On Weak Approximation of Stochastic Differential Equations through Hard Bounds by Mathematical Programming

Upper and lower hard bounds of the expected value on stochastic differential equations can be obtained with the help of the mathematical programming and the Dynkin formula, without recourse to Monte Carlo sample paths simulation. In this paper, we show that feasible solutions of those optimization approaches further provide useful additional information. Namely, feasible solutions provide upper and lower bounds for arbitrary intermediate times and/or different initial states. We also show that the optimization approach can also be applied to stochastic differential equations with a random initial state. Our theoretical analyses are illustrated by numerical results on the survival probability for a square-root diffusion, on the moment estimation of a Doléans-Dade stochastic exponential with jumps, and on pricing of a barrier basket option under the multi-asset Black-Scholes model.