Recursive Stochastic Linear-Quadratic Control with Indefinite Weights for Jump-Diffusion Systems with Random Coefficients

Abstract

We study finite-horizon stochastic linear-quadratic (LQ) control for finite-activity jump diffusions with bounded random coefficients, indefinite weights and a linear recursive cost. The cost is defined by a backward stochastic differential equation (BSDE) whose generator has bounded deterministic coefficients multiplying the Brownian and jump integrands. A Brownian-Poisson change of measure and an integrating factor reduce the criterion to a nonrecursive LQ functional. On the intersection of the physical and transformed square-integrable control spaces, we establish recursive evaluation for L^1 quadratic data under the transformed measure and equality with the full transformed value. Under transformed uniform convexity and invertibility of the uncontrolled jump map, we identify the stochastic Riccati equation from the nonrecursive value kernel. Conditional energy estimates justify transferring its martingale coefficients to the physical measure. Completion of squares gives feedback representation, comparison and uniqueness in the feedback-admissible class. For each initial pair, the recursive value is attained exactly when the transformed optimizer has finite physical control energy. Analytical examples illustrate nonzero feedback and uniform convexity with a negative control weight.