Two-player zero-sum stochastic differential games with random horizon

2019 ◽  
Vol 51 (4) ◽  
pp. 1209-1235
Author(s):  
M. Ferreira ◽  
D. Pinheiro ◽  
S. Pinheiro
Keyword(s):  
Viscosity Solution ◽  
Nonlinear Partial Differential Equation ◽  
Random Variable ◽  
Planning Horizon ◽  
Continuous Random Variable ◽  
State Variable ◽  
Random Horizon ◽  
Hamilton Jacobi Bellman ◽  
Zero Sum ◽  
Random Planning Horizon

AbstractWe consider a two-player zero-sum stochastic differential game with a random planning horizon and diffusive state variable dynamics. The random planning horizon is a function of a non-negative continuous random variable, which is assumed to be independent of the Brownian motion driving the state variable dynamics. We study this game using a combination of dynamic programming and viscosity solution techniques. Under some mild assumptions, we prove that the value of the game exists and is the unique viscosity solution of a certain nonlinear partial differential equation of Hamilton–Jacobi–Bellman–Isaacs type.

scholarly journals Optimal Stopping Rule for the No-Information Duration Problem with Random Horizon

2013 ◽  
Vol 45 (04) ◽  
pp. 1028-1048 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Classical Problem ◽  
Random Order ◽  
Random Variable ◽  
Planning Horizon ◽  
Secretary Problem ◽  
Main Concern ◽  
Asymptotic Results ◽  
Optimal Rule ◽  
Random Horizon

As a version of the secretary problem, Ferguson, Hardwick and Tamaki (1992) considered an optimal stopping problem called the duration problem. The basic duration problem is the classical duration problem, in which the objective is to maximize the time of possession of a relatively best object when a known number of rankable objects appear in random order. In this paper we generalize this classical problem in two directions by allowing the number N (of available objects) to be a random variable with a known upper bound n and also allowing the objects to appear in accordance with Bernoulli trials. Two models can be considered for our random horizon duration problem according to whether the planning horizon is N or n. Since the form of the optimal rule is in general complicated, our main concern is to give to each model a sufficient condition for the optimal rule to be simple. For N having a uniform, generalized uniform, or curtailed geometric distribution, the optimal rule is shown to be simple in the so-called secretary case. The asymptotic results, as n → ∞, will also be given for these priors.

Optimal Production Planning for a Random Horizon

2015 ◽  
Vol 791 ◽  
pp. 63-69
Author(s):  
Arkadiusz Gola ◽  
Edward Kozłowski
Keyword(s):  
Mathematical Model ◽  
Production Control ◽  
Planning Horizon ◽  
Control Model ◽  
Planning Period ◽  
Manufacturing Costs ◽  
Optimal Production ◽  
Random Horizon ◽  
The Mathematical Model ◽  
Random Planning Horizon

The article presents the mathematical model of production control for a random planning horizon. The model assumes that the horizon of control is unknown and connected with reliability of machines, which are used for production. The aim of the control model is to determine the number of products which should be manufactured in each planning period to minimize both manufacturing costs and potential financial penalties for failing to fulfill the order completely.

Optimal Stopping Rule for the No-Information Duration Problem with Random Horizon

2013 ◽  
Vol 45 (4) ◽  
pp. 1028-1048
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Classical Problem ◽  
Random Order ◽  
Random Variable ◽  
Planning Horizon ◽  
Secretary Problem ◽  
Main Concern ◽  
Asymptotic Results ◽  
Optimal Rule ◽  
Random Horizon

As a version of the secretary problem, Ferguson, Hardwick and Tamaki (1992) considered an optimal stopping problem called the duration problem. The basic duration problem is the classical duration problem, in which the objective is to maximize the time of possession of a relatively best object when a known number of rankable objects appear in random order. In this paper we generalize this classical problem in two directions by allowing the number N (of available objects) to be a random variable with a known upper bound n and also allowing the objects to appear in accordance with Bernoulli trials. Two models can be considered for our random horizon duration problem according to whether the planning horizon is N or n. Since the form of the optimal rule is in general complicated, our main concern is to give to each model a sufficient condition for the optimal rule to be simple. For N having a uniform, generalized uniform, or curtailed geometric distribution, the optimal rule is shown to be simple in the so-called secretary case. The asymptotic results, as n → ∞, will also be given for these priors.

