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In this paper, we study the reflected backward stochastic differential equations driven by G-Lévy process with two reflecting obstacles under the Lipschitz condition on the coefficients, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. We give some a priori estimations for the uniqueness and we prove the existence of the solution by penalization method.

In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 \frac{1}{2}<\alpha<1 . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our results is also provided.

Risk management is one of the fundamental concepts in the mathematical finance. We study quadratic-exponential growth Backward Stochastic Volterra Integral Equations with Jumps (QBSVIEJs, for short). We prove the existence and uniqueness of solutions to QBSVIEJ in two steps. First, we use the fact that a QBSVIEJ with the generator independent of 𝑦 can be represented by a (infinite) family of QBSDEJs to get the first simple well-posedness result. And secondly, we use fixed point argument and induction to deal with the general case. We prove a comparison theorem and, as an application, define a convex dynamic risk measure by means of the QBSVIEJs.

The main focus of this work is a result related to multidimensional stochastic differential equations and one-dimensional SDEs (stochastic differential equations), using the Smith’s π-projection approach. This method is a commonly used technique in mathematics to simplify complex systems by projecting them onto lower-dimensional spaces, which can facilitate analysis and resolution. In the context of multidimensional stochastic processes (i.e., systems involving multiple random variables), extending the Smith’s π-projection approach we mention would likely involve defining a multidimensional version of this projection. This could include extending the concepts of projection and dimension reduction to multiple random variables, as well as analyzing how these projections can simplify multidimensional stochastic differential equations. As an application, we will prove the existence of almost periodic solutions to n-dimensional stochastic differential equations using the Smith projection. In fact, we will generalize the result from the article [M.-A. Boudref and A. Berboucha, Existence of almost periodic solutions of stochastic differential equations with periodic coefficients, Random Oper. Stoch. Equ. 25 2017, 1, 57–70] to dimensions n > 1 {n>1} .

The purpose of this work is to use the mixed fractional Brownian motion with parameter H ∈ ( 0 , 1 2 ) {H\in(0,\frac{1}{2})} as a modeling tool in financial mathematic, specifically in stochastic volatility modeling. We employ high frequency data from Oxford-Man Institute of Quantitative Finance Realized Library to estimate the smoothness of log-volatility. We show through empirical experiments that the measured smoothness of log-volatility is similar to that of mixed fractional Brownian motion when 0.09 ≤ H ≤ 0.2 {0.09\leq H\leq 0.2} . We propose to model the log-volatility by a mixed Fractional Brownian motion with H < 1 2 {H<\frac{1}{2}} and we construct a stationary mixed fractional Ornstein–Uhlenbeck process d ⁢ X t = - λ ⁢ ( X t - μ ) ⁢ d ⁢ t + γ ⁢ d ⁢ M t H , t ∈ [ 0 , T ] {dX_{t}=-\lambda(X_{t}-\mu)dt+\gamma dM_{t}^{H},t\in[0,T]} as a stationary model of log-volatility X t {X_{t}} and we show that it tends in distribution to the mixed fractional Brownian motion when λ → 0 {\lambda\rightarrow 0} .

This paper is devoted to presenting an averaging principle for reflected backward doubly stochastic differential equations with jumps (RBDSDEP in short), in which the obstacle process is right continuous with left limits. Under suitable Lipschitz conditions, it is shown that the solutions of RBDSDEP can be approximated by solutions to averaged stochastic systems in the mean-square sense and probability.

This paper is concerned with the existence of mild solutions for stochastic differential equations driven by mixed sub-fractional Brownian motion with Riemann–Liouville fractional derivative. The results are obtained by the fixed point theorem. An example is given to illustrate the obtained results.