In this paper, we investigate the solvability of matrix valued Backward stochastic Riccati equations with jumps (BSREJ), which is associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and Poisson jumps. By dynamic programming principle, Doob-Meyer decomposition and inverse flow technique, the existence and uniqueness of the solution for the BSREJ is established. The difficulties addressed to this issue not only are brought from the high nonlinearity of the generator of the BSREJ like the case driven only by Brownian motion, but also from that i) the inverse flow of the controlled linear stochastic differential equation driven by Poisson jumps may not exist without additional technical condition, and ii) how to show the inverse matrix term involving jump process in the generator is well-defined. Utilizing the structure of the optimal problem, we overcome these difficulties and establish the existence of the solution. In additional, a verification theorem for BSREJ is given which implies the uniqueness of the solution.

The necessary conditions for an optimal control of a stochastic control problem with recursive utilities is investigated. The first order condition is the the well-known Pontryagin type maximum principle. When the optimal control satisfying such first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

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In the paper, we consider the no-explosion condition and pathwise uniqueness for SDEs driven by a Poisson random measure with coefficients that are super-linear and non-Lipschitz. We give a comparison theorem in the one-dimensional case under some additional condition. Moreover, we study the non-contact property and the continuity with respect to the space variable of the stochastic flow. As an application, we will show that there exists a unique strong solution for SDEs with coefficients like x log |x|. (Yuchao Dong).The purpose of this paper is to study the no-explosion condition and pathwise uniqueness for the solutions of (1.1) with super-linear and non-Lipschitz coefficients including the functions like x log |x|. Some other properties of the solutions, including comparison principle, non-contact property and continuity with respect to x of the stochastic flow, are also proved under some additional assumptions. As an application, we prove that the following SDE: