Finite Horizon Optimal Stopping Research Articles

Compound Poisson disorder problem with uniformly distributed disorder time

Suppose that the arrival rate and the jump distribution of a compound Poisson process change suddenly at an unknown and unobservable time. We want to detect the change as quickly as possible to take counteractions, e.g., to assure top quality of products in a production system, or to stop credit card fraud in a banking system. If we have no prior information about future disorder time, then we typically assume that the disorder is equally likely to happen any time – or has uniform distribution – over a long but finite time horizon. We solve this so-called compound Poisson disorder problem for the practically important case of unknown, unobserved, but uniformly distributed disorder time. The solution hinges on the complete separation of information flow from the hard time horizon constraint, by describing the former with an autonomous time-homogeneous one-dimensional Markov process in terms of which the detection problem translates into a finite horizon optimal stopping problem. For any given finite horizon, the solution is two-dimensional. For cases where the horizon is large and one is unwilling to set a fixed value for it, we give a one-dimensional approximation. Also, we discuss an extension where the disorder may not happen on the given interval with a positive probability. In this extended model, if no detection decision is made by the end of the horizon, then a second level hypothesis testing problem is solved to determine the local parameters of the observed process.

Socially Aware Device-to-multi-device User Grouping for Popular Content Distribution

The distribution of popular videos incurs a large amount of traffic at the base stations (BS) of networks. Device-to-multi-device (D2MD) communication has emerged an efficient radio access technology for offloading BS traffic in recent years. However, traditional studies have focused on synchronous user requests whereas asynchronous user requests are more common. Hence, offloading BS traffic in case of asynchronous user requests while considering their time-varying characteristics and the quality of experience (QoE) of video request users (VRUs) is a pressing problem. This paper uses social stability (SS) and video loading duration (VLD)-tolerant property to group VRUs and seed users (SUs) to offload BS traffic. We define the average amount of data transmission (AADT) to measure the network’s capacity for offloading BS traffic. Based on this, we formulate a time-varying bipartite graph matching optimization problem. We decouple the problem into two subproblems which can be solved separately in terms of time and space. Then, we propose the socially aware D2MD user selection (SA-D2MD-S) algorithm based on finite horizon optimal stopping theory, and propose the SA-D2MD user matching (SA-D2MD-M) algorithm to solve the two subproblems. The results of simulations show that our algorithms outperform prevalent algorithms.

Analysis of CDS Spread Fluctuations with an Application to the Negative Basis Arbitrage

In this article, we first construct the empirical measure of spread (EMS) to capture the dynamics of quoted CDS spreads. It is the measure of creditworthiness of a company derived from the asset model in Egami and Kevkhishvili [2017]. We then use the information provided by the EMS to derive a refined negative CDS-bond basis trading strategy. Our contributions are twofold. By analyzing the period encompassing the global financial crisis, we find that the EMS is a forward-looking or leading indicator and can explain the movement of near future CDS spreads (20 or 30 days ahead). We then demonstrate that this feature of the EMS is present in non-crisis periods as well: therefore, the EMS could be used to predict future movement of CDS spreads. The second contribution is to improve efficiency in negative basis arbitrage trading. Specifically, we set up a finite-horizon optimal stopping problem for an investor who uses CDS contracts to hedge a long position in a zero-coupon bond. Using this problem, we provide a trading tool for the investor and show that the information obtained from the EMS can enhance the gain from negative CDS-bond basis arbitrage.

Some extensions of optimal stopping with financial applications

Finite horizon optimal stopping problems for Markov chains are a well researched topic. Frequently they are phrased in terms of cost or return because many financial models are based on Markov chains. In this paper we will apply optimal stopping to certain random walks on binary trees motivated by insurance considerations. The results are direct extensions of known results but the implications for insurance are of interest.

Optimal stopping of a killed exponentially growing process

We consider a finite horizon optimal stopping problem with a gain function equal to the call option’s. The value of the underlying process grows exponentially until a Poisson process jumps for the first time, at which the process jumps to zero and stays there forever. As applications of this model we consider valuing real options and options written on the stock of a start-up company.

