Michael Kohlmann

Relationship Between Backward Stochastic Differential Equations and Stochastic Controls: A Linear-Quadratic Approach

It is well known that backward stochastic differential equations (BSDEs) stem from the study on the Pontryagin type maximum principle for optimal stochastic controls. A solution of a BSDE hits a given terminal value (which is a random variable) by virtue of an it additional martingale term and an indefinite initial state. This paper attempts to explore the relationship between BSDEs and stochastic controls by interpreting BSDEs as some stochastic optimal control problems. More specifically, associated with a BSDE, a new stochastic control problem is introduced with the same dynamics but a definite given initial state. The martingale term in the original BSDE is regarded as the control, and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the BSDE. This problem is solved in a closed form by the stochastic linear-quadratic (LQ) theory developed recently. The general result is then applied to the Black--Scholes model, where an optimal mean-variance hedging portfolio is obtained explicitly in terms of the option price. Finally, a modified model is investigated, where the difference between the state and the expectation of the given terminal value at any time is taken into account.

Connections between optimal stopping and singular stochastic control

We consider an optimal control problem for an Ito diffusion and a related stopping problem. Their value functions satisfy (d/dx)V=u and an optimal control defines an optimal stopping time. Conversely, we construct an optimal control from optimal stopping times, find a representation of V as an integral of u and describe the optimal state as a reflected process.

Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean–variance hedging☆

Backward stochastic Riccati equations are motivated by the solution of general linear quadratic optimal stochastic control problems with random coefficients, and the solution has been open in the general case. One distinguishing difficult feature is that the drift contains a quadratic term of the second unknown variable. In this paper, we obtain the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation. This solves the one-dimensional case of Bismut-Pengs problem which was initially proposed by Bismut (Lecture Notes in Math. 649 (1978) 180). We use an approximation technique by constructing a sequence of monotone drifts and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean-variance hedging problem with general random market conditions.

The partially observed stochastic minimum principle

Using stochastic flows and the generalized differentiation formula of Bismut and Kunita, the change in cost due to a strong variation of an optimal control is explicitly calculated. Differentiating this expression gives a minimum principle in both the partially observed and stochastic open loop situations. In the latter case the equation satisfied by the adjoint process is obtained by applying a martingale representation result.

Minimization of risk and linear quadratic optimal control theory

This article is concerned with the optimal control problem for the linear stochastic system Xt = x + � t 0 (AsXs + Bsus + fs) ds + � t 0 � d i=1 (Ci(s)Xs + Di(s)us + gi(s)) dwi(s) with the convex risk functional J(u )= EM(XT )+ ET 0 G(t, Xt ,u t) dt. In order to guarantee the existence of an optimal control without any(weak) compactness assumption on the admissible control set, we assume that the risk function M is coercive and thatd=1 D ∗ Di is uniformlypositive, rather than to assume like in the control literature that the running risk function G is coercive with respect to the control variable. In this new setting, the running risk function G maybe independent of the control variable, and therefore the so-called singular linear-quadratic (LQ) stochastic control problem is included. A rigorous theoryis developed for the general stochastic LQ problem with random coefficients, and the bounded mean oscillation-martingale theoryis used to account for the concerned integrability. It plays a crucial role in the following exposition: (a) to connect the stochastic LQ problem to two associated backward stochastic differential equations (BSDEs)—one is an n × n symmetric matrix-valued nonlinear Riccati BSDE and the other is an n-dimensional linear BSDE with unbounded coefficients; (b) to show that the latter BSDE has an adapted solution pair of the suitablynecessaryregularity . This seems to be the first application in a stochastic LQ theory of the BMO-martingale theory, which roots in harmonic analysis. Furthermore, with the help of an a priori estimate on the risk functional, existence and uniqueness of the solutions of backward stochastic Riccati differential equations (BSRDEs) in the singular case is reduced to the regular case via a perturbation method, and then a new existence and uniqueness result on BSRDEs is obtained for the singular case.

The second order minimum principle and adjoint process

When the control parameter appears in both the drift and diffusion coefficients, Peng has derived a “second order” minimum principle in terms of an adjoint process . Peng has also characterized this process as the unique, adapted solution of a reverse time stochastic differential equation. Using stochastic flows this paper gives an explicit expression for .

A short proof of a martingale representation result

Using the Ito differentiation rule, the properties of stochastic flows and the unique decomposition of special seminartingales, the integrand in a stochastic integral is quickly identified.

Mean Variance Hedging in a General Jump Model

Abstract We consider the mean-variance hedging of a contingent claim H when the discounted price process S is an -valued quasi-left continuous semimartingale with bounded jumps. We relate the variance-optimal martingale measure (VOMM) to a backward semimartingale equation (BSE) and show that the VOMM is equivalent to the original measure P if and only if the BSE has a solution. For a general contingent claim, we derive an explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by means of another BSE and an appropriate predictable process δ

The variational principle for optimal control of diffusions with partial information

Abstract Strong variations are described for the ϵ-optimal control of a class of control problems for systems described by stochastic diffusion equations. The differentiation process developed identifies the adjoint process.

Change of filtrations and mean–variance hedging

We consider the mean–variance hedging (MVH) problem (under measure P) of two kinds of investors for two different levels of information, described by two filtrations and such that . Under the assumption that there exists a measure such that all -martingales are -martingales, we give the variance-optimal martingale measure (VOMM) with respect to and through a couple of stochastic Riccati equation (SRE)s, which can be viewed as the same SRE with differential terminal value under . Then we derive an explicit form of the optimal mean–variance strategy and the optimal costs with respect to and . We describe the concept of -no-value-to-investment in the means of mean–variance, and for a given contingent claim , we compare their optimal costs with respect to and .