From Jensen's inequality, the above equations in (43) and (44) also imply the following condition Δð Þ *ρ ε*ð Þ ≥1.
