Bujar Gashi

Linear Backward Stochastic Differential Equations of Descriptor Type: Regular Systems

In this article, a class of linear backward stochastic differential equations of descriptor type with time-invariant coefficients are introduced. Necessary and sufficient conditions for their solvability are obtained. It turns out that such equations may not always have a solution, and even when they do, some components of the solution could have a jump at terminal time. Exact controllability of linear descriptor stochastic control systems is also considered.

Risk-sensitive control for a class of nonlinear systems with multiplicative noise

Abstract In this paper, we consider the problem of optimal control for a class of nonlinear stochastic systems with multiplicative noise. The nonlinearity consists of quadratic terms in the state and control variables. The optimality criteria are of a risk-sensitive and generalised risk-sensitive type. The optimal control is found in an explicit closed-form by the completion of squares and the change of measure methods. As applications, we outline two special cases of our results. We show that a subset of the class of models which we consider leads to a generalised quadratic–affine term structure model (QATSM) for interest rates. We also demonstrate how our results lead to generalisation of exponential utility as a criterion in optimal investment.

Stochastic minimum-energy control

We give the solution to the minimum-energy control problem for linear stochastic systems. The problem is as follows: given an exactly controllable system, find the control process with the minimum expected energy that transfers the system from a given initial state to a desired final state. The solution is found in terms of a certain forward-backward stochastic differential equation of Hamiltonian type.

Generalised Risk-Sensitive Control with Full and Partial State Observation

This paper generalises the risk-sensitive cost functional by introducing noise dependent penalties on the state and control variables. The optimal control problems for the full and partial state observation are considered. Using a change of probability measure approach, explicit closed-form solutions are found in both cases. This has resulted in a new risk-sensitive regulator and filter, which are generalisations of the well-known classical results.

Optimal Portfolio Control with Trading Strategies of Finite Variation

We propose a method for portfolio selection with trading strategies constrained to having a finite variation. A linear combination of logarithms of each asset holdings values are used as a criterion, which also includes a penalty on the logarithmic rates of change of trading strategies. A simulation example shows a significant reduction in transaction cost as compared to a log-optimal portfolio.

A derivation of conventional portfolios and a new linear utility method

Four known portfolios are derived using a new control theory approach. These are the mean-variance, the HARA utility, the log-optimal and the exponential utility portfolios. A single HJB equation is derived from the dynamics of the wealth logarithm. The value functions for different portfolios appear as solutions to this single equation, and thus unifying the derivation of the mean-variance and the other three portfolios. A new method for portfolio selection is proposed that uses the linear utility and penalizes the fractions of wealth allocated across the risky assets. Conventional portfolios appear as examples to this more general method

Robust Stabilization and RobustH1 Control of Uncertain Linear Stochastic Systems with Markovian Switching

This paper deals with the problems of robust stabilization and robustH1 control of uncertain linear stochastic systems with multiplicative Brownian noise and Markovian switching. The system coefficients have norm-bounded uncertainties with Markovian switching. In addition, the mode transition rate matrix is assumed to have an interval uncertainty. We give sufficient conditions for the solvability of these two problems in terms of linear matrix inequalities (LMIs).

RISK-SENSITIVE CONTROL FOR A CLASS OF NONLINEAR SQUARE-ROOT PROCESSES

In this paper, we consider the risk-sensitive control problem for a class of nonlinear systems. The nonlinearity consists of quadratic and square-root terms in the state. By using the completion of squares method, the solution to such an optimal control problem is obtained in an explicit closed-form.

Generalised Risk-sensitive Control in Infinite Horizon

In this paper, we consider an infinite horizon version of the generalised risksensitive control problem. Two different infinite horizon criteria are considered. In each case, the solutions are found in an explicit closed-form. The change of measure and the completion of squares methods are used for this purpose.

Controllability and controller-observer design for a class of linear time-varying systems

In this paper a class of linear time-varying control systems is considered. The time variation consists of a scalar time-varying coefficient multiplying the state matrix of an otherwise time-invariant system. Under very weak assumptions of this coefficient, we show that the controllability can be assessed by an algebraic rank condition, Kalman canonical decomposition is possible, and we give a method for designing a linear state-feedback controller and Luenberger observer.