On the forward algorithm for stopping problems on continuous-time Markov chains

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only 𝒞 0 across the optimal boundary when stopping is allowed at t = 0 and 𝒞 2 otherwise, both contradicting the usual 𝒞 1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

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Optimal stopping with random intervention times

Advances in Applied Probability ◽

10.1239/aap/1019160954 ◽

2002 ◽

Vol 34 (1) ◽

pp. 141-157 ◽

Cited By ~ 35

Author(s):

Paul Dupuis ◽

Hui Wang

Keyword(s):

Optimal Stopping ◽

Continuous Time ◽

Value Function ◽

Value Functions ◽

Optimal Exercise Boundary ◽

Exercise Boundary ◽

Time Optimal ◽

Optimal Stopping Problems ◽

Necessary And Sufficient ◽

The Value Function

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed att= 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only𝒞0across the optimal boundary when stopping is allowed att= 0 and𝒞2otherwise, both contradicting the usual𝒞1smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

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American Option Valuation under Continuous-Time Markov Chains

Advances in Applied Probability ◽

10.1239/aap/1435236980 ◽

2015 ◽

Vol 47 (2) ◽

pp. 378-401 ◽

Cited By ~ 6

Author(s):

B. Eriksson ◽

M. R. Pistorius

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Optimal Stopping ◽

Continuous Time ◽

Markov Models ◽

American Options ◽

American Option ◽

Discrete State ◽

Continuous Time Markov Chains ◽

Exponential Lévy Models

This paper is concerned with the solution of the optimal stopping problem associated to the value of American options driven by continuous-time Markov chains. The value-function of an American option in this setting is characterised as the unique solution (in a distributional sense) of a system of variational inequalities. Furthermore, with continuous and smooth fit principles not applicable in this discrete state-space setting, a novel explicit characterisation is provided of the optimal stopping boundary in terms of the generator of the underlying Markov chain. Subsequently, an algorithm is presented for the valuation of American options under Markov chain models. By application to a suitably chosen sequence of Markov chains, the algorithm provides an approximate valuation of an American option under a class of Markov models that includes diffusion models, exponential Lévy models, and stochastic differential equations driven by Lévy processes. Numerical experiments for a range of different models suggest that the approximation algorithm is flexible and accurate. A proof of convergence is also provided.

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American Option Valuation under Continuous-Time Markov Chains

Advances in Applied Probability ◽

10.1017/s0001867800007904 ◽

2015 ◽

Vol 47 (02) ◽

pp. 378-401

Author(s):

B. Eriksson ◽

M. R. Pistorius

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Optimal Stopping ◽

Continuous Time ◽

Markov Models ◽

American Options ◽

American Option ◽

Discrete State ◽

Continuous Time Markov Chains ◽

Exponential Lévy Models

This paper is concerned with the solution of the optimal stopping problem associated to the value of American options driven by continuous-time Markov chains. The value-function of an American option in this setting is characterised as the unique solution (in a distributional sense) of a system of variational inequalities. Furthermore, with continuous and smooth fit principles not applicable in this discrete state-space setting, a novel explicit characterisation is provided of the optimal stopping boundary in terms of the generator of the underlying Markov chain. Subsequently, an algorithm is presented for the valuation of American options under Markov chain models. By application to a suitably chosen sequence of Markov chains, the algorithm provides an approximate valuation of an American option under a class of Markov models that includes diffusion models, exponential Lévy models, and stochastic differential equations driven by Lévy processes. Numerical experiments for a range of different models suggest that the approximation algorithm is flexible and accurate. A proof of convergence is also provided.

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Regularity of the value function and viscosity solutions in optimal stopping problems for general Markov processes

Stochastics and Stochastics Reports ◽

10.1080/1045112021000015331 ◽

2002 ◽

Vol 74 (3-4) ◽

pp. 633-649 ◽

Cited By ~ 7

Author(s):

Bruno Bassan ◽

Claudia Ceci

Keyword(s):

Markov Processes ◽

Viscosity Solutions ◽

Optimal Stopping ◽

Value Function ◽

Optimal Stopping Problems ◽

The Value Function

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On Optimal Stopping of Inhomogeneous Standard Markov Processes

Georgian Mathematical Journal ◽

10.1515/gmj.1994.335 ◽

1995 ◽

Vol 2 (4) ◽

pp. 335-346

Author(s):

B. Dochviri

Keyword(s):

Markov Process ◽

Optimal Stopping ◽

Value Function ◽

Stopping Times ◽

Homogeneous Markov Process ◽

Limit Procedure ◽

Optimal Stopping Problems ◽

Optimal Stopping Times ◽

The Value Function

Abstract 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.

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