## Existence and uniqueness of the solution of a stochastic differential equation, driven by fractional Brownian motion with a stabilizing term.(Ukrainian, English)Zbl 1199.60121

Teor. Jmovirn. Mat. Stat. 76, 117-124 (2007); translation in Theory Probab. Math. Stat. 76, 131-139 (2008).
The authors deal with the stochastic differential equation driven by a Wiener process and fractional Brownian motion $X_t=X_0+ \int_0^ta(s,X_s)ds+ \int_0^tb(s,X_s)dW_s+ \int_0^tc(s,X_s)dB_s^H+ \varepsilon\int_0^tc(s,X_s)dV_s,\,t\in[0,T],$ where $$a,b,c:[0,T]\times\mathbb R\to\mathbb R$$ are measurable functions, $$V$$ and $$W$$ are independent standard Brownian motions, $$\varepsilon>0$$, and $$B^H$$ is a fractional Brownian motion independent of $$W$$ and $$V$$ with the Hurst parameter $$H\in(3/4,1]$$. It is assumed that the coefficients of this equation are such that all stochastic integrals on the right hand side of are well defined. The integral $$\varepsilon\int_0^tc(s,X_s)dV_s$$ is considered as a stabilizing term. This term allows to prove the existence and uniqueness of the solution adapted to the flow $$\mathcal F_t,t\geq0$$, where $$\mathcal F=\sigma\{X_0,W_s,( \varepsilon V_s+B_s^H)| s\in[0,t]\}$$. The case of $$b=0$$ and $$\varepsilon = 0$$ corresponds to the equation containing a fractional Brownian motion only (an equation without a standard Brownian motion). Conditions for the existence and uniqueness of a solution for the latter equation are considered in the paper by D. Nualart and A. Răşcanu [Collect. Math. 53, No. 1, 55–81 (2002; Zbl 1018.60057)]. Conditions for the existence and uniqueness of a solution of a semilinear equation with $$\varepsilon = 0$$, $$b(s,x)=bx$$ and $$c(s,x)=cx$$ are obtained by Y. Krvavych and Y. Mishura [in: Mathematical finance. Workshop of the mathematical finance research project, Basel: Birkhäuser, 230–238 (2001; Zbl 0983.60057)].

### MSC:

 60G15 Gaussian processes 60H05 Stochastic integrals 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

### Citations:

Zbl 1018.60057; Zbl 0983.60057
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