Optimal stopping with discount and observation costs

In some recent work it has been shown how to solve optimal stopping problems approximatively for independent sequences and also for some dependent sequences, when the associated embedded point processes converge to a Poisson process. In this paper we extend these results to the case where the limit is a Poisson cluster process with random or with deterministic cluster. We develop a new method of directly proving convergence of optimal stopping times, stopping curves, and values and to identify the limiting stopping curve by a unique solution of some first order differential equation. In the random cluster case one has to combine the optimal stopping curve of the underlying hidden Poisson process with a statistical prediction procedure for the maximal point in the cluster. We study in detail some finite and infinite moving average processes.

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.

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 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: .

Independent exponential capture times are assumed for each of N prey. The payoff is the number of prey caught less a linear time cost. The optimal stopping time and value are obtained. When N is known an adaptive statistical procedure is proposed for which the expected payoff differs from the value by at most one-half plus a term which tends to zero (of order N –α, a < 1) as N → ∞.

This note concerns a problem of optimally stopping a nondegenerate, two-dimensional, geometric Brownian motion Q t = (x t ,y t ), with the goal of maximizing where the supremum is taken over the class of all stopping times 𝒳 Q , with finite expectation, H:ℝ+2 → ℝ is a measurable function satisfying a certain growth condition, and c > 0 is a positive constant. It is proved that, under certain conditions, the maximal value Φ(.,.) is a logarithmic function, and the optimal stopping time τ* < ∞ admits the form where ψ( · ) ∈ C 2(0,∞), positive non-decreasing solution of a certain second-order nonlinear ordinary differential equation. The present result extends and supplements a class of Wald-type optimal stopping problems in Graversen and Peskir's paper (1997), which treats the case of a one-dimensional Brownian motion.

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.

Solving optimal stopping problems by backward induction in high dimensions is often very complex since the computation of conditional expectations is required. Typically, such computations are based on regression, a method that suffers from the curse of dimensionality. Therefore, the objective of this paper is to establish dimension reduction schemes for large-scale asset price models and to solve related optimal stopping problems (e.g., Bermudan option pricing) in the reduced setting, where regression is feasible. The proposed algorithm is based on an error measure between linear stochastic differential equations. We establish optimality conditions for this error measure with respect to the reduced system coefficients and propose a particular method that satisfies these conditions up to a small deviation. We illustrate the benefit of our approach in several numerical experiments, in which Bermudan option prices are determined.

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.

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.

We consider the optimal stopping problem, based on the compound binomial model with capital injection controlled by optimal dividend strategy. First, the value function V(x) is established. Second, via the Bellman equation satisfied by the value function V(x) which maximizes the discounted value of define between dividend payment and the penalized discounted capital injection, we find the optimal stopping time τ*. Furthermore an optimal stopping model with capital injection in order to maximize the shareholders’ interests is obtained.

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.

On the Robust Optimal Stopping Problem

We study a robust optimal stopping problem with respect to a set ${\cal P}$ of mutually singular probabilities. This can be interpreted as a zero-sum controller-stopper game in which the stopper is trying to maximize its payoff while an adverse player wants to minimize this payoff by choosing an evaluation criteria from ${\cal P}$. We show that the upper Snell envelope $\overline{Z}$ of the reward process $Y$ is a supermartingale with respect to an appropriately defined nonlinear expectation $\underline{\mathscr{E}}$, and $\overline{Z}$ is further an $\underline{\mathscr{E}}$-martingale up to the first time $\tau^*$ when $\overline{Z}$ meets $Y$. Consequently, $\tau^*$ is the optimal stopping time for the robust optimal stopping problem and the corresponding zero-sum game has a value. Although the result seems similar to the one obtained in the classical optimal stopping theory, the mutual singularity of probabilities and the game aspect of the problem give rise to major technical hurdles, which we circum...

This work generalizes the connection between the optimal singular stochastic control problem and the optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the càdlàg space. Such a problem was left open in Haussmann and Suo [SIAM J. Control Optim., 33 (1995), pp. 916–936] and [SIAM J. Control Optim., 33 (1995), pp. 937–959]). In addition, our compactification method can remove the convexity condition on the coefficients used in Dufour and Miller [SIAM J. Control Optim., 43 (2004), pp. 708–730].