Adaptive epidemic dissemination as a finite-horizon optimal stopping problem

Wireless ad hoc networks are characterized by their limited capabilities and their routine deployment in unfavorable environments. This creates the strong requirement to regulate energy expenditure. We present a scheme to regulate energy cost through optimized transmission scheduling in a noisy epidemic dissemination environment. Building on the intrinsically cross-layer nature of the adaptive epidemic dissemination process, we strive to deliver an optimized mechanism, where energy cost is regulated without compromising the network infection. Improvement of data freshness and applicability in routing are also investigated. Extensive simulations are used to support our proposal.

Time-optimized management of IoT nodes

The vision of Internet of Things (IoT) aims to offer a vast infrastructure of numerous interconnected devices usually called IoT nodes. The infrastructure consists of the basis of pervasive computing applications. Applications can be built with the participation of the IoT nodes that interact in very dynamic environments. In this setting, one can identify the need for applying updates in the software/firmware of the autonomous nodes. Updates may include software extensions and patches significant for the efficient functioning of the IoT nodes. Legacy methodologies involve centralized models where complex algorithms and protocols are adopted for the distribution of the updates to the nodes. This paper proposes a distributed approach where each node is responsible to initiate and conclude the update process. We envision that each node monitors specific performance metrics (related to the node itself and/or the network) and based on a time-optimized scheme identifies the appropriate time to perform the update process.We propose the adoption of a finite horizon optimal stopping scheme. Our stopping model originates in the Optimal Stopping Theory (OST) and takes into account multiple performance metrics. The aim is to have the nodes capable of identifying when their performance and the performance of the network are of high quality. In that time, nodes could be able to efficiently conclude the update process. We provide a set of formulations and the analysis of our problem. Extensive experiments and a comparison assessment reveal the advantages of the proposed solution.

On the regularity of American options with regime-switching uncertainty

We study the regularity of the stochastic representation of the solution of a class of initial–boundary value problems related to a regime-switching diffusion. This representation is related to the value function of a finite-horizon optimal stopping problem such as the price of an American-style option in finance. We show continuity and smoothness of the value function using coupling and time-change techniques. As an application, we find the minimal payoff scenario for the holder of an American-style option in the presence of regime-switching uncertainty under the assumption that the transition rates are known to lie within level-dependent compact sets.

Optimal Stopping and Early Exercise: An Eigenfunction Expansion Approach

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-and-death (BD) processes, and jump-diffusions and continuous-time Markov chains obtained by time changing diffusions and BD processes with Levy 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.

Optimal Stopping of a Hilbert Space Valued Diffusion: An Infinite Dimensional Variational Inequality

A finite horizon optimal stopping problem for an infinite dimensional diffusion $X$ is analyzed by means of variational techniques. The diffusion is driven by a SDE on a Hilbert space $\mathcal{H}$ with a non-linear diffusion coefficient $\sigma(X)$ and a generic unbounded operator $A$ in the drift term. When the gain function $\Theta$ is time-dependent and fulfils mild regularity assumptions, the value function $\mathcal{U}$ of the optimal stopping problem is shown to solve an infinite-dimensional, parabolic, degenerate variational inequality on an unbounded domain. Once the coefficient $\sigma(X)$ is specified, the solution of the variational problem is found in a suitable Banach space $\mathcal{V}$ fully characterized in terms of a Gaussian measure $\mu$. This work provides the infinite-dimensional counterpart, in the spirit of Bensoussan and Lions \cite{Ben-Lio82}, of well-known results on optimal stopping theory and variational inequalities in $\mathbb{R}^n$. These results may be useful in several fields, as in mathematical finance when pricing American options in the HJM model.

A Note on the Continuity of Free-Boundaries in Finite-Horizon Optimal Stopping Problems for One-Dimensional Diffusions

We provide sufficient conditions for the continuity of the free-boundary in a general class of finite-horizon optimal stopping problems arising, for instance, in finance and economics. The underlying process is a strong solution of a one-dimensional, time-homogeneous stochastic differential equation (SDE). The proof relies on both analytic and probabilistic arguments and is based on a contradiction scheme inspired by the maximum principle in partial differential equations theory. Mild, local regularity of the coefficients of the SDE and smoothness of the gain function locally at the boundary are required.

