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Markov chain approximation of one-dimensional sticky diffusions

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

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: [email protected]
*Postal address: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong SAR. Email address: [email protected]
**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.

References

Altinisik, N., Kadakal, M. and Mukhtarov, O. S. (2004). Eigenvalues and eigenfunctions of discontinuous Sturm–Liouville problems with eigenparameter-dependent boundary conditions. Acta Math. Hung. 102, 159–175.CrossRefGoogle Scholar
Amir, M. (1991). Sticky Brownian motion as the strong limit of a sequence of random walks. Stoch. Process. Appl. 39, 221–237.CrossRefGoogle Scholar
Ankirchner, S., Kruse, T. and Urusov, M. (2019). Wasserstein convergence rates for random bit approximations of continuous Markov processes. Preprint. Available at https://arxiv.org/abs/1903.07880.Google Scholar
Bass, R. F. (2014). A stochastic differential equation with a sticky point. Electron. J. Prob. 19, 1–22.CrossRefGoogle Scholar
Bonilla, F. A. and Cushman, J. H. (2002). Effect of $\alpha$ -stable sorptive waiting times on microbial transport in microflow cells. Phys. Rev. E 66, 031915.CrossRefGoogle ScholarPubMed
Bonilla, F. A., Kleinfelter, N. and Cushman, J. H. (2007). Microfluidic aspects of adhesive microbial dynamics: a numerical exploration of flow-cell geometry, Brownian dynamics, and sticky boundaries. Adv. Water Resources 30, 1680–1695.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.CrossRefGoogle Scholar
Bou-Rabee, N. and Holmes-Cerfon, M. C. (2020). Sticky Brownian motion and its numerical solution. SIAM Rev. 62, 164–195.CrossRefGoogle Scholar
Cai, N., Song, Y. and Kou, S. (2015). A general framework for pricing Asian options under Markov processes. Operat. Res. 63, 540–554.CrossRefGoogle Scholar
Cui, Z., Kirk, J. L. and Nguyen, D. (2017). A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps. Europ. J. Operat. Res. 262, 381–400.CrossRefGoogle Scholar
Cui, Z., Kirkby, J. L. and Nguyen, D. (2018). A general valuation framework for SABR and stochastic local volatility models. SIAM J. Financial Math. 9, 520–563.CrossRefGoogle Scholar
Cui, Z., Lee, C. and Liu, Y. (2018). Single-transform formulas for pricing Asian options in a general approximation framework under Markov processes. Europ. J. Operat. Res. 266, 1134–1139.CrossRefGoogle Scholar
Davies, M. and Truman, A. (1994). Brownian motion with a sticky boundary and point sources in quantum mechanics. Math. Comput. Modelling 20, 173–193.CrossRefGoogle Scholar
Davydov, D. and Linetsky, V. (2003). Pricing options on scalar diffusions: an eigenfunction expansion approach. Operat. Res. 51, 185–209.CrossRefGoogle Scholar
Dhillon, I. (1997). A new $O(n^2)$ algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem. Doctoral Thesis, University of California, Berkeley.Google Scholar
Engelbert, H.-J. and Peskir, G. (2014). Stochastic differential equations for sticky Brownian motion. Stochastics 86, 993–1021.CrossRefGoogle Scholar
Eriksson, B. and Pistorius, M. R. (2015). American option valuation under continuous-time Markov chains. Adv. Appl. Prob. 47, 378–401.CrossRefGoogle Scholar
Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 2, 468–519.CrossRefGoogle Scholar
Feng, L. and Linetsky, V. (2008). Pricing options in jump-diffusion models: an extrapolation approach. Operat. Res. 56, 304–325.CrossRefGoogle Scholar
Gawedzki, K. and Horvai, P. (2004). Sticky behavior of fluid particles in the compressible Kraichnan model. J. Statist. Phys. 116, 1247–1300.CrossRefGoogle Scholar
Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York.Google Scholar
Graham, C. (1988). The martingale problem with sticky reflection conditions, and a system of particles interacting at the boundary. Ann. Inst. H. Poincaré Prob. Statist. 24, 45–72.Google Scholar
Harrison, J. M. and Lemoine, A. J. (1981). Sticky Brownian motion as the limit of storage processes. J. Appl. Prob. 18, 216–226.CrossRefGoogle Scholar
Higham, N. J. (2005). The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26, 1179–1193.CrossRefGoogle Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Jiang, Y., Song, S. and Wang, Y. (2019). Some explicit results on one kind of sticky diffusion. J. Appl. Prob. 56, 398–415.CrossRefGoogle Scholar
Kalda, J. (2007). Sticky particles in compressible flows: aggregation and Richardson’s law. Phys. Rev. Lett. 98, 064501.CrossRefGoogle ScholarPubMed
