On $$L_p$$-solvability of stochastic integro-differential equations

Abstract

A class of (possibly) degenerate stochastic integro-differential equations of parabolic type is considered, which includes the Zakai equation in nonlinear filtering for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces.2010 Mathematics Subject Classification. Primary 45K05, 60H15, 60H20, 60J75; Secondary 35B65.Its derivatives in x ∈ R d up to order max{ m , 3} exist and are continuous in x ∈ R d such that |D k ξ| ≤ξ k = 0, 1, 2, ..., max{ m , 3} :=m for all (ω, t, x, z) ∈ Ω × H T × Z. Moreover, K −1 ≤ | det(I + θDξ t,z (x))| for all (ω, t, x, z, θ) ∈ Ω × H T × Z × [0, 1], where I is the d × d identity matrix, and Dξ denotes the Jacobian matrix of ξ in x ∈ R d . Assumption 2.3. The function η = (η i ) maps Ω × [0, T ] × R d × Z into R d such that Assumption 2.2 holds with η andη in place of ξ andξ, respectively.

On partially observed jump diffusions II. The filtering density

“…To show the solvability of the linear filtering equation (3.9) with any F 0 -measurable L pvalued initial condition, we want to apply the existence and uniqueness theorem for stochastic integro-differential equations proved in [8]. With this purpose in mind first we assume that the coefficients σ, b, ρ, B, ξ, η are smooth in x P R d , and under this additional assumption we are going to determine the form of the "adjoint" operators L˚, M k˚, J η˚, J ξ˚a nd I ξ˚a s operators acting directly on C 8 0 such that ż…”

On Itô formulas for jump processes

“…3, further details of the proof can be found in [10]. Theorem 3.1 plays a crucial role in proving existence, uniqueness and regularity results in [11] for solutions to stochastic integro-differential equations. In [11], instead of a single random field v t (x) we have to deal with a system of random fields v i t (x) for i = 1, 2, .…”

Regional classification according to PM10 concentrations in the Seoul Metropolitan and Chungcheong Area, Republic of Korea

“…Particularly for γ < 2, these operators can be represented in a form of non-local operators. We refer to [9,14,18,24,26,27,28,37,38,39,40,49,50] for L p -theories with generators of stochastic processes and non-local operators. We also introduce papers [41,42,43] handling general operators with smooth symbols in L p -spaces based on a H ∞ -calculus approach.…”