Optimal stock liquidation in a regime switching model with finite time horizon

This paper is concerned with a finite-horizon optimal selling rule. A set of geometric Brownian motions coupled by a finite-state Markov chain is used to characterize stock price movements. Given a fixed transaction fee, the optimal selling rule can be obtained by solving an optimal stopping problem. The corresponding value function is shown to be the unique viscosity solution to the associated HJB equations. Numerical solutions to these equations and their convergence are obtained. A numerical example is presented to illustrate the results.     

Smooth solutions to portfolio liquidation problems under price-sensitive market impact

We consider the stochastic control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price-dependent impact functions describing the trading costs in the primary market and price-dependent adverse selection costs associated with dark pool trading. We prove that the value function can be characterized in terms of the unique smooth solution to a PDE with singular terminal value, establish its explicit asymptotic behavior at the terminal time, and give the optimal trading strategy in feedback form.     

Substochastic semigroups and densities of piecewise deterministic Markov processes

Necessary and sufficient conditions are given for a substochastic semigroup on L¹ obtained through the Kato-Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise deterministic Markov process, provide a probabilistic interpretation of our results, and apply them to fragmentation equations.     

Some Remark on Optimal Stochastic Control with Partial Information

Abstract

We are interested in the control problem of a partially observable diffusion process, which is initialized at a fixed point of ? ⁿ , and we want to characterize the associated value function. To resort to the theory of viscosity solutions depends on the possibility to translate such a problem on Hilbert spaces like L ²(? ⁿ ), and so it can not be used here. Nevertheless, a result of N. Bouleau and F. Hirsch allows us to introduce a broadened problem which fulfills the condition. The fact remains to link these two control problems.     

Optimal impulsive control of piecewise deterministic Markov processes

In this paper, we study the infinite-horizon expected discounted continuous-time optimal control problem for Piecewise Deterministic Markov Processes with both impulsive and gradual (also called continuous) controls. The set of admissible control strategies is supposed to be formed by policies possibly randomized and depending on the past-history of the process. We assume that the gradual control acts on the jump intensity and on the transition measure, but not on the flow. The so-called Hamilton–Jacobi–Bellman (HJB) equation associated to this optimization problem is analyzed. We provide sufficient conditions for the existence of a solution to the HJB equation and show that the solution is in fact unique and coincides with the value function of the control problem. Moreover, the existence of an optimal control strategy is proven having the property to be stationary and non-randomized.     

Large deviations of Markov chains with multiple time-scales

For Markov processes evolving on multiple time-scales a combination of large component scalings and averaging of rapid fluctuations can lead to useful limits for model approximation. A general approach to proving a law of large numbers to a deterministic limit and a central limit theorem around it have already been proven in Kang and Kurtz (2013) and Kang et al. (2014). We present here a general approach to proving a large deviation principle in path space for such multi-scale Markov processes. Motivated by models arising in systems biology, we apply these large deviation results to general chemical reaction systems which exhibit multiple time-scales, and provide explicit calculations for several relevant examples.     

Smoothness of certain functions in two kinds of risk models with a barrier dividend strategy

In this paper,we study the smoothness of certain functions in two kinds of risk models with a barrier dividend strategy.Mainly using technique from the piecewise deterministic Markov processes theory,we prove that the function is continuously differentiable in the first risk model.Using the weak infinitesimal generator method of Markov processes,we prove that the function is twice continuously differentiable in the second risk model.Intego-differential equations satisfied by them are derived.     

Optimal risk and dividend control of an insurance company with exponential premium principle and liquidation value

Assume that an insurer can control it’s surplus by paying dividends, purchasing reinsurance and injecting capital. The exponential premium principle is used when pricing insurance contract instead of the expected value principle. Under the objective of maximizing the company’s value, we identify the optimal strategies with liquidation value and transaction costs. The results illustrate that the insurer should buy less reinsurance when the surplus increases, capital injection should be considered if and only if the transaction costs and the liquidation value are relatively low, dividends are paid according to barrier strategy if the dividend rate is unrestricted or threshold strategy if the dividend rate is bounded.     

Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova & Rakhlin,2013; Farmer et?al., 2013; Donier et?al., 2015; Tóth (2016).Mathematically, the Hamilton–Jacobi–Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.     