A class of infinite-horizon stochastic delay optimal control problems and a viscosity solution to the associated HJB equation

2018 ◽  
Vol 24 (2) ◽  
pp. 639-676
Author(s):  
Jianjun Zhou
Keyword(s):  
Optimal Control ◽  
Viscosity Solution ◽  
Optimal Control Problems ◽  
Nonlinear Partial Differential Equation ◽  
Infinite Horizon ◽  
Hjb Equation ◽  
Second Order ◽  
Control Problems ◽  
Stochastic Delay ◽  
Hamilton Jacobi Bellman

In this paper, we investigate a class of infinite-horizon optimal control problems for stochastic differential equations with delays for which the associated second order Hamilton−Jacobi−Bellman (HJB) equation is a nonlinear partial differential equation with delays. We propose a new concept for the viscosity solution including timetand identify the value function of the optimal control problems as a unique viscosity solution to the associated second order HJB equation.

Optimal stopping rule for the full-information duration problem with random horizon

2016 ◽  
Vol 48 (1) ◽  
pp. 52-68 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Continuous Distribution ◽  
Random Variable ◽  
Planning Horizon ◽  
Stopping Rule ◽  
Full Information ◽  
Optimal Rule ◽  
Relative Maximum ◽  
Optimal Stopping Rule ◽  
Random Horizon ◽  
Necessary And Sufficient

Abstract The full-information duration problem with a random number N of objects is considered. These objects appear sequentially and their values Xk are observed, where Xk, independent of N, are independent and identically distributed random variables from a known continuous distribution. The objective of the problem is to find a stopping rule that maximizes the duration of holding a relative maximum (e.g. the kth object is a relative maximum if Xk = max{X1, X2, . . ., Xk}). We assume that N is a random variable with a known upper bound n, so two models, Model 1 and Model 2, can be considered according to whether the planning horizon is N or n. The structure of the optimal rule, which depends on the prior distribution assumed on N, is examined. The monotone rule is defined and a necessary and sufficient condition for the optimal rule to be monotone is given for both models. Special attention is paid to the class of priors such that N / n converges, as n → ∞, to a random variable Vm having density fVm(v) = m(1 - v)m-1, 0 ≤ v ≤ 1 for a positive integer m. An interesting feature is that the optimal duration (relative to n) for Model 2 is just (m + 1) times as large as that for Model 1 asymptotically.

OPTIMIZATION OF MARKOV SYSTEMS OF QUEUEING WITH WAITING TIME IN THE MATLAB SYSTEM

2017 ◽  
pp. 39-47
Author(s):  
Viktor Afonin ◽  
Vladimir Valer'evich Nikulin
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Random Variable ◽  
Model Systems ◽  
Queuing Systems ◽  
Input Flow ◽  
Continuous Random Variable ◽  
Input Stream ◽  
Time Intervals ◽  
Markov Systems

The article focuses on attempt to optimize two well-known Markov systems of queueing: a multichannel queueing system with finite storage, and a multichannel queueing system with limited queue time. In the Markov queuing systems, the intensity of the input stream of requests (requirements, calls, customers, demands) is subject to the Poisson law of the probability distribution of the number of applications in the stream; the intensity of service, as well as the intensity of leaving the application queue is subject to exponential distribution. In a Poisson flow, the time intervals between requirements are subject to the exponential law of a continuous random variable. In the context of Markov queueing systems, there have been obtained significant results, which are expressed in the form of analytical dependencies. These dependencies are used for setting up and numerical solution of the problem stated. The probability of failure in service is taken as a task function; it should be minimized and depends on the intensity of input flow of requests, on the intensity of service, and on the intensity of requests leaving the queue. This, in turn, allows to calculate the maximum relative throughput of a given queuing system. The mentioned algorithm was realized in MATLAB system. The results obtained in the form of descriptive algorithms can be used for testing queueing model systems during peak (unchanged) loads.

scholarly journals Distributional Replication

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 1063
Author(s):  
Brendan K. Beare
Keyword(s):  
Hedge Fund ◽  
Minimum Cost ◽  
Random Variable ◽  
Optimal Approximation ◽  
Regularity Conditions ◽  
Return Distribution ◽  
Continuous Random Variable ◽  
Hedge Fund Replication ◽  
Market Index ◽  
Fund Return

A function which transforms a continuous random variable such that it has a specified distribution is called a replicating function. We suppose that functions may be assigned a price, and study an optimization problem in which the cheapest approximation to a replicating function is sought. Under suitable regularity conditions, including a bound on the entropy of the set of candidate approximations, we show that the optimal approximation comes close to achieving distributional replication, and close to achieving the minimum cost among replicating functions. We discuss the relevance of our results to the financial literature on hedge fund replication; in this case, the optimal approximation corresponds to the cheapest portfolio of market index options which delivers the hedge fund return distribution.

