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

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

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)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

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

Keywords:

stochastic differential equation; Wiener process; fractional Brownian motion

Citations:

Zbl 1018.60057; Zbl 0983.60057