  In this paper, we propose a new approach for stochastic control problems
arising from utility maximization. The main idea is to directly start from the
dynamical programming equation and compute the conditional expectation using a
novel representation of the conditional density function through the Dirac
Delta function and the corresponding series representation. We obtain an
explicit series representation of the value function, whose coefficients are
expressed through integration of the value function at a later time point
against a chosen basis function. Thus we are able to set up a recursive
integration time-stepping scheme to compute the optimal value function given
the known terminal condition, e.g. utility function. Due to tensor
decomposition property of the Dirac Delta function in high dimensions, it is
straightforward to extend our approach to solving high-dimensional stochastic
control problems. The backward recursive nature of the method also allows for
solving stochastic control and stopping problems, i.e. mixed control problems.
We illustrate the method through solving some two-dimensional stochastic
control (and stopping) problems, including the case under the classical and
rough Heston stochastic volatility models, and stochastic local volatility
models such as the stochastic alpha beta rho (SABR) model.
