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.
Highlights
As far as we know Theorem 2.1 below is the first result on solvability in L p-spaces of stochastic integro-differential equations (SIDEs) without any non-degeneracy conditions
To formulate our assumptions we fix a constant K, a non-negative real number m, an exponent p ∈ [2, ∞), and non-negative Z-measurable real-valued functions ηand ξon Z such that they are bounded by K and
Theorem 4.1 (i) If A0, A1 and B0, B1 are two interpolation couples and S : A0 + A1 → B0+ B1 is a linear operator such that its restriction onto Ai is a continuous operator into Bi with operator norm Ci for i = 0, 1, its restriction onto Aθ = [ A0, A1]θ is a continuous operator into Bθ = [B0, B1]θ with operator norm C01−θ C1θ for every θ ∈ (0, 1)
Summary
Theorem 2.1 states that under the stochastic parabolicity condition on the operators (L, M), N ξ , N η, and appropriate regularity conditions on their coefficients and on the initial and free data, the Cauchy problem (1.1)–(1.2) has a unique generalised solution u = (ut )t∈[0,T ] for any given T. As far as we know Theorem 2.1 below is the first result on solvability in L p-spaces of stochastic integro-differential equations (SIDEs) without any non-degeneracy conditions It generalises the main result of [9] on deterministic integro-differential equations to SIDEs. Our motivation to study Eq (1.1) comes from nonlinear filtering of jump-diffusion processes, and we want to apply Theorem 2.1 to filtering problems in a continuation of the present paper.
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