Open AccessIntegration methods of mixed-type stochastic differential equations with fractional Brownian motions
Author(s) -
М. М. Васьковский,
И. В. Качан
Publication year - 2019
Publication title -
vescì nacyânalʹnaj akadèmìì navuk belarusì. seryâ fìzìka-matèmatyčnyh navuk
Language(s) - English
Resource type - Journals
eISSN - 2524-2415
pISSN - 1561-2430
DOI - 10.29235/1561-2430-2019-55-2-135-151
Subject(s) - mathematics , fractional brownian motion , stochastic differential equation , hurst exponent , mathematical analysis , stochastic partial differential equation , brownian motion , geometric brownian motion , differential equation , type (biology) , bernoulli's principle , diffusion process , physics , computer science , ecology , knowledge management , statistics , innovation diffusion , biology , thermodynamics
In the present, article new methods of exact integration of mixed-type stochastic differential equations with standard Brownian motion, fractional Brownian motion with the Hurst exponent H > 1/2 and the drift term have been constructed. Solutions of these equations are understood in integral sense where, in turn, the standard Brownian motion integral is the Ito integral and the fractional Brownian motion integral is the pathwise Young integral. The constucted integration methods can be attributed to two types. The first-type methods are based on reducing the equations to simpler equations, in particular – to the simplest equations and the linear inhomogeneous equations. In the article, necessary and sufficient conditions of reducing the equations applicable to one-dimensional equations have been obtained and the examples particularly covering the stochastic Bernoulli-type equations have been given. The second-type method is based on going to the Stratonovich equation and is applicable to multidimensional equations. In addition to the mentioned integration methods, the analogues of the differential Kolmogorov equation have been obtained for mathematical expectations and the solution probability density, assuming that coefficients of the mixed-type stochastic differential equation generate commuting flows.