Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-07T00:04:29.641Z Has data issue: true hasContentIssue true
  • English
  • Français

Some explicit results on one kind of sticky diffusion

Part of: Markov processes Mathematical finance

Published online by Cambridge University Press:  30 July 2019

Yiming Jiang ,
Shiyu Song  and
Yongjin Wang
Yiming Jiang*
Affiliation:
Nankai University
Shiyu Song*
Affiliation:
Tianjin University
Yongjin Wang*
Affiliation:
Nankai University
*
*Postal address: School of Mathematical Sciences, Nankai University, Tianjin 300071, China.
**Postal address: School of Mathematics, Tianjin University, Tianjin 300354, China.
****Postal address: School of Business, Nankai University, Tianjin 300071, China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper we derive several explicit results on one special sticky diffusion process which is constructed as a time-changed version of a diffusion with no sticky points. A theorem concerning the process-related Green operators defined on some nonnegative piecewise continuous functions is provided. Then, based on this theorem, we explore the distributional properties of the sticky diffusion. A financial application is presented where we compute the value of the European vanilla call option written on the underlying with sticky price dynamics.

Keywords

Sticky diffusionfirst hitting timeGreen’s operatordistributional propertiesoption pricing

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 91G20: Derivative securities

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 398 - 415
Copyright
© Applied Probability Trust 2019 

References

Amir, M. (1991). Sticky Brownian motion as the strong limit of a sequence of random walks. Stoch. Process. Appl. 39, 221–237.CrossRefGoogle Scholar
Appuhamillage, T., Bokil, V., Thomann, E., Waymire, E. and Wood, B. (2011). Occupation and local times for skew Brownian motion with applications to dispersion across an interface. Ann. Appl. Prob. 21, 183–214.CrossRefGoogle Scholar
Ball, C. A., Torous, W. N. and Tschoegl, A. E. (1985). The degree of price resolution: the case of the gold market. J. Futures Mark. 5, 29–43.CrossRefGoogle Scholar
Bass, R. (2014). A stochastic differential equation with a sticky point. Electron. J. Probab. 19, 1–22.CrossRefGoogle Scholar
Bharati, R., Crain, S. J. and Kaminski, V. (2012). Clustering in crude oil prices and the target pricing zone hypothesis. Energy Econ. 34, 1115–1123.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2015). Handbook of Brownian Motion: Facts and Formulae, 2nd edn. 2nd corr. print. Birkhäuser-Verlag, Basel.Google Scholar
Chi, Z., Dong, F. and Wong, H. Y. (2017). Option pricing with threshold mean reversion. J. Futures Mark. 37, 107–131.CrossRefGoogle Scholar
Crocce, F. and Mordecki, E. (2014). Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions. Stochastics 86, 491–509.CrossRefGoogle Scholar
Davydov, D. and Linetsky, V. (2001). Pricing and hedging path-dependent options under the CEV process. Manag. Sci. 47, 949–965.CrossRefGoogle Scholar
Davydov, D. and Linetsky, V. (2003). Pricing options on scalar diffusions: an eigenfunction expansion approach. Oper. Res. 51, 185–209.CrossRefGoogle Scholar
Engelbert, H. J. and Peskir, G. (2014). Stochastic differential equations for sticky Brownian motion. Stochastics 86, 993–1021.CrossRefGoogle Scholar
Engelbert, H. J. and Schmidt, W. (1991). Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations (Part III). Math. Nachr. 151, 149–197.CrossRefGoogle Scholar
Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55, 468–519.CrossRefGoogle Scholar
Feller, W. (1954). Diffusion processes in one dimension. T. Am. Math. Soc. 77, 1–31.CrossRefGoogle Scholar
Feller, W. (1957). Generalized second order differential operators and their lateral conditions. Illinois J. Math. 1, 459–504.CrossRefGoogle Scholar
Freedman, D. (1971). Brownian Motion and Diffusion. Holden-Day, San Francisco.Google Scholar
Grothaus, M. and Vosshall, R. (2018). Strong Feller property of sticky reflected distorted Brownian motion. J. Theor. Probab. 31, 827–852.CrossRefGoogle Scholar
Hajri, H., Caglar, M. and Arnaudon, M. (2017). Application of stochastic flows to the sticky Brownian motion equation. Electron. Commun. Probab. 22, 1–10.CrossRefGoogle Scholar
Hull, J. C. (2012). Options, Futures, and Other Derivatives, 8th edn. Prentice-Hall, Upper Saddle River.Google Scholar
Itô, K. and McKean, H. P. (1974). Diffusion Processes and their Sample Paths. Springer-Verlag, New York.Google Scholar
Jang, B. G., Kim, C., Kim, K. T., Lee, S. and Shin, D. H. (2015). Psychological barriers and option pricing. J. Futures Mark. 35, 52–74.Google Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer-Verlag, New York.Google Scholar
Karlin, S. and Taylor, H. M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Le Gall, J. -F. (1984). One-dimensional stochastic differential equations involving the local times of the unknown process. Stochastic Analysis and Applications, Lecture Notes in Math. 1095, 51–82.CrossRefGoogle Scholar
Lemoine, A. J. (1975). Limit theorems for generalized single server queues: the exceptional system. SIAM J. Appl. Math. 28, 596–606.CrossRefGoogle Scholar
Protter, P. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer-Verlag, New York.Google Scholar
Rácz, M. Z. and Shkolnikov, M. (2015). Multidimensional sticky Brownian motions as limits of exclusion processes. Ann. Appl. Prob. 25, 1155–1188.CrossRefGoogle Scholar
Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales 2: Itô Calculus, 2nd edn. Wiley, New York.Google Scholar
Salins, M. and Spiliopoulos, K. (2017). Markov processes with spatial delay: path space characterization, occupation time and properties. Stoch. Dynam. 17, 1750042 (21 pages).CrossRefGoogle Scholar
Salminen, P. and Ta, B. (2015). Differentiability of excessive functions of one-dimensional diffusions and the principle of smooth fit. Banach Center Publications 104, 181–199.CrossRefGoogle Scholar
Sonnemans, J. (2006). Price clustering and natural resistance points in the Dutch stock market: a natural experiment. Eur. Econ. Rev. 50, 1937–1950.CrossRefGoogle Scholar
Valko, P. P. and Abate, J. (2004). Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion. Comput. Math. Appl. 48, 629–636.CrossRefGoogle Scholar
Warren, J. (1999). On the joining of sticky Brownian motion. Séminaire de Probabilités XXXIII 1709, 257–266.CrossRefGoogle Scholar
Warren, J. (2002). The noise made by a Poisson snake. Electron. J. Prob. 7, 1–21.CrossRefGoogle Scholar
Warren, J. (2015). Sticky particles and stochastic flows. In Memoriam Marc Yor: Séminaire de Probabilités XLVII. 2137, 17–35.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