On parabolic stochastic integro-differential equations: existence, regularity and numerics
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In this thesis, we study the existence, uniqueness, and regularity of systems of degenerate linear stochastic integro-differential equations (SIDEs) of parabolic type with adapted coefficients in the whole space. We also investigate explicit and implicit finite difference schemes for SIDEs with non-degenerate diffusion. The class of equations we consider arise in non-linear filtering of semimartingales with jumps. In Chapter 2, we derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by Lévy driven stochastic differential equations (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 Weiner driven SDEs, we prove the existence and uniqueness of classical solutions of linear parabolic second order stochastic partial differential equations (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. Chapter 3 is dedicated to the proof of existence and uniqueness of classical solutions of degenerate SIDEs using the method of stochastic characteristics. More precisely, we use Feynman-Kac transformations, conditioning, and the interlacing of space inverses of stochastic flows generated by SDEs with jumps to construct solutions. In Chapter 4, 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, we establish the existence and uniqueness of solutions of degenerate linear stochastic integro-differential equations in the L2-Sobolev scale. Finite difference schemes for non-degenerate SIDEs are considered in Chapter 5. Specifically, we study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear SIDEs and show that the rate is of order one in space and order one-half in time.
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