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Mixed stochastic differential equations with long-range dependence: existence, uniqueness and convergence of solutions. (English) Zbl 1268.60088
Summary: For a mixed stochastic differential equation involving standard Brownian motion and an almost surely Hölder continuous process $Z$ with Hölder exponent $\gamma >1/2$, we establish a new result on its unique solvability. We also establish an estimate for difference of solutions to such equations with different processes $Z$ and deduce a corresponding limit theorem. As a by-product, we obtain a result on existence of moments of a solution to a mixed equation under an assumption that $Z$ has certain exponential moments.

60H10Stochastic ordinary differential equations
60G22Fractional processes, including fractional Brownian motion
35R60PDEs with randomness, stochastic PDE
Full Text: DOI arXiv
[1] Kubilius, K.: The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type, Stochastic process. Appl. 98, No. 2, 289-315 (2002) · Zbl 1059.60068 · doi:10.1016/S0304-4149(01)00145-4
[2] Mishura, Y.: Stochastic calculus for fractional Brownian motion and related processes, (2008) · Zbl 1138.60006
[3] Guerra, J.; Nualart, D.: Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Stoch. anal. Appl. 26, No. 5, 1053-1075 (2008) · Zbl 1151.60028 · doi:10.1080/07362990802286483
[4] Mishura, Y. S.; Shevchenko, G. M.: Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index H1/2, Comm. statist. Theory methods 40, No. 19--20, 3492-3508 (2011) · Zbl 1315.60071
[5] Mishura, Y. S.; Shevchenko, G. M.: Rate of convergence of Euler approximations of solution to mixed stochastic differential equation involving Brownian motion and fractional Brownian motion, Random oper. Stoch. equ. 20, No. 4, 387-406 (2011) · Zbl 1290.60069
[6] Nualart, D.; Răşcanu, A.: Differential equations driven by fractional Brownian motion, Collect. math. 53, No. 1, 55-81 (2002) · Zbl 1018.60057
[7] Zähle, M.: Integration with respect to fractal functions and stochastic calculus. I, Probab. theory related fields 111, No. 3, 333-374 (1998) · Zbl 0918.60037 · doi:10.1007/s004400050171
[8] Garsia, A. M.; Rodemich, E.: Monotonicity of certain functionals under rearrangement, Ann. inst. Fourier (Grenoble) 24, No. 2, 67-116 (1974) · Zbl 0274.26006 · doi:10.5802/aif.507 · numdam:AIF_1974__24_2_67_0