##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics Seminar

## Konstantinos Spiliopoulos

#### Brown University & Boston University

## Large Deviations and Monte Carlo Methods for Problems with Multiple Scales

##### Abstract:

The need to simulate rare events occurs in many application areas, including telecommunication, finance, insurance, computational physics and chemistry. However, virtually any simulation problem involving rare events will have a number of mathematical and computational challenges. As it is well known, standard Monte Carlo sampling techniques perform very poorly in that the relative errors under a fixed computational effort grow rapidly as the event becomes more rare. In this talk, I will discuss large deviations, rare events and Monte Carlo methods for systems that have multiple scales and that are stochastically perturbed by small noise. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. Using stochastic control arguments we identify the large deviations principle for each regime of interaction. Furthermore, we derive a control (equivalently a change of measure) that allows to design asymptotically efficient importance sampling schemes for the estimation of associated rare event probabilities and expectations of functionals of interest. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies. Applications of these results in chemistry problems and in mathematical finance will also be discussed.

### May 24, 2012

### 10:00 AM

### AP&M 7321

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