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

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.