Optimal expulsion and optimal confinement of a Brownian particle with a switching cost

We solve two stochastic control problems in which a player tries to minimize or maximize the exit time from an interval of a Brownian particle, by controlling its drift. The player can change from one drift to another but is subject to a switching cost. In each problem, the value function is written as the solution of a free boundary problem involving second order ordinary differential equations, in which the unknown boundaries are found by applying the principle of smooth fit. For both problems, we compute the value function, we exhibit the optimal strategy and we prove its generic uniqueness.     

Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which is determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with a finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration.     

A Fokker–Planck control framework for multidimensional stochastic processes

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang–Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka–Volterra model and a noised limit cycle model.     

Sequential tracking of a hidden Markov chain using point process observations

We study finite horizon optimal switching problems for hidden Markov chain models with point process observations. The controller possesses a finite range of strategies and attempts to track the state of the unobserved state variable using Bayesian updates over the discrete observations. Such a model has applications in economic policy making, staffing under variable demand levels and generalized Poisson disorder problems. We show regularity of the value function and explicitly characterize an optimal strategy. We also provide an efficient numerical scheme and illustrate our results with several computational examples.     

Explicit characterization of the super-replication strategy in financial markets with partial transaction costs

We consider a continuous time multivariate financial market with proportional transaction costs and study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. The model is similar to the one considered in Bouchard and Touzi [B. Bouchard, N. Touzi, Explicit solution of the multivariate super-replication problem under transaction costs, The Annals of Applied Probability 10 (3) (2000) 685–708] except that some of the assets can be exchanged freely, i.e. without paying transaction costs. In this context, we generalize the result of the above paper and prove that the super-replication price is given by the cost of the cheapest hedging strategy in which the number of non-freely exchangeable assets is kept constant over time. Our proof relies on the introduction of a new auxiliary control problem whose value function can be interpreted as the super-hedging price in a model with unbounded stochastic volatility (in the directions where transaction costs are non-zero). In particular, it confirms the usual intuition that transaction costs play a similar role to stochastic volatility.     

Representation of asymptotic values for nonexpansive stochastic control systems

In ergodic stochastic problems the limit of the value function $V_{\lambda }$ of the associated discounted cost functional with infinite time horizon is studied, when the discounted factor $\lambda$ tends to zero. These problems have been well studied in the literature and the used assumptions guarantee that the value function $\lambda V_{\lambda }$ converges uniformly to a constant as $\lambda \to 0$. The objective of this work consists in studying these problems under the assumption, namely, the nonexpansivity assumption, under which the limit function is not necessarily constant. Our discussion goes beyond the case of the stochastic control problem with infinite time horizon and discusses also $V_{\lambda }$ given by a Hamilton–Jacobi–Bellman equation of second order which is not necessarily associated with a stochastic control problem. On the other hand, the stochastic control case generalizes considerably earlier works by considering cost functionals defined through a backward stochastic differential equation with infinite time horizon and we give an explicit representation formula for the limit of $\lambda V_{\lambda }$, as $\lambda \to 0$.     

带补偿的随机二次规划的近似Lagrange-Newton方法

In this paper, two-stage stochastic quadratic programming problems with equality constraints are considered. By Monte Carlo simulation-based approximations of the objective function and its first (second)derivative,an inexact Lagrange-Newton type method is proposed.It is showed that this method is globally convergent with probability one. In particular, the convergence is local superlinear under an integral approximation error bound condition.Moreover, this method can be easily extended to solve stochastic quadratic programming problems with inequality constraints.     

Forward–backward SDEs with random terminal time and applications to pricing special European-type options for a large investor

In this paper, we first discuss the solvability of coupled forward–backward stochastic differential equations (FBSDEs, for short) with random terminal time. We prove the existence and uniqueness of adapted solution to such FBSDEs under some natural assumptions. The method of proof adopted is to construct a contraction mapping related to the solutions of a sequence of backward SDEs. Our monotonicity-type assumptions are different from those in Hu and Peng (1995) [4], Peng and Shi (2000) [11], and so on. As a corollary of our main result, the solvability of FBSDEs with a finite time horizon is discussed. Finally, the existence and uniqueness theorem of the solution to FBSDEs with a finite time horizon is applied to price special European-type options for a large investor.     

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. Received October 24, 2000 / final version received July 24, 2001?Published online November 27, 2001     

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.     

A stochastic partially reversible investment problem on a finite time-horizon: Free-boundary analysis

We study a continuous-time, finite horizon, stochastic partially reversible investment problem for a firm producing a single good in a market with frictions. The production capacity is modeled as a one-dimensional, time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment–disinvestment strategy. We associate to the investment–disinvestment problem a zero-sum optimal stopping game and characterize its value function through a free-boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment–disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.     

