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Markov chain approximation of one-dimensional sticky diffusions

Part of: Markov processes Ordinary differential operators Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  01 July 2021

Christian Meier ,
Lingfei Li  and
Gongqiu Zhang
Christian Meier*
Affiliation:
The Chinese University of Hong Kong
Lingfei Li*
Affiliation:
The Chinese University of Hong Kong
Gongqiu Zhang*
Affiliation:
The Chinese University of Hong Kong, Shenzhen
*
*Postal address: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong SAR. Email address: lfli@se.cuhk.edu.hk
*Postal address: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong SAR. Email address: lfli@se.cuhk.edu.hk
**Postal address: School of Science and Engineering, The Chinese University of Hong Kong (Shenzhen), China.
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Abstract

We develop a continuous-time Markov chain (CTMC) approximation of one-dimensional diffusions with sticky boundary or interior points. Approximate solutions to the action of the Feynman–Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matrix exponentials efficiently and prove that a carefully designed scheme achieves second-order convergence. We also propose a scheme based on CTMC approximation for the simulation of sticky diffusions, for which the Euler scheme may completely fail. The efficiency of our method and its advantages over alternative approaches are illustrated in the context of bond pricing in a sticky short-rate model for a low-interest environment and option pricing under a geometric Brownian motion price model with a sticky interior point.

Keywords

Diffusionssticky boundaryMarkov chain approximationsimulationSturm–Liouville problem

MSC classification

Primary: 65C40: Computational Markov chains 60J60: Diffusion processes 65C05: Monte Carlo methods
Secondary: 34L10: Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions 34L16: Numerical approximation of eigenvalues and of other parts of the spectrum
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 335 - 369
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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Footnotes

The supplementary material for this article can be found at http://doi.org/10.1017/apr.2020.65.

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