Posashkov S. Mixed Brownian-fractional Brownian models and their applications in filtration theory and financial mathematics

The thesis is devoted to mixed Brownian-fractional Brownian models. Existence and uniqueness are proved for the solution of stochastic differential equation driven by mixture of standard and fractional Brownian motions. Filtration problem is considered in system with signal process driven by mixture of sBm and fBm. The equation is obtained for optimal filter of signal process with the help of observation process in general case. The particular attention is put to linear case: the closed system of equations is obtained for conditional mean of optimal filter and for variation of filtration error. The absence of arbitrage and incompleteness are proved for $(B,S)$-market with volatility driven by either fractional Brownian motion or mixture of fractional and standard Brownian motions. The formula for price of European contingent claim is obtained using minimal martingal measure. The partial differential equation is obtained for European contingent claim price in case when volatility is driven by mixtureof sBm and fBm.