A Weak Approximation of Stochastic Differential Equations with Jumps Through Tempered Polynomial Optimization

Kenji Kashima,Reiichiro Kawai

doi:10.1080/15326349.2011.542721

Abstract

We present an optimization approach to the weak approximation of a general class of stochastic differential equations with jumps, in particular, when value functions with compact support are considered. Our approach employs a mathematical programming technique yielding upper and lower bounds of the expectation, without Monte Carlo sample paths simulations, based upon the exponential tempering of bounding polynomial functions to avoid their explosion at infinity. The resulting tempered polynomial optimization problems can be transformed into a solvable polynomial programming after a minor approximation. The exponential tempering widens the class of stochastic differential equations for which our methodology is well defined. The analysis is supported by numerical results on the tail probability of a stable subordinator and the survival probability of Ornstein-Uhlenbeck processes driven by a stable subordinator, both of which can be formulated with value functions with compact support and are not applicable in our framework without exponential tempering.

Highlights

Stochastic differential equations have long been used to build realistic models in economics, finance, biology, the social sciences, chemistry, physics and other fields
We have shown that the tempered polynomial optimization can be transformed into a polynomial optimization problem after the polynomial approximation of the exponential function on a compact set
Exponential tempering widens the class of stochastic differential equations to which our methodology is applicable

Summary

Introduction

Stochastic differential equations have long been used to build realistic models in economics, finance, biology, the social sciences, chemistry, physics and other fields. In [5], bounding functions must be in polynomial form to arrive at a polynomial programming, while in principle, any polynomial function necessarily explodes at infinity whenever it is constrained to be either non-positive or non-negative Due to this explosion at infinity, bounds are likely to be very far from the true value in particular when considering stochastic differential equations with very heavy tails and a value function with compact support. Such situations are often of practical interest, for example, the tail probability estimation of a stochastic differential equation with jumps To address this issue, we introduce the exponential tempering of bounding polynomial functions so that the explosion never occurs at infinity. We provide a brief sketch of the method-of-moments approach, that is dual to polynomial optimization

Motivation

Exponential Tempering of Bounding Functions

Numerical Illustrations

Tail Probability of Stable Subordinator

Survival Probability of Ornstein-Uhlenbeck Processes

Concluding Remarks