In this paper, we present an efficient method for determining the solution of the stochastic second kind Volterra integral equations (SVIE) by using the Taylor expansion method. This method transforms the SVIE to a linear stochastic ordinary differential equation which needs specified boundary conditions. For determining boundary conditions, we use the integration technique. This technique give...

In this paper, we consider the problem of approximate solutions of set-valued stochastic differential equations. We firstly prove an inequality of set-valued Itô integrals, which is related to classical Itô isometry, and an inequality of set-valued Lebesgue integrals. Both of the inequalities play an important role to discuss set-valued stochastic differential equations. Then we mainly state th...

Motivated by a problematic coming from mathematical finance, the paper deals with existing and additional results on the continuity and the differentiability of the Itô map associated to rough differential equations. These regularity results together with the Malliavin calculus are applied to the sensitivities analysis of stochastic differential equations driven by multidimensional Gaussian pro...

The paper is concerned with a mixed stochastic delay differential equation involving both a Wiener process and a γ-Hölder continuous process with γ > 1/2 (e.g. a fractional Brownian motion with Hurst parameter greater than 1/2). It is shown that its solution depends continuously on the coefficients and the initial data. Two applications of this result are given: the convergence of solutions to ...

In order to give a fully rigorous account of the modern theory of diffusion processes, one must first develop a considerable amount of rather technical machinery. The measure-theoretic foundations of the theory of diffusion processes provide a rich field of mathematical inquiry, but are arguably of minimal relevance to practitioners wishing to use these processes to build models of real-world p...

The Stochastic differential equations, especially diffusion processes, have been widely used in physical and biological sciences and in financial economics. In mathematical finance the success of the diffusion process can be attributed to its many attractive properties. However, all models involve unknown parameters or functions, which need to be estimated from observations of the process. The ...

Gradient estimates and a Harnack inequality are established for the semigroup associated to stochastic differential equations driven by Poisson processes. As applications, estimates of the transition probability density, the compactness and ultraboundedness of the semigroup are studied in terms of the corresponding invariant measure.

The expressions of solutions for general n × m matrix-valued inhomogeneous linear stochastic differential equations are derived. This generalizes a result of Jaschke (2003) for scalar inhomogeneous linear stochastic differential equations. As an application, some IR vector-valued inhomogeneous nonlinear stochastic differential equations are reduced to random differential equations, facilitating...

where W is a Wiener process, Z is a Hölder continuous process with Hölder exponent greater than 1/2, the coefficients a, b, c depend on the past of the process X . The integral with respect to W is understood in the usual Itô sense, while the one with respect to Z is understood in the pathwise sense. (A precise definition of all objects is given in Section 2.) We will call this equation a mixed...