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Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type. (English) Zbl 1287.60076

The paper deals with a \(d\)-dimensional stochastic differential equation of jump type (jump-diffusion) of the form \[ dX_t = b_t(X_t)dt+\sigma_t(X_t)dW_t+\int_{\mathbb R^d\backslash\{0\}}f_t(X_{t-},y) \tilde N(dt,dy), \] where \(W\) is a \(d\)-dimensional standard Wiener process and \(\tilde N\) is a compensated Poisson random measure. The latter is such that all moments of solutions to this equation exist. The author then studies the Fokker-Planck type equation connected to this jump-diffusion where the interest is in extending existence and uniqueness results to irregular coefficients. Under the assumptions that the drift \(b\) is in some (appropriate) Sobolev space and that \(\sigma\) and \(f\) are Lipschitz continuous, it is proven that a unique generalized stochastic flow exists. This result then allows to obtain the existence and uniqueness of \(L^p\)-valued or measure-valued solutions to the Fokker-Planck equation.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J75 Jump processes (MSC2010)
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