The authors consider a financial market where the risky assets are jump diffusion processes and where the utility function of an agent is a power utility function \(U(x) = \frac{x^{\delta}}{\delta}\), \(x \in \mathbb{R}_+\), for some \(\delta \in (0,1)\). They impose constraints on the portfolio by prohibiting borrowing and short selling and are interested in maximizing the expected utility \(v(t,x,\lambda)\) from the terminal wealth.
They transform the corresponding Hamilton-Jacobi-Bellman equation into the semi-linear PDE \[ - \frac{\partial \phi}{\partial t} - \frac{1}{2} \Delta \phi + H(\lambda,D\phi) = 0, \quad (t,\lambda) \in [0,T) \times \mathbb{R}^d \] with terminal condition \[ \phi(T,\lambda) = 0, \quad \lambda \in \mathbb{R}^d, \] where \(H\) denotes the Hamiltonian, and they prove the existence of a smooth solution of this equation.
Afterwards, the authors derive a backward stochastic differential equation which is related to the semi-linear PDE and which turns out to be Lipschitz, and they present a numerical scheme in order to simulate this BSDE. Some numerical studies illustrate their results.