QoS-Compliant Sequential Channel Sensing for Cognitive Radios

In this paper, we study the quality-of-service (QoS) support for realtime traffic in cognitive radio (CR) networks when spectrum availability and quality is not known a priori. A resource-constrained CR relies on sequential channel sensing and probing to resolve spectrum uncertainty and search for good transmission opportunities in real time. We are interested in maximizing the effective throughput the CR can achieve with a desired confidence (success probability) under a spectrum access delay constraint. This quantity can be interpreted as the QoS-compliant (e.g., min-rate and delay) capacity of the CR link under the uncertain spectrum environment. The optimization is formulated as a finite-horizon optimal stopping problem under the objective of maximizing a given percentile of the rate of return at the stopping time. This formulation cannot be directly solved by classical optimal stopping theory, because the latter only supports a mean-reward objective function. A novel transformation is developed to convert the problem into solving a series of sub problems, each of which optimizes a transformed mean-reward of the original problem and therefore can be solved using classical optimal stopping method. We prove the monotonicity of the sub problems, based on which we develop a fast algorithm to efficiently find the unique solution to the original problem. To account for different MAC mechanisms used in practice, our analysis considers both non-recall and recall channel decision strategies. Extensive simulations are performed to verify the effectiveness and significance of the optimization. We show that significant gains (e.g., over 30%) on the QoS-compliant capacity can be achieved by the proposed algorithm when compared with the counterparts.

In This Issue

In This Issue

Optimal Stopping and Early Exercise: An Eigenfunction Expansion Approach

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.

Optimal stopping of expected profit and cost yields in an investment under uncertainty

We consider a finite horizon optimal stopping problem related to trade-off strategies between expected profit and cost cash flows of an investment under uncertainty. The optimal problem is first formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We then construct both a minimal solution and a maximal solution using an approximation scheme of the associated system of reflected backward stochastic differential equations (SDEs). We also address the question of uniqueness of solutions of this system of SDEs. When the dependence of the cash flows on the sources of uncertainty, such as fluctuation market prices, assumed to evolve according to a diffusion process, is made explicit, we obtain a connection between these solutions and viscosity solutions of a system of variational inequalities with interconnected obstacles.

Incomplete markets, ambiguity, and irreversible investment

The problem of irreversible investment with idiosyncratic risk is studied by interpreting market incompleteness as a source of ambiguity over the appropriate no-arbitrage discount factor. The maxmin utility over multiple priors framework is used to model and solve the irreversible investment problem. Multiple priors are modeled using the notion of κ ‐ignorance . This set-up is used to analyze finitely lived options. For infinitely lived options the notion of constant κ ‐ignorance is introduced. For these sets of density generators the corresponding optimal stopping problem is solved for general (in-)finite horizon optimal stopping problems driven by geometric Brownian motion. It is argued that an increase in the set of priors delays investment, whereas an increase in the degree of market completeness can have a non-monotonic effect on investment.

Finite Horizon Optimal Stopping of Time-Discontinuous Functionals with Applications to Impulse Control with Delay

We study finite horizon optimal stopping problems for continuous-time Feller–Markov processes. The functional depends on time, state, and external parameters and may exhibit discontinuities with respect to the time variable. Both left- and right-hand discontinuities are considered. We investigate the dependence of the value function on the parameters, on the initial state of the process, and on the stopping horizon. We construct $\varepsilon$-optimal stopping times and provide conditions under which an optimal stopping time exists. We demonstrate how to approximate this optimal stopping time by solutions to discrete-time problems. Our results are applied to the study of impulse control problems with finite time horizon, decision lag, and execution delay.

Optimal Stopping of a Brownian Bridge

We study several optimal stopping problems in which the gains process is a Brownian bridge or a functional of a Brownian bridge. Our examples constitute natural finite-horizon optimal stopping problems with explicit solutions.

Optimal Stopping of a Brownian Bridge

We study several optimal stopping problems in which the gains process is a Brownian bridge or a functional of a Brownian bridge. Our examples constitute natural finite-horizon optimal stopping problems with explicit solutions.

Incomplete Markets, Knightian Uncertainty, and Irreversible Investment

The problem of irreversible investment with idiosyncratic risk is studied by interpreting market incompleteness as a source of Knightian uncertainty over the appropriate discount factor. Maxmin utility over multiple priors is used to solve the irreversible investment problem. The notion of priors with kappa-ignorance are used to analyse finitely lived options. For infinitely lived options the notion of constant kappa-ignorance is introduced. For these sets of density generators the corresponding optimal stopping problem is solved for general (in-)finite horizon optimal stopping problems driven by geometric Brownian motions. It is argued that an increase in the level of ambiguity delays investment, whereas an increase in the degree of market completeness can have a non-monotonic effect.