Strong solutions and pathwise control for non-linear stochastic system with Poisson jumps in \(n\)-dimensional space (Q1175585)

To prove the existence of strong solutions for stochastic differential equations normally requires some monotone condition on the whole coefficient and some continuous, less than linear growth condition on the drift coefficients. Moreover, as the author mentiones, there is a lack of results with respect to the existence of strong solutions to stochastic differential equations with Poisson jump in \(n\)-dimensional space without the assumption on the continuity for the drift coefficients \(\sigma(t,x)\) or integrability of \((\partial\sigma/\partial x)^ 2\). The results in the paper weaken such condition on \(\sigma\) and the usual monotone condition on the coefficients. It excludes the continuous assumption on drift and also replaces the less than linear growth condition by a much weaker one. The result is also used to obtain the existence of a pathwise optimal Bang-Bang stochastic control for such nonlinear stochastic systems with Poisson jumps in \(n\)-dimensional case. Finally, the result is also applied to derive a maximum likelihood estimate of parameter for some continuous stochastic system with non-Lipschitz coefficients in \(n\)- dimensional space.