Criticality of viscous Hamilton–Jacobi equations and stochastic ergodic control

This paper deals with nonlinear additive eigenvalue problems for viscous Hamilton–Jacobi equations which appear in stochastic ergodic control. Certain qualitative properties of principal eigenvalues and associated eigenfunctions are studied. Such analysis plays a key role in studying the recurrence and transience of feedback diffusions for the corresponding stochastic control problems. Our results can be regarded as a nonlinear extension of the criticality theory for Schrödinger operators with decaying potentials.     

Mirror and synchronous couplings of geometric Brownian motions

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.     

On the link between infinite horizon control and quasi-stationary distributions

We study infinite horizon control of continuous-time non-linear branching processes with almost sure extinction for general (positive or negative) discount. Our main goal is to study the link between infinite horizon control of these processes and an optimization problem involving their quasi-stationary distributions and the corresponding extinction rates. More precisely, we obtain an equivalent of the value function when the discount parameter is close to the threshold where the value function becomes infinite, and we characterize the optimal Markov control in this limit. To achieve this, we present a new proof of the dynamic programming principle based upon a pseudo-Markov property for controlled jump processes. We also prove the convergence to a unique quasi-stationary distribution of non-linear branching processes controlled by a Markov control conditioned on non-extinction.     

Fractional-order regularization and wavelet approximation to the inverse estimation problem for random fields

The least-squares linear inverse estimation problem for random fields is studied in a fractional generalized framework. First, the second-order regularity properties of the random fields involved in this problem are analysed in terms of the fractional Sobolev norms. Second, the incorporation of prior information in the form of a fractional stochastic model, with covariance operator bicontinuous with respect to a certain fractional Sobolev norm, leads to a regularization of this problem. Third, a multiresolution approximation to the class of linear inverse problems considered is obtained from a wavelet-based orthogonal expansion of the input and output random models. The least-squares linear estimate of the input random field is then computed using these orthogonal wavelet decompositions. The results are applied to solving two important cases of linear inverse problems defined in terms of fractional integral operators.     

Deterministic and Stochastic Control of Navier—Stokes Equation with Linear, Monotone, and Hyperviscosities

This paper deals with the optimal control of space—time statistical behavior of turbulent fields. We provide a unified treatment of optimal control problems for the deterministic and stochastic Navier—Stokes equation with linear and nonlinear constitutive relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case with monotone viscosity we use the Minty—Browder technique to prove the existence of optimal controls. For the stochastic case with monotone viscosity, we combine the Minty—Browder technique with the martingale problem formulation of Stroock and Varadhan to establish existence of optimal controls. The deterministic models given in this paper also cover some simple eddy viscosity type turbulence closure models. Accepted 7 June 1999     

Generalized stochastic evolution equations

An approach to generalized stochastic evolution equations is presented which is based on a generalized Ito formula. This allows the consideration of interesting examples which are stochastic generalizations of evolution equations of mixed type or second order in time hyperbolic equations. It includes more standard material involving a Gelfand triple of spaces as a special case. Several examples are given which illustrate the use of the abstract theory presented.     

Stochastic Linear Quadratic Optimal Control Problems

This paper is concerned with the stochastic linear quadratic optimal control problem (LQ problem, for short) for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. Some intrinsic relations among the LQ problem, the stochastic maximum principle, and the (linear) forward—backward stochastic differential equations are established. Some results involving Riccati equation are discussed as well. Accepted 15 May 2000. Online publication 1 December 2000     

General existence results for reflected BSDE and BSDE

In this paper, we are concerned with the problem of existence of solutions for generalized reflected backward stochastic differential equations (GRBSDEs for short) and generalized backward stochastic differential equations (GBSDEs for short) when the generator is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. We deal with the case of a bounded terminal condition ξ and a bounded barrier L as well as the case of unbounded ones. This is done by using the notion of generalized BSDEs with two reflecting barriers studied in Essaky and Hassani (submitted for publication) [14]. The work is suggested by the interest the results might have in finance, control and game theory.     

A singular control model with application to the goodwill problem

We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Itô diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The control problem that we address aims at maximising a performance criterion that rewards high values of the utility derived from the system’s controlled state but penalises any expenditure of control effort. This problem has been motivated by applications such as the so-called goodwill problem in which the system’s state is used to represent the image that a product has in a market, while control expenditure is associated with raising the product’s image, e.g., through advertising. We obtain the solution to the optimisation problem that we consider in a closed analytic form under rather general assumptions. Also, our analysis establishes a number of results that are concerned with analytic as well as probabilistic expressions for the first derivative of the solution to a second-order linear non-homogeneous ordinary differential equation. These results have independent interest and can potentially be of use to the solution of other one-dimensional stochastic control problems.