Solving High-Dimensional Optimal Stopping Problems Using Optimization Based Model Order Reduction

We consider a optimal dividend and stopping problem for two-dimensional compound Poisson risk model. The two companies allow to help each other by transffering payments. The goal of the companies is to maximize the expected cumulative discounted dividends payments up to the time of ruin. The problem is formulated as a singular control and optimal stopping problem. We explicitly construct the value function and the optimal strategy when claims are exponentially distributed and present two numerical examples.

In this paper, we consider the optimal stopping problems on semi-Markov processes (sMPs) with finite horizon and aim to establish the existence and algorithm of optimal stopping times. The key method is the equivalence between optimal stopping problems on sMPs and a special class of semi-Markov decision processes (sMDPs). We first introduce the optimality equation and show the existence of the optimal policies of finite-horizon sMDPs with additional terminal costs. Based on the optimal stopping problems on sMPs, we give an explicit construction of sMDPs such that the optimal stopping times of sMPs are equivalent to the optimal policies of the constructed sMDPs. Then, using the results of sMDPs developed here, we not only prove the existence of the optimal stopping times of sMPs, but also provide an algorithm for computing the optimal stopping times of sMPs. Moreover, we show that the optimal and \(\varepsilon \)-optimal stopping time can be characterized by the hitting time of some special sets. Finally, we give an example to illustrate the effectiveness of our conclusions.

High-order approximation of the finite horizon control problem via a tree structure algorithm

Optimal stopping is a fundamental class of stochastic dynamic optimization problems with numerous applications in finance and operations management. In “A nonparametric algorithm for optimal stopping based on robust optimization,” the author introduces an approach based on robust optimization for computing near-optimal solutions for computationally demanding stochastic optimal stopping problems with known probability distributions. Through this new use of robust optimization, the paper develops new algorithms for solving stochastic optimal stopping problems that are practical and can outperform state-of-the-art simulation-based algorithms in the context of pricing high-dimensional financial options.

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

The connection between the optimal stopping problems for inhomogeneous standard Markov process and the corresponding homogeneous Markov process constructed in the extended state space is established. An excessive characterization of the value-function and the limit procedure for its construction in the problem of optimal stopping of an inhomogeneous standard Markov process is given. The form of ε-optimal (optimal) stopping times is also found.

The main aim of this chapter is to present and analyze two methods for the pricing of multi-asset American – in practice Bermudan – options: the regression methods “a la Longstaff-Schwarz” and the quantization methods. The pricing of American options is a typical example of an optimal stopping problem. So, we start from such a general optimal stopping problem but for a diffusion model, we introduce the Snell envelope of a payoff or reward process and its value function in ou Markov framework. We give some rate results for various “levels” of time discretization of such Snell envelope. Then we consider a discrete time optimal stopping problem in a Markov framework. We introduce the backward dynamic programming principle and its variant based on optimal stopping times. We describe in detail the algorithmic aspects of both the regression and the quantization method. The Monte Carlo error (Central Limit Theorem) induced by the regression method is described. Finally, a theoretical analysis of the convergence rate of the quantization method is carried out.

The optimal stopping problem with incomplete data for the Kalman–Bucy scheme is reduced to the optimal stopping problem with complete data. In the case of a continuous gain function it is proven that price convergence has an order $(\varepsilon_1+\varepsilon_2)$ when small perturbation parameters $\varepsilon_1$, $\varepsilon_2$ of an observable process tend to zero.

Residential demand response is one of the key enabling technologies which plays an important role in managing the load demand of prosumers. However, the load scheduling problem becomes quite challenging due to the involvement of dynamic parameters and renewable energy resources. This work has proposed a bi-level load scheduling mechanism with dynamic electricity pricing integrated with renewable energy and storage system to overcome this problem. The first level involves the formulation of load scheduling and optimization problems as optimal stopping problems with the objective of energy consumption and delay cost minimization. This problem involved the real-time electricity pricing signal, customers load scheduling priority, machine learning (ML) based forecasted load demand, and renewable & storage unit profiles, which is solved using mathematical programming with branch-and-cut & branch-and-bound algorithms. Since the first-level optimization problem is formulated as a stopping problem, the optimal time slots are obtained using a one-step lookahead rule to schedule the load with the ability to handle the uncertainties. The second level is used to further model the load scheduling problem through the dynamic electricity pricing signal. The cost minimization objective function is then solved using the genetic algorithm (GA), where the input parameters are obtained from the first-level optimization solution. Furthermore, the impact of load prioritization in terms of time factor and electricity price is also modeled to allow the end-users to control their load. Analytical and simulation results are conducted using solar-home electricity data, Ausgrid, AUS to validate the proposed model. Results show that the proposed model can handle uncertainties involved in the load scheduling process along with a cost-effective solution in terms of cost and discomfort reduction. Furthermore, the bi-level process ensures cost minimization with end-user satisfaction regarding the dynamic electricity price signal.

