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.


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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[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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