Stochastic Differential Equations with Boundary Conditions

The stochastic calculus with anticipating integrands has been recently developed by several authors (see in particular [5,6,9] and the references therein). This new theory allows to study different types of stochastic differential equations driven by a d—dimensional Brownian motion {W(t), 0 ≤ t≤ 1}, where the solutions turn out to be non necessarily adapted to the filtration generated by W. We refer the reader to [12] for a survey of the applications of the anticipating stochastic calculus to stochastic differential equations. In particular one can consider stochastic differential equations of the form $$\matrix{ {d{X_t} = f\left( {{X_t}} \right) + \sum\limits_{i = 1}^k {{g_i}} \left( {{X_t}} \right)odW_t^i,} & {0 \le t \le 1,} \cr }$$ and, instead of giving the value of the process at time zero, we impose a boundary condition of the form h(X₀, X₁)= h₀. In general, the solution {X _t , 0 ≤ t≤ 1} will not be an adapted process, and the stochastic integral $$f_0^t{g_i}\left( {{X_s}} \right)o\,dW_s^i$$ is taken in the extended Stratonovich sense. The existence and uniqueness of a solution for an equation of this type has been investigated in some particular cases, and the Markov property of the solution has been studied. More precisely the following particular situations have been considered: (a)The functions f, g and h are affine. (Ocone-Pardoux [11])(b)k = d and the function g is a constant equal to the identity matrix. (Nualart—Pardoux [7])(c)k = d = 1, and h is linear. (Donati—Martin [2])(d)Second order stochastic differential equations in dimension one of the following type $$\matrix{ {{{\ddot X}_t} + f\left( {{X_t},{{\dot X}_t}} \right) = {{\dot W}_t},} & {0 \le t \le 1,} \cr } $$ with Dirichlet boundary conditions X₀ = a, X₁ = b. (Nualart—Pardoux [8)).