Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.     

A Note on Stochastic Burgers’ System of Equations

Abstract

We consider a stochastic version of a system of two equations formulated by Burgers [Burgers, J.M. Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Verh. Kon. Nerderl. Akad. Weten-Schappen Amsterdam, Afdeel Natuurkunde, 1939, 17 (2), 1–53] with the aim to describe the laminar and turbulent motions of a fluid in a channel. The existence and uniqueness theorem for a global solution is established. The paper generalizes the result from the paper by Da Prato and Ga¸tarek [Da Prato, G.; Ga¸tarek, D. Stochastic Burgers equation with correlated noise. Stochastics Stochastics Rep. 1995, 52, 29–41] dealing with the equation describing only the turbulent motion.     

Almost sure exponential stability for stochastic neutral partial functional differential equations

In this paper, we initiate a study on stochastic neutral partial functional differential equations in a real separable Hilbert space. Our goal here is to study the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths. The results obtained here generalize the main results from [Taniguchi, Stochastics and Stochastics Reports, 53, (1995) 41–52], [Taniguchi, Stochastic Analysis and Applications, 16, (1998) 965–975] and [Liu and Truman, Statistics Probability Letters, 50, (2000) 273–278]. An example is given to illustrate the theory.     

Single source single-commodity stochastic network design

Stochastics affects the optimal design of a network. This paper examines the single-source single-commodity stochastic network design problem. We characterize the optimal designs under demand uncertainty and compare with the deterministic counterparts to outline the basic structural differences. We do this partly as a basis for developing better algorithms than are available today, partly to simply understand what constitutes robust network designs.     

Optimal Market Making in the Foreign Exchange Market

Abstract

This paper is concerned with optimal market making in the foreign exchange market. The market maker's holdings in the different currencies are modelled as stochastic processes that are influenced by both the stochastic exchange rates and the stochastic customer buy and sell orders. The market maker can control their own bid and ask price quotes and, additionally, can buy and sell at other market participants' quotes. The resulting stochastic control problem consists of a controlled diffusion problem for the optimal quotes and a singular control problem for optimal trades at other market participants' quotes. A Markov chain approximation is used to derive optimal strategies.     

Some Solvable Stochastic Control Problems With Delay

We consider optimal harvesting of systems described by stochastic differential equations with delay. We focus on those situations where the value function of the harvesting problem depends on the initial path of the process in a simple way, namely through its value at 0 and through some weighted averages

A verification theorem of variational inequality type is proved. This is applied to solve explicitly some classes of optimal harvesting delay problems     

Optimal Harvesting for a Logistic Population Dynamics Driven by a Lévy Process

The optimal harvesting problem for a stochastic logistic jump-diffusion process is studied in this paper. Two kinds of environmental noises are considered in the model. One is called white noise which is described by a standard Brownian motion, and the other is called jumping noise which is described by a Lévy process. For three types of yield functions (time averaging yield, expected yield and sustainable yield), the optimal harvesting efforts, the corresponding maximum yields and the steady states of population mean under optimal harvesting strategy are respectively given. A new equivalent relationship among these three different objective functions is showed by the ergodic method. This method provides a new approach to the optimal harvesting problem. Results in this paper show that environmental noises have important effect on the optimal harvesting problem.     

Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process

In this paper, the stochastic harvesting problem is regarded as a mathematical formulation of finding the maximum sustained yield and the corresponding best sustainable harvesting strategies under uncertainty. We use a new method to solve this problem, and prove the equivalency between this method and previous methods. This paper is the first attempt to apply the ergodic theory on the optimal harvesting problem, to the best of our knowledge.     

Stochastic Control for Mean-Field Stochastic Partial Differential Equations with Jumps

We study optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in case of partial information control. One important novelty of our problem is represented by the introduction of general mean-field operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove the existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We apply our results to find the explicit optimal control for an optimal harvesting problem.     

Necessary conditions for optimality in relaxed stochastic control problems

In this paper, we are concerned with optimal control problems where the system is driven by a stochastic differential equation of the Ito type. We study the relaxed model for which an optimal solution exists. This is an extension of the initial control problem, where admissible controls are measure valued processes. Using Ekeland's variational principle and some stability properties of the corresponding state equation and adjoint processes, we establish necessary conditions for optimality satisfied by an optimal relaxed control. This is the first version of the stochastic maximum principle that covers relaxed controls.     

Optimal Harvesting for a Stochastic Predator-prey Model with S-type Distributed Time Delays

In this paper, the optimization problem of harvesting for a stochastic predator-prey model with S-type distributed time delays (which contain both discrete time delays and continuously distributed time delays) is studied by using ergodic method. Sufficient and necessary conditions for the existence of optimal harvesting strategy are obtained. Moreover, the optimal harvesting effort (OHE, for short) and the maximum of expectation of sustainable yield (MESY, for short) are given. Some numerical simulations are introduced to illustrate our main results.     

