In this study, we consider the exponential utility maximization problem in the context of a jump-diffusion model. To solve this problem, we rely on the dynamic programming principle and we derive from it a quadratic BSDE with jumps. Since this quadratic BSDE 2 is driven both by a Wiener process and a Poisson random measure having a Levy measure with infinite mass, our main work consists in establishing a new existence result for the specific BSDE introduced.

We consider a finite horizon optimal stopping problem related to trade-off strategies between expected profit and cost cash-flows of an investment under uncertainty. The optimal problem is first formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We then construct both a minimal and a maximal solutions using an approximation scheme of the associated system of reflected backward SDEs. We also address the question of uniqueness of solutions of this system of SDEs. When the dependence of the cash-flows on the sources of uncertainty, such as fluctuation market prices, assumed to evolve according to a diffusion process, is made explicit, we also obtain a connection between these solutions and viscosity solutions of a system of variational inequalities (VI) with interconnected obstacles.In other words, the cost Y 2 − a and the profit Y 1 + b act as obstacles that define the exit strategy.The main result of the paper is to show existence of the pair (Y 1 , Y 2 ) that solves the system of equations (1.1) and (1.4) and also to prove that τ * and σ * given respectively by (1.2) and (1.5) are

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewiseconstant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton-Jacobi-Bellman equation in our non-Markovian framework.