In this paper, we propose a new weak second-order numerical scheme for solving stochastic differential equations with jumps. By using trapezoidal rule and the integration-by-parts formula of Malliavin calculus, theoretically prove that has convergence rate. To demonstrate effectiveness rate, three experiments are given.

We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Major (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and...

Stein’s method is a method of probability approximation which hinges on the solution of a functional equation. For normal approximation, the functional equation is a first-order differential equation. Malliavin calculus is an infinite-dimensional differential calculus whose operators act on functionals of general Gaussian processes. Nourdin and Peccati (Probab. Theory Relat. Fields 145(1–2), 75...

This paper considers a controlled Itô-Lévy process the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly nonMarkovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explic...

Given a continuous Gaussian process x which gives rise to p-geometric rough path for p∈(2,3), and general y controlled by x, under proper conditions we establish the relationship between Skorohod integral ∫0tysd♢xs Stratonovich ∫0tysdxs. Our strategy is employ tools from paths theory Malliavin calculus analyze discrete sums of integrals.

By using the heat kernel parameter expansion with respect to frozen SDEs, intrinsic derivative is estimated for law of Mckean-Vlasov SDEs initial distribution. As an application, total variation distance between laws two solutions bounded by Wasserstein distributions. These extend some recent results proved distribution-free noise coupling method and Malliavin calculus.

The tools of the stochastic calculus of variations are constructed for Poisson processes on Lie group, and the corresponding analysis on the Lie-Wiener space is recovered as the limiting case of this construction.

We derive conditional Edgeworth-type expansions for Skorohod and Itô integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. As a consequence we obtain conditional Stein approximation bounds for multiple stochastic integrals and quadratic Brownian functionals.

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let p t,x (y) be the density of the law of the solution u(t, x) of such an equation at points (t, x) ∈]0, T ] × R 3. We prove that the mapping (t, x) → p t,x (y) owns the same regularit...