arXiv subjects

Remigijus Mikulevicius

Publications and source records attributed to Remigijus Mikulevicius.

3 recordsLinked to original sources

arXiv2015

On Degenerate Linear Stochastic Evolution Equations Driven by Jump Processes

We prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application of this result, we derive the existence and uniqueness of solutions of degenerate parabolic linear stochastic integro-differential equations (SIDEs) in the Sobolev scale. The SIDEs that we consider arise in the theory of non-linear filtering as the equations governing the conditional density of a degenerate jump-diffusion signal given a jump-diffusion observation, possibly with correlated noise.

math.PR↗

arXiv2014

On Classical Solutions of Linear Stochastic Integro-Differential Equations

We prove the existence of classical solutions to parabolic linear stochastic integro-differential equations with adapted coefficients using Feynman-Kac transformations, conditioning, and the interlacing of space-inverses of stochastic flows associated with the equations. The equations are forward and the derivation of existence does not use the "general theory" of SPDEs. Uniqueness is proved in the class of classical solutions with polynomial growth.

math.PR↗

arXiv2014

On Some Properties of Space Inverses of Stochastic Flows

We derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by jump SDEs with adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of continuous SDEs, we derive the existence and uniqueness of classical solutions of linear parabolic second order SPDEs by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case.

math.PR↗