Stochastic Target Problems Research Articles

Martingale representation theorem for the [formula omitted]-expectation

This paper considers the nonlinear theory of G -martingales as introduced by Peng (2007) in [16,17]. A martingale representation theorem for this theory is proved by using the techniques and the results established in Soner et al. (2009) [20] for the second-order stochastic target problems and the second-order backward stochastic differential equations. In particular, this representation provides a hedging strategy in a market with an uncertain volatility.

The Obstacle Version of the Geometric Dynamic Programming Principle: Application to the Pricing of American Options Under Constraints

We provide an obstacle version of the Geometric Dynamic Programming Principle of Soner and Touzi (J. Eur. Math. Soc. 4:201–236, 2002) for stochastic target problems. This opens the doors to a wide range of applications, particularly in risk control in finance and insurance, in which a controlled stochastic process has to be maintained in a given set on a time interval [0,T]. As an example of application, we show how it can be used to provide a viscosity characterization of the super-hedging cost of American options under portfolio constraints, without appealing to the standard dual formulation from mathematical finance. In particular, we allow for a degenerate volatility, a case which does not seem to have been studied so far in this context.

A stochastic target formulation for optimal switching problems in finite horizon

We consider a general optimal switching problem for a controlled diffusion and show that its value coincides with the value of a well-suited stochastic target problem associated to a diffusion with jumps. The proof consists in showing that the Hamilton–Jacobi–Bellman equations of both problems are the same and in proving a comparison principle for this equation. This provides a new family of lower bounds for the optimal switching problem, which can be computed by Monte-Carlo methods. This result has also a nice economical interpretation in terms of a firm's valuation.

The Dynamic Programming Equation for Second Order Stochastic Target Problems

Motivated by applications in mathematical finance [U. Cetin, H. M. Soner, and N. Touzi, “Options hedging for small investors under liquidity costs,” Finance Stoch., to appear] we continue our study of second order backward stochastic equations. In this paper, we derive the dynamic programming equation for a certain class of problems which we call the second order stochastic target problems. In contrast with previous formulations of similar problems, we restrict control processes to be continuous. This new framework enables us to apply our results to a larger class of models. Also the resulting derivation is more transparent. The main technical tool is the geometric dynamic programming principle in this context, and it is proved by using the framework developed in [H. M. Soner and N. Touzi, J. Eur. Math. Soc. (JEMS), 8 (2002), pp. 201–236].

Stochastic Target Problems with Controlled Loss

We consider the problem of finding the minimal initial data of a controlled process which guarantees to reach a controlled target with a given probability of success or, more generally, with a given level of expected loss. By suitably increasing the state space and the controls, we show that this problem can be converted into a stochastic target problem, i.e., finding the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one. Unlike in the existing literature on stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls, together with the appropriate boundary conditions. These results are applied to the problem of quantile hedging in financial mathematics and are shown to recover the explicit solution of Follmer and Leukert [Finance Stoch., 3 (1999), pp. 251-273].

A general stochastic target problem with jump diffusion and an application to a hedging problem for large investors

Let $Z(t,z)$ be a $\mathbb{R}^d$-valued controlled jump diffusion starting from the point $z$ at time $t$. The aim of this paper is to characterize the set $V(t)$ of initial conditions $z$ such that $Z(t,z)$ can be driven into a given target at a given time. We do this by proving that the characteristic function of the complement $V(t)$ satisfies some partial differential equation in the viscosity sense. As an application, we study the problem of hedging in a financial market with a large investor.

A stochastic representation for mean curvature type geometric flows

A smooth solution {Γ(t)}t∈[0,T]⊂\Rd of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem. In this control problem the controller tries to steer the state process into a given deterministic set \Tc with probability one. The reachability set, V(t), for the target problem is the set of all initial data x from which the state process \xx(t)∈\Tc for some control process ν. This representation is proved by studying the squared distance function to Γ(t). For the codimension k mean curvature flow, the state process is dX(t)=2√PdW(t), where W(t) is a d-dimensional Brownian motion, and the control P is any projection matrix onto a (d−k)-dimensional plane. Smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given.

Dynamic programming for stochastic target problems and geometric flows

Given a controlled stochastic process, the reachability set is the collection of all initial data from which the state process can be driven into a target set at a specified time. Differential properties of these sets are studied by the dynamic programming principle which is proved by the Jankov-von Neumann measurable selection theorem. This principle implies that the reachability sets satisfy a geometric partial differential equation, which is the analogue of the Hamilton-Jacobi-Bellman equation for this problem. By appropriately choosing the controlled process, this connection provides a stochastic representation for mean curvature type geometric flows. Another application is the super-replication problem in financial mathematics. Several applications in this direction are also discussed.

A STOCHASTIC REPRESENTATION FOR THE LEVEL SET EQUATIONS

ABSTRACT A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data.

Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions

In this paper, we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward SDEs and forward-backward SDEs. The controlled process $(X^\nu,Y^\nu)$ takes values in ${\mathbb R}^d \times {\mathbb R}$ and a given initial data for $X^{\nu}(0)$. Then the control problem is to find the minimal initial data for $Y^{\nu}$ so that it reaches a stochastic target at a specified terminal time T. The main application is from financial mathematics, in which the process $X^{\nu}$ is related to stock price, $Y^{\nu}$ is the wealth process, and $\nu$ is the portfolio. We introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation. The boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense. This provides a unique characterization of the value function which is the minimal initial data for $Y^{\nu}$.