An Imperfect Production Model for Breakable Multi-Item with Dynamic Demand and Learning Effect on Rework over Random Planning Horizon

2021 ◽  
Vol 14 (12) ◽  
pp. 574
Author(s):  
Amalesh Kumar Manna ◽  
Leopoldo Eduardo Cárdenas-Barrón ◽  
Barun Das ◽  
Ali Akbar Shaikh ◽  
Armando Céspedes-Mota ◽  
...  
Keyword(s):  
Learning Effect ◽  
Planning Horizon ◽  
Inventory Model ◽  
Inventory System ◽  
Production Model ◽  
Inventory Models ◽  
Imperfect Production ◽  
Production Inventory ◽  
Random Planning Horizon ◽  
Production Inventory System

In recent times, in the literature of inventory management there exists a notorious interest in production-inventory models focused on imperfect production processes with a deterministic time horizon. Nevertheless, it is well-known that there is a high influence and impact caused by the learning effect on the production-inventory models in the random planning horizon. This research work formulates a mathematical model for a re-workable multi-item production-inventory system, in which the demand of the items depends on the accessible stock and selling revenue. The production-inventory model allows shortages and these are partial backlogged over a random planning horizon. Also, the learning effect on the rework policy, inflation, and the time value of money are considered. The main aim is to determine the optimum production rates that minimize the expected total cost of the multi-item production-inventory system. A numerical example is solved and a detailed sensitivity analysis is conducted in order to study the production-inventory model.

Urn sampling distributions giving alternate correspondences between two optimal stopping problems

2016 ◽  
Vol 48 (3) ◽  
pp. 726-743 ◽  
Author(s):  
Mitsushi Tamaki
Keyword(s):  
Optimal Stopping ◽  
Random Variable ◽  
Planning Horizon ◽  
Information Model ◽  
Secretary Problem ◽  
Choice Problem ◽  
Sampling Distributions ◽  
Best Choice Problem ◽  
Optimal Stopping Problems ◽  
Bounded Random Variable

Abstract The best-choice problem and the duration problem, known as versions of the secretary problem, are concerned with choosing an object from those that appear sequentially. Let (B,p) denote the best-choice problem and (D,p) the duration problem when the total number N of objects is a bounded random variable with prior p=(p1, p2,...,pn) for a known upper bound n. Gnedin (2005) discovered the correspondence relation between these two quite different optimal stopping problems. That is, for any given prior p, there exists another prior q such that (D,p) is equivalent to (B,q). In this paper, motivated by his discovery, we attempt to find the alternate correspondence {p(m),m≥0}, i.e. an infinite sequence of priors such that (D,p(m-1)) is equivalent to (B,p(m)) for all m≥1, starting with p(0)=(0,...,0,1). To be more precise, the duration problem is distinguished into (D1,p) or (D2,p), referred to as model 1 or model 2, depending on whether the planning horizon is N or n. The aforementioned problem is model 1. For model 2 as well, we can find the similar alternate correspondence {p[m],m≥ 0}. We treat both the no-information model and the full-information model and examine the limiting behaviors of their optimal rules and optimal values related to the alternate correspondences as n→∞. A generalization of the no-information model is given. It is worth mentioning that the alternate correspondences for model 1 and model 2 are respectively related to the urn sampling models without replacement and with replacement.

scholarly journals Some non monotone schemes for Hamilton-Jacobi-Bellman equations

2019 ◽  
Vol 65 ◽  
pp. 476-497
Author(s):  
Xavier Warin
Keyword(s):  
Viscosity Solution ◽  
Discrete Problem ◽  
Numerical Schemes ◽  
Bellman Equations ◽  
Monotone Schemes ◽  
Hamilton Jacobi Bellman

We extend the theory of Barles Jakobsen [3] for a class of almost monotone schemes to solve stationary Hamilton Jacobi Bellman equations. We show that the monotonicity of the schemes can be relaxed still leading to the convergence to the viscosity solution of the equation even if the discrete problem can only be solved with some error. We give some examples of such numerical schemes and show that the bounds obtained by the framework developed are not tight. At last we test the schemes.

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