Abstract
We establish n-th-order Fréchet differentiability with respect to the initial datum of mild solutions to a class of jump diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz-continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order Gâteaux differentiability of their solutions with respect to the initial datum, extending previous results by Marinelli, Prévôt, and Röckner in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.
Highlights
Our goal is to obtain existence and uniqueness of mild solutions, and, especially, their differentiability with respect to the initial datum, to a class of stochastic evolution equations on Hilbert spaces of the form⎧ ⎪⎪⎪⎨ du(t) + Au(t) dt = f (t, u(t)) dt + B(t, u(t)) dW (t) ⎪⎪⎪⎩+ G(t, Z u(0) = u0. z, u (t −)) μ, (1.1)Here, A is a linear m-accretive operator, W is a cylindrical Wiener process, μis a compensated integer-valued quasi-left-continuous random measure, and the coefficients f, B, G satisfy suitable measurability and Lipschitz continuity conditions
Our ultimate goal is the extension of the results in [18] to non-local Kolmogorov equations associated with stochastic evolution equations with jumps in a generalized variational setting as considered in [17]
We show existence and uniqueness of a mild solution, as well as its continuous dependence on the initial datum, in spaces of processes with finite moments of order p ∈ ]0, +∞[
Summary
Our goal is to obtain existence and uniqueness of mild solutions, and, especially, their differentiability with respect to the initial datum, to a class of stochastic evolution equations on Hilbert spaces of the form. A typical approach is, roughly speaking, to regularize the coefficients of the equation, obtaining a family of approximating Kolmogorov equations that are sufficiently simple to have classical solutions, and to obtain a solution to the original problem passing to the limit, in an appropriate sense, with respect to the regularization parameter In this spirit, our ultimate goal is the extension of the results in [18] to non-local Kolmogorov equations associated with stochastic evolution equations with jumps in a generalized variational setting as considered in [17].
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