Integration methods of mixed-type stochastic differential equations with fractional Brownian motions
https://doi.org/10.29235/1561-2430-2019-55-2-135-151
Abstract
In the present, article new methods of exact integration of mixed-type stochastic differential equations with standard Brownian motion, fractional Brownian motion with the Hurst exponent H> 1/2 and the drift term have been constructed. Solutions of these equations are understood in integral sense where, in turn, the standard Brownian motion integral is the Ito integral and the fractional Brownian motion integral is the pathwise Young integral. The constucted integration methods can be attributed to two types. The first-type methods are based on reducing the equations to simpler equations, in particular – to the simplest equations and the linear inhomogeneous equations. In the article, necessary and sufficient conditions of reducing the equations applicable to one-dimensional equations have been obtained and the examples particularly covering the stochastic Bernoulli-type equations have been given. The second-type method is based on going to the Stratonovich equation and is applicable to multidimensional equations. In addition to the mentioned integration methods, the analogues of the differential Kolmogorov equation have been obtained for mathematical expectations and the solution probability density, assuming that coefficients of the mixed-type stochastic differential equation generate commuting flows.
About the Authors
M. M. Vas’kovskii
Belarusian State University
Belarus
Belarus
Maksim M. Vas’kovskii – Ph. D. (Physics and Mathematics), Associate Professor of the Department of Higher Mathematics
4, Nezavisimosti Ave., 220072, Minsk, Republic of Belarus
I. V. Kachan
Belarusian State University
Belarus
Belarus
Ilya V. Kachan – Postgraduate Student, Assistant of the Department of Higher Mathematics
4, Nezavisimosti Ave., 220072, Minsk, Republic of Belarus
References
1. Biagini F., Hu Y., Oksendal B., Zhang T. Stochastic Calculus for Fractional Brownian Motion and Applications. London, Springer-Verlag, 2008. 330 p. https://doi.org/10.1007/978-1-84628-797-8
2. Cheridito P. Regularizing Fractional Brownian Motion with a View Towards Stock Price Modeling. Zurich, ETH, 2001. 121 p.
3. Guerra J., Nualart D. Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Stochastic Analysis and Applications, 2008, vol. 26, no. 5, pp. 1053–1075. https://doi.org/10.1080/07362990802286483
4. Mishura Y. S. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Berlin, Heidelberg, Springer-Verlag, 2008. 411 p. https://doi.org/10.1007/978-3-540-75873-0
5. Mishura Y. S., Shevchenko G. M. Existence and uniqueness of the solution of stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index H > 1/2. Communications in Statistics – Theory and Methods, 2011, vol. 40, no. 19–20, pp. 3492–3508. https://doi.org/10.1080/03610926.2011.581174
6. Shevchenko G. Mixed stochastic delay differential equations. Theory of Probability and Mathematical Statistics, 2014, vol. 89, pp. 181–195. https://doi.org/10.1090/s0094-9000-2015-00944-3
7. Levakov A. A., Vas'kovskii M. M. Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motions and with discontinuous coefficients. Differential Equations, 2014, vol. 50, no. 2, pp. 189–202. https://doi.org/10.1134/s0012266114020062
8. Levakov A. A., Vas'kovskii M. M. Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion, discontinuous coefficients, and a partly degenerate diffusion operator. Differential Equations, 2014, vol. 50, no. 8, pp. 1053–1069. https://doi.org/10.1134/s0012266114080059
9. Vas'kovskii M. M. Existence of weak solutions of stochastic delay differential equations driven by standard and fractional Brownian motions. Vestsi Natsyianal'nai akademii navuk Belarusi. Seryia fizika-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2015, no. 1, pp. 22–34 (in Russian).
10. Levakov A. A., Vas'kovskii M. M. Existence of solutions of stochastic differential inclusions with standard and fractional Brownian motions. Differential Equations, 2015, vol. 51, no. 8, pp. 991–997. https://doi.org/10.1134/s0012266115080030
11. Levakov A. A., Vas'kovskii M. M. Properties of solutions of stochastic differential equations with standard and fractional Brownian motions. Differential Equations, 2016. vol. 52, no. 8, pp. 972–980. https://doi.org/10.1134/s0012266116080024
12. Vas'kovskii M. M. Stability and attraction of solutions of nonlinear stochastic differential equations with standard and fractional Brownian motions. Differential Equations, 2017, vol. 53, no. 2, pp. 157–170. https://doi.org/10.1134/s0012266117020021
13. Gard T. C. Introduction to Stochastic Differential Equations.New York, Basel, Marcel Dekker Inc., 1988. 234 p.
14. Levakov A. A. Stochastic Differential Equations. Minsk, BSU, 2009. 231 p. (in Russian).
15. Russo F., Vallois P. Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics and Stochastic Reports, 2000, vol. 70, no. 1–2, pp. 1–40. https://doi.org/10.1080/17442500008834244
16. Oksendal B. Stochastic Differential Equations. An Introduction with Applications. Berlin, Heidelberg, SpringerVerlag, 2003. 379 p. https://doi.org/10.1007/978-3-642-14394-6_5
17. Baudoin F., Coutin L. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Processes and their Applications, 2007, vol. 117, no. 5, pp. 550–574. https://doi.org/10.1016/j.spa.2006.09.004
18. Vaskouski M., Kachan I. Asymptotic expansions of solutions of stochastic differential equations driven by multivariate fractional Brownian motions having Hurst indices greater than 1/3. Stochastic Analysis and Applications, 2018, vol. 36, no. 6, pp. 909–931. https://doi.org/10.1080/07362994.2018.1483247
19. Vyoral M. Kolmogorov equation and large-time behavior for fractional Brownian motion driven linear SDE's. Applications of Mathematics, 2005, vol. 50, no. 1, pp. 63–81. https://doi.org/10.1007/s10492-005-0004-4
Review
Views: 1759
Email this article 