On $$L_{p}$$ - theory for stochastic parabolic integro-differential equations

Abstract

The existence and uniqueness in fractional Sobolev spaces of the Cauchy problem to a stochastic parabolic integro-differential equation is investigated. A model problem with coefficients independent of space variable is considered. The equation arises in a filtering problem with a jump signal and jump observation process.

On solvability of integro-differential equations

“…Results in this direction can be found, for example, in [11], [12], [15], [25], [27], [28], [30] and [37], and for nonlinear equations of the type (1.1), arising in the theory of stochastic control of random processes with jumps, we refer to [12] and [38]. Extensions of the L p -theory of Krylov [16] to stochastic equations and systems of stochastic equations with integral operators of the type M and N above are developed in [7], [6], [17], [18] and [29].…”

On the Cauchy problem for stochastic integro-differential equations with radially O-regularly varying Lévy measure

“…We prove the following based on Lemma 12 from [16]. J s Φ (r, •, z) J −s ϕdx q (dr, dz) , 0 ≤ t ≤ T.…”

On Solvability of Integro-Differential Equations

“…Results in this direction can be found, for example, in [12,13,17,27,29,30,32] and [39], and for nonlinear equations of the type (1.1), arising in the theory of stochastic control of random processes with jumps, we refer to [13] and [40]. Extensions of the L p -theory of Krylov [18] to stochastic equations and systems of stochastic equations with integral operators of the type M and N above are developed in [6,7,19,20,28] and [31].…”