Anticipating Linear Stochastic Differential Equations with Adapted Coefficients

Stochastic differential equations with adapted integrands and initial conditions are well studied within Itô’s theory. However, such a general theory is not known for corresponding equations with anticipation. We use examples to illustrate essential ideas of the Ayed–Kuo integral and techniques for dealing with anticipating stochastic differential equations. We prove the general form of the solution for a class of linear stochastic differential equations with adapted coefficients and anticipating initial condition, which in this case is an analytic function of a Wiener integral. We show that for such equations, the conditional expectation of the solution is not the same as the solution of the corresponding stochastic differential equation with the initial condition as the expectation of the original initial condition. In particular, we show that there is an extra term in the stochastic differential equation, and give the exact form of this term.