×

zbMATH — the first resource for mathematics

Skew Ornstein-Uhlenbeck processes and their financial applications. (English) Zbl 1304.60046
The paper establishes the existence and uniqueness of the skew Ornstein-Uhlenbeck process satisfying a certain stochastic differential equation with local time, and calculates its transition densities and first hitting time. As applications, the authors propose an effective price dynamics in a regulated market and relate their theoretical results to financial markets combined with default risk.

MSC:
60G15 Gaussian processes
91G40 Credit risk
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Itô, K.; McKean, H. P., Diffusion processes and their sample paths, (1974), Springer Verlag · Zbl 0285.60063
[2] Walsh, J. B., A diffusion with a discontinuous local time, Astérisque, 52, 37-45, (1978)
[3] Harrison, J. M.; Shepp, L. A., On skew Brownian motion, Ann. Probab., 9, 309-313, (1981) · Zbl 0462.60076
[4] Le Gall, J. F., Temps Locaux et Equations Différentielles Stochastiques, vol. 986, (1983), Springer Verlag Berlin
[5] Barlow, M. T., Skew Brownian motion and a one dimensional stochastic differential equation, Stochastics, 25, 1-2, (1988) · Zbl 0657.60075
[6] Decamps, M.; Goovaerts, M.; Schoutens, W., Asymmetric skew Bessel processes and their applications to finance, J. Comput. Appl. Math., 186, 130-147, (2006) · Zbl 1087.91022
[7] Blei, S., On symmetric and skew Bessel processes, Stochastic Process. Appl., 122, 3262-3287, (2012) · Zbl 1248.60060
[8] Trutnau, G., Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve, Stochastic Process. Appl., 120, 381-402, (2010) · Zbl 1195.60085
[9] Trutnau, G., Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve, Stochastic Process. Appl., 121, 1845-1863, (2011) · Zbl 1228.60071
[10] Decamps, M.; Goovaerts, M.; Schoutens, W., Self exciting threshold interest rates models, Int. J. Theor. Appl. Finance, 9, 1093-1122, (2006) · Zbl 1140.91384
[11] Corns, T. R.A.; Satchell, S. E., Skew Brownian motion and pricing European options, Eur. J. Financ., 13, 523-544, (2007)
[12] Lejay, A., On the constructions of the skew Brownian motion, Probab. Surv., 3, 413-466, (2006) · Zbl 1189.60145
[13] Ricciardi, L. M.; Sato, S., First-passage-time density and moments of the Ornstein-Uhlenbeck process, J. Appl. Probab., 25, 43-57, (1988) · Zbl 0651.60080
[14] Fang, S.; Zhang, T. S., On the small time behavior of Ornstein-Uhlenbeck processes with unbounded linear drifts, Probab. Theory Related, 114, 487-504, (1999) · Zbl 0932.60071
[15] Bercu, B.; Rouault, A., Sharp large deviations for the Ornstein-Uhlenbeck process, Theory Probab. Appl., 46, 1-19, (2002) · Zbl 1101.60320
[16] Lindner, A.; Maller, R., Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes, Stochastic Process. Appl., 115, 1701-1722, (2005) · Zbl 1080.60056
[17] Hull, J.; White, A., Pricing interest-rate-derivative securities, Rev. Finance Stud., 3, 573-592, (1990) · Zbl 1386.91152
[18] Schwartz, E. S., The stochastic behavior of commodity prices: implications for valuation and hedging, J. Finance, 52, 923-973, (1997)
[19] Barndorff-Nielsen, O. E.; Shephard, N., Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. B, 63, 167-241, (2001) · Zbl 0983.60028
[20] Nicolato, E.; Venardos, E., Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type, Math. Finance, 13, 445-466, (2003) · Zbl 1105.91020
[21] Yi, C., On the first passage time distribution of an Ornstein-Uhlenbeck process, Quant. Finance, 10, 957-960, (2010) · Zbl 1221.60114
[22] Ward, A. R.; Glynn, P. W., Properties of the reflected Ornstein-Uhlenbeck process, Queueing Syst., 44, 109-123, (2003) · Zbl 1026.60106
[23] Linetsky, V., On the transition densities for reflected diffusions, Adv. Appl. Probab., 37, 435-460, (2005) · Zbl 1073.60074