In sequential analysis and starting with the seminal treatment of the optimality of the SPRT due to Wald and Wolfowitz [Wald, A.; Wolfowitz, J. Bayes solutions of sequential decision problems. Ann. Math. Statist. 1950, 21, 82–99] various problems of optimal stopping with linear costs of observation arise. Here, we shall describe a new method for solving such problems. For a payoff g(X t ) − ct we propose a linear representation of the form where M t is a local martingale, and the function h and the local martingale depend on a parameter λ. In the case of a diffusion process X t we shall show that, for a proper choice of λ, the boundary points of the optimal stopping region can be obtained from those points of the state space where the maxima of h are located. This method is inspired by a method of Beibel and Lerche [Beibel, M.; Lerche, H.R. A new look at optimal stopping problems related to mathematical finance. Statistica Sinica 1997, 7, 93–108; Beibel, M.; Lerche, H.R. Optimal stopping of regular diffusions under random discounting. Theory Probab. Appl. 2001, 45 (4), 547–557] who, for optimal stopping problems with discounted payoff, used a multiplicative representation e −rt g(X t ) = h(X t )M t with a suitable martingale.

In this note we point out that the problem studied in the comment written by Majumdar and Bouchard (2008) on our paper “Thou Shalt Buy and Hold”, Shiryaev et al. (2008), is fundamentally different (and technically much easier) than the one in our original paper, although the difference between the two problems may appear deceivingly little to non-specialists in optimal stopping. That said, we acknowledge that generally speaking the path integral methods being promoted in the comment could indeed be a useful tool in treating some problems in quantitative finance. The problem (4), considered in the comment by Majumdar and Bouchard (2008), is a one-dimensional deterministic optimisation problem where the optimal time τ ∗ to be determined is known to be deterministic a priori. In contrast, the problem studied in the original paper Shiryaev et al. (2008) is an optimal stopping problem where the decision variable τ is a random time. The scale of the difference and difficulty of the latter problem compared with the former is, shall we say, enormous. A deterministic optimisation problem (sometimes called a mathematical programme especially if there are various constraints involved) can be solved by simple calculus, whereas optimal stopping remains an area where what we know is far less than what we do not; see Shiryaev (1978) and Peskir and Shiryaev (2006) for an account of what we know. One should note that determining a random time as in optimal stopping is a necessity in many decision-making problems facing uncertainties, because the optimal timing may need to respond to the We thank the Editors in Chief for inviting us to respond to the comment of S.N. Majumdar and J.-P. Bouchard on our paper “Thou Shalt Buy and Hold”. Shiryaev acknowledges financial support from Grants RFBI 08-01-00740 and 08-01-91205-YaF, while Zhou acknowledges financial support from a start-up fund of the University of Oxford. Steklov Mathematical Institute, Gubkina str.8, 119991, Moscow, Russia. Mathematical Institute, The University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK. Email: . Nomura Centre for Mathematical Finance, and Oxford–Man Institute of Quantitative Finance, The University of Oxford, 24–29 St Giles, Oxford OX1 3LB, UK, and Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Email: .

This paper proposes a new approach to solve finite-horizon optimal stopping problems for a class of Markov processes that includes one-dimensional diffusions, birth–death processes, and jump diffusions and continuous-time Markov chains obtained by time-changing diffusions and birth-and-death processes with Lévy subordinators. When the expectation operator has a purely discrete spectrum in the Hilbert space of square-integrable payoffs, the value function of a discrete optimal stopping problem has an expansion in the eigenfunctions of the expectation operator. The Bellman's dynamic programming for the value function then reduces to an explicit recursion for the expansion coefficients. The value function of the continuous optimal stopping problem is then obtained by extrapolating the value function of the discrete problem to the limit via Richardson extrapolation. To illustrate the method, the paper develops two applications: American-style commodity futures options and Bermudan-style abandonment and capacity expansion options in commodity extraction projects under the subordinate Ornstein-Uhlenbeck model with mean-reverting jumps with the value function given by an expansion in Hermite polynomials.

Finite Horizon Portfolio Selection with Durable Goods

We discuss some of the main ideas underlying neuro-dynamic programming. This methodology has significant potential as a general approach to approximately solving a wide variety of complex stochastic control problems. However, though neuro-dynamic programming algorithms have generated promising results in a number of applications, the algorithms that have been most successful are not well-understood at a theoretical level. As a case study in the development of theory in support of such algorithms, we propose an algorithm for solving optimal stopping problems, and we provide theoretical results concerning convergence and approximation error. Though this algorithm is customized for optimal stopping problems, it retains the key features seen in the most popular neuro-dynamic programming methods, and its analysis might therefore serve as a starting point for the study of methods of broader scope. The applicability of the algorithm is illustrated through a computational case study involving the pricing of a path-dependent financial derivative security that gives rise to an optimal stopping problem with a one-hundred-dimensional state space.