A BSDE-based approach for the optimal reinsurance problem under partial information

We investigate the optimal reinsurance problem under the criterion of maximizing the expected utility of terminal wealth when the insurance company has restricted information on the loss process. We propose a risk model with claim arrival intensity and claim sizes distribution affected by an unobservable environmental stochastic factor. By filtering techniques (with marked point process observations), we reduce the original problem to an equivalent stochastic control problem under full information. Since the classical Hamilton–Jacobi–Bellman approach does not apply, due to the infinite dimensionality of the filter, we choose an alternative approach based on Backward Stochastic Differential Equations (BSDEs). Precisely, we characterize the value process and the optimal reinsurance strategy in terms of the unique solution to a BSDE driven by a marked point process.     

通胀环境下考虑随机微分效用的最优消费和投资问题研究

本文研究了投资者在通胀环境下基于随机微分效用的最优消费和投资问题．首先对投资机会集进行描述．并用随机微分效用函数刻画了投资者的偏好．其次利用动态规划原理,考虑带通胀的最优消费和投资问题,并建立相应的HJB方程．接下来,根据假设的效用函数,推导出最优消费和投资策略,并分析参数对投资策略的影响．     

Optimal control of stochastic functional differential equations with a bounded memory

This paper treats a finite time horizon optimal control problem in which the controlled state dynamics are governed by a general system of stochastic functional differential equations with a bounded memory. An infinite dimensional Hamilton–Jacobi–Bellman (HJB) equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.     

Optimal stopping problems with discontinous reward: Regularity of the value function and viscosity solutions

In this paper we show that the solution of a stochastic boundary value problem with additive noise and with a completely nonlinear drift is a Markov field if only if the boundary condition is an initial or a final type condition     

The impact of partial information sharing in a two-echelon supply chain

We consider a simple two-echelon supply chain composed of a manufacturer and a retailer in which the demand process of the retailer is an AR(1) where the random component is a function of both sides’ information. We focus on partial information sharing under which each side informs the other of an interval in which the exact value of its own component of demand lies. These various levels of information sharing can reduce the supply chain costs.     

Ruin theory with excess of loss reinsurance and reinstatements

The present paper studies the probability of ruin of an insurer, if excess of loss reinsurance with reinstatements is applied. In the setting of the classical Cramér-Lundberg risk model, piecewise deterministic Markov processes are used to describe the free surplus process in this more general situation. It is shown that the finite-time ruin probability is both the solution of a partial integro-differential equation and the fixed point of a contractive integral operator. We exploit the latter representation to develop and implement a recursive algorithm for numerical approximation of the ruin probability that involves high-dimensional integration. Furthermore we study the behavior of the finite-time ruin probability under various levels of initial surplus and security loadings and compare the efficiency of the numerical algorithm with the computational alternative of stochastic simulation of the risk process.     

On the Stochastic Maximum Principle in Optimal Control of Degenerate Diffusions with Lipschitz Coefficients

We establish a stochastic maximum principle in optimal control of a general class of degenerate diffusion processes with global Lipschitz coefficients, generalizing the existing results on stochastic control of diffusion processes. We use distributional derivatives of the coefficients and the Bouleau Hirsh flow property, in order to define the adjoint process on an extension of the initial probability space. This work is partially supported by MENA Swedish Algerian Research Partnership Program (348-2002-6874) and by French Algerian Cooperation, Accord Programme Tassili, 07 MDU 0705.     

Optimal Consumption in a Growth Model with the CES Production Function

Abstract

This article studies the optimization problem of maximizing the expected discounted present value of lifetime utility of consumption in the framework of one-sector neoclassical growth model with the Constant Elasticity of Substitution (CES) production function. We establish the existence of a classical solution of the Hamilton–Jacobi–Bellman (HJB) equation associated with this problem by the technique of viscosity solutions under the strict concavity of the utility function, and hence derive an optimal consumption from the optimality conditions in the HJB equation.     

Stochastic partial differential equations with unbounded and degenerate coefficients

In this article, using DiPerna-Lions theory (DiPerna and Lions, 1989) [1], we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and uniqueness. Moreover, we also prove the L¹-integrability and a general maximal principle for generalized solutions of SPDEs. As applications, we study nonlinear filtering problem and also obtain the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE.