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On mixed fractional stochastic differential equations with discontinuous drift coefficient

Part of: Probabilistic methods, simulation and stochastic differential equations Stochastic analysis

Published online by Cambridge University Press:  01 December 2022

Ercan Sönmez
Ercan Sönmez*
Affiliation:
University of Klagenfurt
*
*Postal address: Department of Statistics, Universitätsstrase 65–67, 9020 Klagenfurt, Austria. Email: ercan.soenmez@aau.at
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Abstract

We prove existence and uniqueness for the solution of a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with an absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient.

Keywords

Mixed stochastic differential equationdiscontinuous driftlong-range dependenceItô formulaabsolute continuity

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 65C30: Stochastic differential and integral equations 60H99: None of the above, but in this section

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 589 - 606
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Fan, X. and Zhang, S. (2021). Moment estimates and applications for SDEs driven by fractional Brownian motions with irregular drifts. Bull. Sci. Math. 170, 1–33.CrossRefGoogle Scholar
Guerra, J. and Nualart, D. (2008). Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Stoch. Anal. Appl. 26, 1053–1075.CrossRefGoogle Scholar
Hu, Y., Liu, Y. and Nualart, D. (2016). Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions. Ann. Appl. Prob. 26, 1147–1207.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1998). Brownian Motion and Stochastic Calculus. Springer, New York, pp. 47–127.CrossRefGoogle Scholar
Kleptsyna, M. L., Kloeden, P. E. and Anh, V. V. (1998). Existence and uniqueness theorems for stochastic differential equations with fractal Brownian motion. Probl. Peredachi Inf. 34, 51–61.Google Scholar
Kubilius, K. (2000). The existence and uniqueness of the solution of the integral equation driven by fractional Brownian motion. Lith. Math. J 40, 104–110.Google Scholar
Kubilius, K. (2002). The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type. Stoch. Process. Appl. 98, 289–315.CrossRefGoogle Scholar
Leobacher, G., Reisinger, C. and Stockinger, W. Well-posedness and numerical schemes for McKean–Vlasov equations and interacting particle systems with discontinuous drift. Preprint, arXiv:2006.14892.Google Scholar
Leobacher, G. and Szölgyenyi, M. (2016). A numerical method for SDEs with discontinuous drift. BIT Numer. Math. 56, 151–162.CrossRefGoogle Scholar
Leobacher, G. and Szölgyenyi, M. (2017). A strong order $1/2$ method for multidimensional SDEs with discontinuous drift. Ann. Appl. Prob. 27, 2383–2418.Google Scholar
Lin, S. J. (1995). Stochastic analysis of fractional Brownian motions. Stochastics 55, 121–140.Google Scholar
Lyons, T. (1994). Differential equations driven by rough signals (I): An extension of an inequality of LC Young. Math. Res. Lett. 1, 451–464.CrossRefGoogle Scholar
Mishura, Y. (2008). Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, New York.CrossRefGoogle Scholar
Mishura, Y. and Nualart, D. (2004). Weak solutions for stochastic differential equations with additive fractional noise. Statist. Prob. Lett. 70, 253–261.CrossRefGoogle Scholar
Mishura, Y. and Posashkov, S. (2007). Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and Wiener process. Theory Stoch. Process. 13, 152–165.Google Scholar
Mishura, Y. and Shevchenko, G. (2011). Rate of convergence of Euler approximations of solution to mixed stochastic differential equation involving Brownian motion and fractional Brownian motion. Random Oper. Stoch. Eq. 19.Google Scholar
Mishura, Y. and Shevchenko, G. (2012). Mixed stochastic differential equations with long-range dependence: Existence, uniqueness and convergence of solutions. Comput. Math. Appl. 64, 3217–3227.Google Scholar
Müller-Gronbach, T. and Yaroslavtseva, L. (2020). On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient. Ann. Inst. H. Poincaré Prob. Statist. 56, 1162–1178.CrossRefGoogle Scholar
Nualart, D. (2006). The Malliavin Calculus and Related Topics. Springer, New York.Google Scholar
Nualart, D. and Ouknine, Y. (2002). Regularization of differential equations by fractional noise. Stoch. Process. Appl. 102, 103–116.CrossRefGoogle Scholar
Nualart, D. and Ouknine, Y. (2003). Stochastic differential equations with additive fractional noise and locally unbounded drift. In Stochastic Inequalities and Applications, eds. E. Giné, C. Houdré and D. Nualart. Springer, New York, pp. 353–365.CrossRefGoogle Scholar
Nualart, D. and Răşcanu, A. (2002). Differential equations driven by fractional Brownian motion. Collect. Math. 53, 55–81.Google Scholar
Romito, M. (2018). A simple method for the existence of a density for stochastic evolutions with rough coefficients. Electron. J. Prob. 23, 1–43.CrossRefGoogle Scholar
Ruzmaikina, A. A. (2000). Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100, 1049–1069.CrossRefGoogle Scholar
Suo, Y. and Yuan, C. Weak convergence of path-dependent SDEs driven by fractional Brownian motion with irregular coefficients. Preprint, arXiv:1907.02293.Google Scholar
Veretennikov, A. J. (1981). On strong solutions and explicit formulas for solutions of stochastic integral equations. Sbornik: Math. 39, 387–403.CrossRefGoogle Scholar
Veretennikov, A. J. (1984). On stochastic equations with degenerate diffusion with respect to some of the variables. Math. USSR-Izvestiya 22, 173–180.CrossRefGoogle Scholar
Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67, 251–282.CrossRefGoogle Scholar
Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Prob. Theory Relat. Fields 111, 333–374.CrossRefGoogle Scholar
Zähle, M. (1999). On the link between fractional and stochastic calculus. In Stochastic Dynamics, eds. H. Crauel and M. Gundlach. Springer, New York, pp. 305–325 CrossRefGoogle Scholar
Zvonkin, A. K. (1974), A transformation of the phase space of a diffusion process that removes the drift. Math. USSR-Sbornik 22, 129–149.CrossRefGoogle Scholar