Computing a Stationary Base-Stock Policy for a Finite Horizon Stochastic Inventory Problem with Non-linear Shortage Costs

Abstract

We consider a periodic-review stochastic inventory problem in which demands for a single product in each of a finite number of periods are independent and identically distributed random variables. We analyze the case where shortages (stockouts) are penalized via fixed and proportional costs simultaneously. For this problem, due to the finiteness of the planning horizon and non-linearity of the shortage costs, computing the optimal inventory policy requires a substantial effort as noted in the previous literature. Hence, our paper is aimed at reducing this computational burden. As a resolution, we propose to compute “the best stationary policy.” To this end, we restrict our attention to the class of stationary base-stock policies, and show that the multi-period, stochastic, dynamic problem at hand can be reduced to a deterministic, static equivalent. Using this important result, we introduce a model for computing an optimal stationary base-stock policy for the finite horizon problem under consideration. Fundamental analytic conclusions, some numerical examples, and related research findings are also discussed.     

The Valuation of American Options with Stochastic Stopping Time Constraints

Abstract

This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff–Schwartz algorithm to solve the stochastic Cauchy–Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.     

Quantitative Stability Analysis of Two-Stage Stochastic Linear Programs with Full Random Recourse

Abstract

In this paper, we apply the parametric linear programing technique and pseudo metrics to study the quantitative stability of the two-stage stochastic linear programing problem with full random recourse. Under the simultaneous perturbation of the cost vector, coefficient matrix, and right-hand side vector, we first establish the locally Lipschitz continuity of the optimal value function and the boundedness of optimal solutions of parametric linear programs. On the basis of these results, we deduce the locally Lipschitz continuity and the upper bound estimation of the objective function of the two-stage stochastic linear programing problem with full random recourse. Then by adopting different pseudo metrics, we obtain the quantitative stability results of two-stage stochastic linear programs with full random recourse which improve the current results under the partial randomness in the second stage problem. Finally, we apply these stability results to the empirical approximation of the two-stage stochastic programing model, and the rate of convergence is presented.     

A Stochastic Linear Quadratic Optimal Control Problem with Generalized Expectation

Abstract

In this article, we initiate a study on optimal control problem for linear stochastic differential equations with quadratic cost functionals under generalized expectation via backward stochastic differential equations.     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

     

Quantitative stability of full random two-stage problems with quadratic recourse

ABSTRACT

In this paper, we discuss quantitative stability of two-stage stochastic programs with quadratic recourse where all parameters in the second-stage problem are random. By establishing the Lipschitz continuity of the feasible set mapping of the restricted Wolfe dual of the second-stage quadratic programming in terms of the Hausdorff distance, we prove the local Lipschitz continuity of the integrand of the objective function of the two-stage stochastic programming problem and then establish quantitative stability results of the optimal values and the optimal solution sets when the underlying probability distribution varies under the Fortet–Mourier metric. Finally, the obtained results are applied to study the asymptotic behaviour of the empirical approximation of the model.     

The optimal harvesting problem under price uncertainty

In this paper we study the exploitation of a one species forest plantation when timber price is governed by a stochastic process. The work focuses on providing closed expressions for the optimal harvesting policy in terms of the parameters of the price process and the discount factor, with finite and infinite time horizon. We assume that harvest is restricted to mature trees older than a certain age and that growth and natural mortality after maturity are neglected. We use stochastic dynamic programming techniques to characterize the optimal policy and we model price using a geometric Brownian motion and an Ornstein–Uhlenbeck process. In the first case we completely characterize the optimal policy for all possible choices of the parameters. In the second case we provide sufficient conditions, based on explicit expressions for reservation prices, assuring that harvesting everything available is optimal. In addition, for the Ornstein–Uhlenbeck case we propose a policy based on a reservation price that performs well in numerical simulations. In both cases we solve the problem for every initial condition and the best policy is obtained endogenously, that is, without imposing any ad hoc restrictions such as maximum sustained yield or convergence to a predefined final state.     

Conditioning of convex piecewise linear stochastic programs

 In this paper we consider stochastic programming problems where the objective function is given as an expected value of a convex piecewise linear random function. With an optimal solution of such a problem we associate a condition number which characterizes well or ill conditioning of the problem. Using theory of Large Deviations we show that the sample size needed to calculate the optimal solution of such problem with a given probability is approximately proportional to the condition number. Received: May 2000 / Accepted: May 2002-07-16 Published online: September 5, 2002 RID="★" The research of this author was supported, in part, by grant DMS-0073770 from the National Science Foundation Key Words. stochastic programming – Monte Carlo simulation – large deviations theory – ill-conditioned problems