Importance Sampling for Multiscale Diffusions

Abstract

We construct importance sampling schemes for stochastic differential equations with small noise and fast oscillating coefficients. Standard Monte Carlo methods perform poorly for these problems in the small noise limit. With multiscale processes there are additional complications, and indeed the straightforward adaptation of methods for standard small noise diffusions will not produce efficient schemes. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Examples and simulation results are provided.

Nonasymptotic performance analysis of importance sampling schemes for small noise diffusions

“…Nevertheless, we shall assume it because it guarantees that the feedback control used in importance sampling is uniformly bounded, which means that a number of technicalities are avoided. The connection between subsolutions and performance of importance sampling schemes has been established in several papers, such as [8,5]. In the present setting, we have the following Theorem regarding asymptotic optimality (Theorem 4.1 in [5]).…”

“…The connection between subsolutions and performance of importance sampling schemes has been established in several papers, such as [8,5]. In the present setting, we have the following Theorem regarding asymptotic optimality (Theorem 4.1 in [5]). …”

“…As it is demonstrated in [8,5] the asymptotically optimal change of measure is associated with subsolutions to a certain class of Hamilton-Jacobi-Bellman (HJB) equations of the type (2.7)…”

“…For these reasons, a theory has been developed in [7,8,5] that gives precise bounds on asymptotic performance for schemes that are based on subsolutions to this HJB equation rather than to the solution itself; see also [3] for the closely related concept of the associated Lyapunov inequalities, that can be also used to find bounds of the performance measures in the prelimit. Such results have been used to test the efficiency of state-dependent importance sampling schemes in the heavy tail and discrete setting, e.g., [1,2,4,8].…”

Maximum likelihood estimation for small noise multiscale diffusions

“…When both parameters ǫ and δ go to zero together, then we need to consider three different regimes depending on how fast ǫ goes to zero relative to δ: We mention here that asymptotic problems for models like (1.1) have a long history in the mathematical literature. We refer the interested reader to classical manuscripts such as [4,13,24] for averaging and homogenization results and to the more recent articles [7,12] for large deviations results and [9,8] for importance sampling results on related rare event estimation problems.…”