[24] Bo, L.; Zhang, L.; Wang, Y., On the first passage times of reflected OU processes with two-sided barriers, Queueing Syst., 54, 313-316, (2006) · Zbl 1114.60047
[25] Giorno, V.; Nobile, A. G.; Di Cesare, R., On the reflected Ornstein-Uhlenbeck process with catastrophes, Appl. Math. Comput., 218, 11570-11582, (2012) · Zbl 1286.60076
[26] Kent, J. T., The spectral decomposition of a diffusion hitting time, Ann. Probab., 10, 207-219, (1982) · Zbl 0483.60075
[27] Carmona, R.; Klein, A., Exponential moments for hitting times of uniformly ergodic Markov processes, Ann. Probab., 11, 648-655, (1983) · Zbl 0523.60064
[28] Göing-Jaeschke, A.; Yor, M., A clarification note about hitting times densities for Ornstein-Uhlenbeck processes, Finance Stoch., 7, 413-415, (2003) · Zbl 1064.60026
[29] Alili, L.; Patie, P.; Pedersen, J. L., Representations of the first hitting time density of an Ornstein-Uhlenbeck process, Stoch. Models, 21, 967-980, (2005) · Zbl 1083.60064
[30] McKean, H. P., Elementary solutions for certain parabolic partial differential equations, Trans. Amer. Math. Soc., 82, 519-548, (1956) · Zbl 0070.32003
[31] Linetsky, V., Computing hitting time densities for CIR and OU diffusions: applications to mean-reverting models, J. Comput. Finance, 7, 1-22, (2004)
[32] Linetsky, V., Lookback options and diffusion hitting times: a spectral expansion approach, Finance Stoch., 8, 373-398, (2004) · Zbl 1065.60105
[33] Linetsky, V., The spectral decomposition of the option value, Int. J. Theor. Appl. Finance, 7, 337-384, (2004) · Zbl 1107.91051
[34] Revuz, D.; Yor, M., Continuous martingales and Brownian motion, (1999), Springer · Zbl 0917.60006
[35] Engelbert, H. J.; Schmidt, W., Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations (part III), Math. Nachr., 151, 149-197, (1991) · Zbl 0731.60053
[36] Lions, P. L.; Sznitman, A. S., Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math., 37, 511-537, (1984) · Zbl 0598.60060
[37] Schmidt, W., On stochastic differential equations with reflecting barriers, Math. Nachr., 142, 135-148, (1989) · Zbl 0699.60043
[38] Karatzas, I.; Shreve, S. E., Brownian motion and stochastic calculus, (1991), Springer · Zbl 0734.60060
[39] Karlin, S.; Taylor, H. M., A second course in stochastic processes, (1981), Academic Press · Zbl 0469.60001
[40] Gorovoi, V.; Linetsky, V., Black’s model of interest rates as options, eigenfunction expansions and Japanese interest rates, Math. Finance, 14, 49-78, (2004) · Zbl 1097.91041
[41] Buchholz, H.; Lichtblau, H.; Wetzel, K., The confluent hypergeometric function: with special emphasis on its applications, (1969), Springer
[42] Slater, L. J., Confluent hypergeometric functions, (1960), Cambridge University Press · Zbl 0086.27502
[43] Borodin, A. N.; Salminen, P., Handbook of Brownian motion: facts and formulae, (1996), Birkhäuser Basel · Zbl 0859.60001
[44] Valko, P. P.; Abate, J., Comparison of sequence accelerators forthe gaver method of numerical Laplace transform inversion, Comput. Math. Appl., 48, 629-636, (2004) · Zbl 1064.65152
[45] Y. Li, Y. Wang, X. Yang, On the hitting time density for reflected OU processes: with an application to the regulated market, Working paper, Nankai University, 2012.
[46] Duffie, D.; Lando, D., Term structures of credit spreads with incomplete accounting information, Econometrica, 69, 633-664, (2003) · Zbl 1019.91022
[47] Bo, L.; Tang, D.; Wang, Y.; Yang, X., On the conditional default probability in a regulated market: a structural approach, Quant. Finance, 11, 1695-1702, (2011) · Zbl 1277.91181
[48] Protter, P., Stochastic integration and differential equations, (2004), Springer · Zbl 1041.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.