In this paper, we propose a class of nonlinear expectations induced by backward stochastic differential equations and reflected backward stochastic differential equations and prove the law of large numbers under the nonlinear expectation.

In this paper, the comparison theorem of stochastic differential delay equations with Markovian switching has been investigated. In order to develop our theory, a new proof of existence and uniqueness of stochastic differential equations with Markovian switching is given.

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Itô formula is derived. When a standard Brownian motion is the original semimartingale, classical Itô stochastic differen...

We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations with jumps. This is a type of equations which appear as adjoint equations in the maximum principle approach to optimal control of systems described by stochastic partial differential equations driven by Lévy processes.

To analyze the stability of Itô stochastic differential equations with multiplicative noise, we introduce the stochastic logarithmic norm. The logarithmic norm was originally introduced by G. Dahlquist in 1958 as a tool to study the growth of solutions to ordinary differential equations and for estimating the error growth in discretization methods for their approximate solutions. We extend the ...

2 Young’s integrals and stochastic differential equations driven by fractional Brownian motions 4 2.1 Young’s integral and basic estimates . . . . . . . . . . . . . . . . . . 4 2.2 Stochastic differential equations driven by a Hölder path . . . . . . . 7 2.3 Multidimensional extension . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Fractional calculus . . . . . . . . . . . . . . . . . . . ...

An equivalence of the exponential stability concerning stochastic differential equations (SDEs), delay (SDDEs), and their corresponding Euler-Maruyama (EM) methods, is established. We show that for these four processes can be deduced from each other, provided or step size small enough. Using this relationship, we obtain between SDEs (or SDDEs) numerical methods difference) delay-free equations....

In this paper, we consider the problem of mean-variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by Brownian motion and doubly stochastic Poisson processes. This problem is formulated as a stochastic control problem and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and th...

In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation o...