Deep Optimal Stopping

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.

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 deal with the optimal stopping problem based on two types of runs under random drawings with replacement (from an urn of known composition,containing three types of balls. Our model extends that of Starr [3], to incorporate the possibility of two types of runs in the analysis of optimal stopping. In Section 2, the case when no cost is involved has been considered. In Section 3, the case when a known constant cost is incurred for each trial has been considered.Some of the results in Section 3 are new and some improve the results of Starr. We have also considered the infinite horizon case in Section 3·4. We have presented eight tables of values under the optimal stopping rule at the end of Section 2 and Section 3.

The article provides solutions for optimal stopping problems for finite and infinite horizons in the case of random walk with exponential utility function. For these problems we found: i) explicit solution of the corresponding Bellman equation; ii) solution of free boundary problem; iii) optimal stopping rule.

Robbins' problem of optimal stopping is that of minimising the expected rank of an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the value of the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.

Cognitive radio is an innovative technique for spectrum management which uses Dynamic Spectrum Access (DSA). DSA has the potential to mitigate the spectrum scarcity by letting the Secondary User (SU) access the idle spectrum unused by the Primary User (PU). DSA improves the spectrum efficiency via spectrum sensing and optimal spectrum access. Spectrum sensing should be perfect so that the SU does not cause interference to the PU. Moreover, the channel selected should be the best for maximising throughput of the SU. In this paper we consider a best channel selection problem in Cognitive Radio Networks (CRNs). The problem is modelled as optimal stopping problem where the controller at each state can stop or continue sensing based on the channel gains achieved. There is a Markovian relation given between the channel occupancies of PU. The stopping rule comes out to be simple threshold rule. The controller stops sensing when the channel gain exceeds the given threshold.

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.

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.

We consider the optimal stopping problem of maximizing the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length n, where m and n are predetermined integers such that 1 ≤ m &lt; n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.

Peskir, (and also Meilijson and Obłój) considered the following optimal stopping problem: find, for an increasing function F and a positive function λ, where S is the maximum process of Brownian motion. In this article, we are interested in the converse: find, for an increasing function F and a suitable function λ, In the non-degenerate cases the optimal stopping rule is of the form stop the first time that reaches γ or falls below where γ, a positive constant, and g, a negative function, are both to be chosen. The optimal function g is characterised as the solution to non-linear differential equation, which is very similar to that used by Peskir to characterise the solution to equation (1), however we derive this differential equation in a completely different way.

Optimal stopping for Brownian motion with applications to sequential analysis and option pricing

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.

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.

With the wide deployment of mobile network, it’s very important to utilize mobile networks resource efficiently and decrease energy consumption of wireless terminals for constructing green mobile computing. This paper mainly studies minimization problem of energy consumption in data transmission under the given data generation rate and transmission delay demand in wireless links. Firstly, a finite horizon optimal stopping problem with transmission data quantity constraint and energy consumption minimization is constructed, and then we prove the existence of optimal stopping rule and offer the solutions and processes. At last, we obtain the optimal transmission rate threshold of the sending terminal for each channel detection slot time, so as to form the optimization strategy of energy consumption for data transmission based on optimal stopping theory. The simulation results show that the strategy proposed by this paper has lower average energy consumption and higher average delivery success ratio than other strategies, and achieves better optimization effect in energy consumption.

In this paper an optimal stopping problem is formulated, concerned with maximizing the probability of a certain event. Necessary and sufficient conditions for existence of an optimal stopping rule are obtained. The results are then applied to a version of the discrete-time “disorder problem”