## 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.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G22 Fractional processes, including fractional Brownian motion 35R60 PDEs with randomness, stochastic partial differential equations
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### References:

 [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, 2, 289-315, (2002) · Zbl 1059.60068 [2] Mishura, Y., Stochastic calculus for fractional Brownian motion and related processes, (2008), Springer Berlin · Zbl 1138.60006 [3] Guerra, J.; Nualart, D., Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Stoch. anal. appl., 26, 5, 1053-1075, (2008) · Zbl 1151.60028 [4] Mishura, Y.S.; Shevchenko, G.M., Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index $$H > 1 / 2$$, Comm. statist. theory methods, 40, 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, 4, 387-406, (2011) · Zbl 1290.60069 [6] Nualart, D.; Răşcanu, A., Differential equations driven by fractional Brownian motion, Collect. math., 53, 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, 3, 333-374, (1998) · Zbl 0918.60037 [8] Garsia, A.M.; Rodemich, E., Monotonicity of certain functionals under rearrangement, Ann. inst. Fourier (Grenoble), 24, 2, 67-116, (1974), vi, colloque International sur les Processus Gaussiens et les Distributions Aléatoires (Colloque Internat. du CNRS, No. 222, Strasbourg, 1973) · Zbl 0274.26006
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