Karlin, S. and Taylor, H. E. (1981). A Second Course in Stochastic Processes. Elsevier, New York.Google Scholar
Kloeden, P. E. and Platen, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.Google Scholar
Li, J., Li, L. and Zhang, G. (2017). Pure jump models for pricing and hedging VIX derivatives. J. Econom. Dynamics Control 74, 28–55.CrossRefGoogle Scholar
Li, L. and Linetsky, V. (2013). Optimal stopping and early exercise: an eigenfunction expansion approach. Operat. Res. 61, 625–643.CrossRefGoogle Scholar
Li, L. and Linetsky, V. (2014). Time-changed Ornstein–Uhlenbeck processes and their applications in commodity derivative models. Math. Finance 24, 289–330.CrossRefGoogle Scholar
Li, L. and Linetsky, V. (2015). Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach. Finance Stoch. 19, 941–977.CrossRefGoogle Scholar
Li, L. and Zhang, G. (2016). Option pricing in some non-Lévy jump models. SIAM J. Sci. Comput. 38, B539–B569.CrossRefGoogle Scholar
Li, L. and Zhang, G. (2018). Error analysis of finite difference and Markov chain approximations for option pricing. Math. Finance 28, 877–919.CrossRefGoogle Scholar
Linetsky, V. (2008). Spectral methods in derivatives pricing. In Financial Engineering, 1st edn, eds J. R. Birge and V. Linetsky, Elsevier, Amsterdam, pp. 223–299.Google Scholar
Lions, P. L. and Sznitman, A. S. (1984). Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511–537.CrossRefGoogle Scholar
McKean, H. P. (1956). Elementary solutions for certain parabolic partial differential equations. Trans. Amer. Math. Soc. 82, 519–548.CrossRefGoogle Scholar
Meier, C., Li, L. and Zhang, G. (2021). Markov chain approximation of one-dimensional sticky diffusions: supplementary material. Adv. Appl. Prob. Available at http://doi.org/10.1017/[TO BE SET].Google Scholar
MijatoviĆ, A. and Pistorius, M. (2013). Continuously monitored barrier options under Markov processes. Math. Finance 23, 1–38.CrossRefGoogle Scholar
Mukhtarov, O. S., Kadakal, M. and Muhtarov, F. S. (2004). On discontinuous Sturm–Liouville problems with transmission conditions. J. Math. Kyoto Univ. 44, 779–798.Google Scholar
Nie, Y. (2017). Term structure modeling at the zero lower bound. Doctoral Thesis, Northwestern University.Google Scholar
Nie, Y. and Linetsky, V. (2020). Sticky reflecting Ornstein–Uhlenbeck diffusions and the Vasicek interest rate model with the sticky zero lower bound. Stoch. Models 36, 1–19.CrossRefGoogle Scholar
Parashar, R. and Cushman, J. H. (2008). Scaling the fractional advective–dispersive equation for numerical evaluation of microbial dynamics in confined geometries with sticky boundaries. J. Comput. Phys. 227, 6598–6611.CrossRefGoogle Scholar
Peskir, G. (2015). On boundary behaviour of one-dimensional diffusions: from Brown to Feller and beyond. In William Feller, Selected Papers II, Springer, Cham, pp. 77–93.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (2005). Continuous Martingales and Brownian Motion. Springer, Berlin.Google Scholar
Skorokhod, A. V. (1961). Stochastic equations for diffusion processes in a bounded region. Theory Prob. Appl. 6, 264–274.CrossRefGoogle Scholar
SŁomínski, L. (1993). On existence, uniqueness and stability of solutions of multidimensional SDE’s with reflecting boundary conditions. Ann. Inst. H. Poincaré Prob. Statist. 29, 163–198.Google Scholar
Song, Y., Cai, N. and Kou, S. (2019). A unified framework for regime-switching models. Preprint. Available at https://ssrn.com/abstract=3310365.Google Scholar
Tanaka, H. (1979). Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, 163–177.CrossRefGoogle Scholar
Warren, J. (1997). Branching processes, the Ray–Knight theorem, and sticky Brownian motion. In Séminaire de Probabilités XXXI, eds Azéma, J., Yor, M. and Emery, M., Springer, Berlin, pp. 1–15.CrossRefGoogle Scholar
Watanabe, S. (1971). On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions. J. Math. Kyoto Univ. 11, 169–180.Google Scholar
Yamada, K. (1994). Reflecting or sticky Markov processes with Lévy generators as the limit of storage processes. Stoch. Process. Appl. 52, 135–164.CrossRefGoogle Scholar
Zaslavsky, G. M. and Edelman, M. (2005). Polynomial dispersion of trajectories in sticky dynamics. Phys. Rev. E 72, 036204.CrossRefGoogle ScholarPubMed
Zhang, G. and Li, L. (2019). Analysis of Markov chain approximation for option pricing and hedging: grid design and convergence behavior. Operat. Res. 67, 407–427.Google Scholar
Zhang, G. and Li, L. (2019). A general method for valuation of drawdown risk under Markovian models. Working paper.Google Scholar
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