A numerical method for survival probability of diffusion processes using semidefinite programming

Recently, some mathematical programming approaches have been proposed for numerical analysis of stochastic processes. In this paper, we deal with the survival probability of diffusion processes, which is defined as the probability that a sample path of the diffusion process does not exceed a given value during a given time period. First, we formulate a semidefinite programming problem for the survival probability of a given diffusion process. Maximizing or minimizing the objective function representing the survival probability, we can compute upper and lower bounds for it, respectively. The constraints of the problem consist of equations derived from fundamental properties of the diffusion process and positive semidefinite conditions on moments with respect to some measures associated with the first hitting problem of stochastic processes. We confirm effectiveness of the proposed method through numerical experiments on some examples of diffusion processes.