Dynamic programming for multidimensional stochastic control problems (Q1819110)

The value function of a general multidimensional control problem with state process \[ X_t=x+\int_s^t b(r,X_r,u_r) dr + \int_s^t \sigma(r,X_r,u_r) dW_r + \xi_t \] and cost functional \[ \begin{multlined} J_{s,x}(u,\xi)=E\left [ \int_s^T h(r,X_r,u_r) dr+ \int_s^T f^u(r)\|\dot{\xi}^{ac}_r\|_1 dr + \int_s^T f^s(r)|d\xi^{sc}_r|\right.+\\ \left.+\sum_{r \in S_{\xi}[s,T]} l(r,\triangle\xi_r) + g(X_T)\right ]\end{multlined} \] for \(u\) a regular and \(\xi\) a singular control, \(S_{\xi}\) the set of times of jumps of \(\xi\), is shown to be a viscosity solution of a \(HJB\)-quasi-variational inequality. To show this the authors introduce a general dynamic programming approach which is able to cover singular and non-singular situations. The uniqueness of this inequality is proved.