Optional decompositions under constraints

Summary. Motivated by a hedging problem in mathematical finance, El Karoui and Quenez [7] and Kramkov [14] have developed optional versions of the Doob-Meyer decomposition which hold simultaneously for all equivalent martingale measures. We investigate the general structure of such optional decompositions, both in additive and in multiplicative form, and under constraints corresponding to different classes of equivalent measures. As an application, we extend results of Karatzas and Cvitanić [3] on hedging problems with constrained portfolios. Received: 6 August 1996/In revised form: 5 March 1997     

Risk-Sensitive Dynamic Asset Management

This paper develops a continuous time portfolio optimization model where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors such as dividend yields, a firm's return on equity, interest rates, and unemployment rates. In particular, the factors are Gaussian processes, and the drift coefficients for the securities are affine functions of these factors. We employ methods of risk-sensitive control theory, thereby using an infinite horizon objective that is natural and features the long run expected growth rate, the asymptotic variance, and a single risk-aversion parameter. Even with constraints on the admissible trading strategies, it is shown that the optimal trading strategy has a simple characterization in terms of the factor levels. For particular factor levels, the optimal trading positions can be obtained as the solution of a quadratic program. The optimal objective value, as a function of the risk-aversion parameter, is shown to be the solution of a partial differential equation. A simple asset allocation example, featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity, provides some additional insight about the risk-sensitive criterion in the context of dynamic asset management. Accepted 10 December 1997     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

   Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

A Barrier Option of American Type

We obtain closed-form expressions for the prices and optimal hedging strategies of American put-options in the presence of an ``up-and-out" barrier , both with and without constraints on the short-selling of stock. The constrained case leads to a stochastic optimization problem of mixed optimal stopping/singular control type. This is reduced to a variational inequality which is then solved explicitly in two qualitatively separate cases, according to a certain compatibility condition among the market coefficients and the constraint. Accepted 18 May 2000. Online publication 13 November 2000.     

Patterns and structure in systems governed by linear second-order differential equations

This article represents a survey of transmutation ideas and their interaction with typical physical problems. For linear second-order differential operatorsP andQ one deals with canonical connectionsB:PQ (transmutations) satisfyingQB=BP and the related transport of structure between the theories ofP andQ. One can study in an intrinsic manner, e.g., Parseval formulas, eigenfunction expansions, integral transform, special functions, inverse problems, integral equations, and related stochastic filtering and estimation problems, etc. There are applications in virtually any area where such operators arise.