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Mishura, Yu. S.; Posashkova, S. V.; Posashkov, S. V.

Continuous dependence of solutions of stochastic differential equations driven by standard and fractional Brownian motion on a parameter. (English. Russian original) Zbl 1254.60060

Theory Probab. Math. Stat. 83, 111-126 (2011); translation from Teor. Jmovirn. Mat. Stat. 83, 92-105 (2010).
A one-dimensional stochastic differential equation driven by both a standard Brownian motion and a fractional Brownian motion with Hurst parameter \(H\in(1/2,1)\) is considered. The coefficients of the equation are assumed to be nonhomogeneous. The coefficients as well as the random initial condition \(X_0^u\) depend on a certain parameter \(u\in[0,u_0]\). Assuming that \(X_0^u\) converges in probability to \(X_0^0\), conditions on the coefficients as functions of the parameter are found under which the solutions \(\{X_t^u,t\in[0,T]\}\) converge to \(\{X_t^0,t\in[0,T]\}\) uniformly in probability as \(u\rightarrow 0\).
Reviewer: Dominique Lepingle (Orléans)

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G22 Fractional processes, including fractional Brownian motion
60J65 Brownian motion

Keywords:

fractional Brownian motion; standard Brownian motion; stochastic differential equation
PDFBibTeX XMLCite
\textit{Yu. S. Mishura} et al., Theory Probab. Math. Stat. 83, 111--126 (2011; Zbl 1254.60060); translation from Teor. Jmovirn. Mat. Stat. 83, 92--105 (2010)
Full Text: DOI

References:

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