Approximation and convergence of solutions to semilinear stochastic evolution equations with jumps

Abstract

We prove that the mild solution to a semilinear stochastic evolution equation on a Hilbert space, driven by either a square integrable martingale or a Poisson random measure, is (jointly) continuous, in a suitable topology, with respect to the initial datum and all coefficients. In particular, if the leading linear operators are maximal (quasi-)monotone and converge in the strong resolvent sense, the drift and diffusion coefficients are uniformly Lipschitz continuous and converge pointwise, and the initial data converge, then the solutions converge.

A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk and Collateralization

“…According to the recent literature, it is also important to note that the BSDEs techniques investigated in the present paper are also strongly connected to theoretical and, respectively, applied, questions (see, e.g., [43,44]) that can lead to highly interesting developments, also from the statistical point of view.…”

Polynomial Chaos Expansion Approach to Malliavin Calculus Analysis of Bond Options Sensitivity

“…To clarify the notation, from now on we indicate the elements of the gPC basis as {Ψ n (x)} n∈N the family of Hermite or Legendre polynomials. We would like to underline that, from a more theoretical point of view, other (approximations) approaches can also be pursued, as, e.g., exploiting techniques outlined in [19], or studying related asymptotics for the involved financial quantities, see, e.g., [2,10] and references therein. The latter approach having the advantage to treat directly small perturbations arising in financial markets, particularly with respect to the consideration of limiting cases for the parameters, as, e.g., w.r.t.…”

Continuous Dependence on the Coefficients for Mean-Field Fractional Stochastic Delay Evolution Equations

“…We would like to mention that this problem for the stochastic differential equations without delay have been studied intensively. Da Prato and Zabczyk [9] studied the dependence of the solution on the initial datum ξ, Marinelli et al [15,16] studied the problem for the case of Poisson noise. The dependence of the solution on the coefficients f and g were considered by Peszat and Zabczyk [18] and Seidler [19].…”