Bounds on mean variance hedging in jump diffusion

We compare the maximum principle and the linear quadratic regulator approach (LQR)/well-posedness criterion to mean variance hedging (MVH) when the wealth process follows a jump diffusion. The comparison is made possible via a measurability assumption on the coefficients of the process. Its application to determine an interval range for the MVH is explained. More precisely, in the MVH setup we show that $$0 \leq \inf _{u \in U}\tfrac{1}{2} E\Big[\int_{0}^T{y’_sy_s}\,ds+y’_Ty_T\Big] = \tfrac{1}{2}y’P^0_0y+f(P^0_0) \leq \tfrac{1}{2}y’P_0y+f(P_0),$$ where $P^0$ and $P$ satisfy a backward stochastic differential equation (BSDE) and $f$ is a measurable function affine in its only argument. The upper bound holds under the measurability assumption that all coefficients including the intensity of the jumps that drive $P$ are in fact predictable with respect to the filtration generated only by the Brownian motion. The lower bound is achieved expectedly under perfect hedging when the Föllmer–Schweizer minimal martingale probability measure is equivalent to the physical measure.