Existence and uniqueness of solution, change of measure and applications in finance of semilinear stochastic differential equations that contain fractional Brownian motion. (Ukrainian, English) Zbl 1026.60074

Krvavych, Yu. V.

Teor. Jmovirn. Mat. Stat. 65, 79-90 (2001); translation in Theory Probab. Math. Stat. 65, 89-100 (2002).

Let \((\Omega,F,(F_t)_{t\geq 0},P)\) be a complete probability space with filtration \((F_t)_{t\geq 0}\), let \((B^H_t,\displaystyle(F_t)_{t\geq 0},P)\) be a fractional Brownian motion with Hurst parameter \(H\in(1/2,1)\). The author investigates properties of the stochastic differential equation \[ dX_t=cX_t dB_t^H+b(t,X_t) dt, \quad t\geq t_0, \quad X_{t=t_0}=X_0. \] He proposes conditions upon a function \(b(t,x)\) under which the equation has a unique solution. Applications of the considered equations to the analysis of financial markets are discussed.

Reviewer: Mikhail Moklyachuk (Kyïv)

MSC:

60H05	Stochastic integrals
60G17	Sample path properties
60G44	Martingales with continuous parameter
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60G15	Gaussian processes
62P05	Applications of statistics to actuarial sciences and financial mathematics

Keywords:

fractional Brownian motion; stochastic differential equation