Boundary value problems for a class of stochastic nonlinear fractional order differential equations

Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation D α u ( t ) = λ √ I β [ σ 2 ( t , u ( t ) ) ] ˙ w ( t ) , 0 < t < 1 with boundary conditions u ( 0 ) = 0 , u ′ ( 0 ) = u ′ ( 1 ) = 0 , where λ > 0 is a level of the noise term, σ : [ 0 , 1 ] × R → R is continuous, ˙ w ( t ) is a generalized derivative of Wiener process (Gaussian white noise), D α is the Riemann-Liouville fractional differential operator of order α ∈ ( 3 , 4 ) and I β , β > 0 is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for α = 2 and β = 0 with u ( 0 ) = u ( 1 ) = 0 is also studied.