Ulrich G. Haussmann

Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain

Abstract.We consider a multi-stock market model where prices satisfy a stochastic differential equation with instantaneous rates of return modeled as a continuous time Markov chain with finitely many states. Partial observation means that only the prices are observable. For the investor’s objective of maximizing the expected utility of the terminal wealth we derive an explicit representation of the optimal trading strategy in terms of the unnormalized filter of the drift process, using HMM filtering results and Malliavin calculus. The optimal strategy can be determined numerically and parameters can be estimated using the EM algorithm. The results are applied to historical prices.

The stochastic maximum principle for a singular control problem

We consider a stochastic control problem in which the control has two components: the first being absolutely continuous, and the second singular. We assume linear dynamics, convex cost criterion and convex state constraint, and we allow the coefficients of the system to be random and the absolutely continuous component of the control to enter both the drift and diffusion coefficients. We do not impose any Lp -bounds on the control. We obtain for this model a Stochastic Maximum Principle in integral form. This is the first version of the Stochastic Maximum Principle that covers the Stochastic Singular Control Problem. When we assume, as in other versions of the Stochastic Maximum Principle, that the admissible controls are square-integrable, we obtain not only a necessary but also a sufficient condition for optimality. The mathematical tools are those of Stochastic Calculus and Convex Analysis.

Singular Optimal Stochastic Controls I: Existence

We apply the compactification method to study the control problem where the state is governed by an Ito stochastic differential equation allowing both classical and singular control. The problem is reformulated as a martingale problem on an appropriate canonical space after the relaxed form of the classical control is introduced. Under some mild continuity hypotheses on the data, it is shown by purely probabilistic arguments that an optimal control for the problem exists. The value function is shown to be Borel measurable.

Singular Optimal Stochastic Controls II: Dynamic Programming

The dynamic programming principle for a multidimensional singular stochastic control problem is established in this paper. When assuming Lipschitz continuity on the data, it is shown that the value function is continuous and is the unique viscosity solution of the corresponding Hamilton--Jacobi--Bellman equation.

The maximum principle for optimal control of diffusions with partial information

We derive necessary conditions for the optimal control of a system that satisfies an Ito equation with control entering the drift term. The control is a function of a noise corrupted observation of the state, and the cost is the expectation of an integral and a final term. The robust form of the Zakai equation from nonlinear filtering is used to compute the variation of the cost due to a strong or “needle” variation of a control. This gives rise to an explicit formula for the adjoint process, much as in the case with complete information.

Stochastic variational inequalities of parabolic type

Existence and uniqueness of strong solutions of stochastic partial differential equations of parabolic type with reflection (e.g., the solutions are never allowed to be negative) is proved. The problem is formulated as a stochastic variational inequality and then compactness is used to derive the result, but the method requires the space dimension to be one.

Stochastic adaptive control with small observation noise

Controlled diffusions depending on an unknown parameter and with small system perturbation are considered in this paper. Two parameter identification methods are proposed and error probabilities are estimated in terms of the small perturbation parameter. These methods are then used to choose among competing filters on successive time intervals. Asymptotically optimal controls based on partial observations are found on successive time intervals by using the best filter identified on the previous time intervals.

On a Stochastic, Irreversible Investment Problem

The productive sector of the economy, represented by a single firm employing labor to produce the consumption good, is studied in a stochastic continuous time model on a finite time interval. The firm must choose the optimal level of employment and capital investment in order to maximize its expected total profits. In this stochastic control problem the firms capacity is modeled as an Ito process controlled by a monotone process, possibly singular, that represents the cumulative real investment. It is optimal to invest when the shadow value of installed capital exceeds the capitals replacement cost; this threshold is the free boundary of a related optimal stopping problem which we recast as a stopping problem without integral cost, similar to the American option problem. Then, under a regularity condition, we characterize the free boundary as the unique solution of a nonlinear integral equation.

Optimal Control of Inflation: A Central Bank Problem

This paper models the action of the central bank on the dynamics of the nominal interest rate with the aim of controlling inflation. The problem is set up as a two-dimensional bounded variation control problem; it is shown that its variational formulation leads to a stochastic differential game with stopping times between the conservative and the expansionist tendencies of the bank.

The Free Boundary of the Monotone Follower

This paper identifies the free boundary arising in the two- dimensional monotone follower, cheap control problem. It proves that if a region of inaction