We consider an optimal control problem for linear stochastic systems with Markovian-switching coefficients, state-delay, additive and multiplicative noise. By developing a generalized quadratic-linear cost functional that includes Markovian-switching coefficients, and employing the method of completing the squares, we succeed in solving the aforementioned problem in an explicit closed-form solution through a system of coupled Riccati and partial differential equations. The optimal control law is of an affine feedback form with respect to the system state, the delayed state, and the integral of the previous system state values.

We consider the problem of optimal investment in a multi-asset market consisting of a bond, two stocks, possibility of borrowing, unbounded random coefficients, and the power utility from terminal wealth. The resulting optimization problem, due to the higher interest rate for borrowing than for lending, is a two-input stochastic optimal control problem with a nonlinear system dynamics and unbounded random coefficients. A certain two-dimensional piece-wise completion of squares method and a linear backward stochastic differential equation are used to find an explicit closed-form solution as a linear state-feedback control, the gain of which can have up to five different random regimes.

We consider the optimal investment problem with power utility from terminal wealth in a market with borrowing and a Heston stochastic volatility model. This is an optimal stochastic control problem with a nonlinear system dynamics due to the higher interest rate for borrowing than for lending and the square-root nonlinearity of the Heston model. Using a certain piece-wise compeltion of squares method, the unique explicit closed-form solution to this problem is obtained as a linear state-feedback control, the gain of which can have up to three different regimes.

We consider an optimal control problem for linear stochastic systems with state-delay, additive and multiplicative noise, and random time-horizon. We obtain an explicit closed-form solution to this problem for a general quadratic-linear cost functional through a system of coupled Riccati and partial differential equations. The optimal control law is of an affine feedback form with respect to the system state, the delayed state, and the integral of past system state values. An application to the optimal investment problem with a logarithmic utility and an interest rate with delayed factor process is also given.

We consider the problem of optimal investment in a market with borrowing, a stochastic interest rate, and the power utility from terminal wealth. A certain combined Hull-White and quadratic-affine interest rate model is introduced, which in particular renders the market incomplete in general. The resulting problem, due to the higher interest rate for borrowing than for lending, is an optimal stochastic control problem with a nonlinear system dynamics and an unbounded coefficient. The explicit closed-form solution is found as a linear state-feedback control the gain of which can have up to three different regimes.

We consider the optimal control problem with an indefinite generalised risk-sensitive criterion, and a class of stochastic control systems with multiplicative noise that have a quadratic type nonlinearity in the state and control variables. All solutions to such an optimal control problem are obtained in an explicit closed-form as affine state-feedback controls with possibly random gains. An application to the optimal investment problem is also given.

We consider the problem of optimally tracking a financial benchmark in a market with random and possibly unbounded coefficients. This is an example of a stochastic linear-quadratic control problem with unbounded system and cost functional coefficients. We obtain the unique solution in an explicit closed-form as an affine state-feedback control. The coefficients of the control law are given in terms of two linear backward stochastic differential equations with unbounded coefficients, the solvability of which is also proved.

We consider the problem of optimal investment in a market with random interest rate for borrowing. This represents an optimal stochastic control problem with a nonlinear system dynamics. We solve this problem under the assumption of coupled Hull-White type model for the interest rates. An explicit closed-form solution is derived for the power and logarithmic utility functions from terminal wealth.