Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Abstract

Several results concerning the existence and uniqueness of solutions of Ito SDE's in a real separable Hubert space have recently been reported. In this work we first obtain the existence and uniqueness of strong solutions to (not necessarily) Ito SDE's in a Hubert space under Lipschitz-type conditions on the coefficients. We assume usual continuity and linear growth conditions. For non-Lipschitz coefficients an approximation technique of Gikhman and Skorokhod is then used to prove the existence of weak solutions taking values in a larger Hubert space H-\. This result depends on an assumption that H can be compactly embedded in H-\, such that the coefficients satisfy regularity conditions with respect to H-\. This assumption is not a limitation to our method as it is necessary even in the deterministic case. Additionally, we prove the existence of martingale solutions to infinite dimensional semilinear SDE's. In both cases, coefficients F(£, ·) and J9(£, ·) may depend on the entire past of X £ C([0, T], H) and not on the value of x(t) alone.Brought to you by | provisional account Unauthenticated Download Date | 7/2/15 1:26 AM

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

“…also to [15,17,23,24,26,27,[30][31][32][33] and references therein for other interesting existence results on mild solutions of infinite-dimensional SDEs with Gaussian or Poisson noise. In particular we mention results by Knoche [31][32][33] which concern SDEs of the type (1) on Sobolev spaces, where properties of the differential dependence of the resolvent on the initial data are discussed.…”

Limit Theorems for Cylindrical Martingale Problems Associated with Lévy Generators

“…Lipschitz conditions for the existence of (pathwise) unique solutions to SDEs of the type (4.1) can be found in [9,22]. A version of Theorem 3 for SDEs driven by Wiener noise is given in [11]. We are not aware of any result in the direction of Theorem 3 which allows Lévy noise.…”

On the existence of weak variational solutions to stochastic differential equations

“…Now we will use the compact embedding argument, as in [2]. We point out the critical calculations in the second part of the proof adapted from [3].…”

“…We use the ideas in [2] and instead of the embeddings being Hilbert-Schmidt operators as in [3], we only assume their compactness. It should be noted that the weak solution X that we construct is in C([0, T ], H) with X ∈ L 2 ([0, T ] × Ω, V ), and it satisfies E sup…”