The Time Change Method and SDEs with Nonnegative Drift

Abstract

Using the time change method we show how to construct a solution to the stochastic equation dXt = b(Xt–)dZt + a(Xt )dt with a nonnegative drift a provided there exists a solution to the auxililary equation dLt = [a–1/αb](Lt–) + dt where Z, ¯ are two symmetric stable processes of the same index α ∈ (0, 2]. This approach allows us to prove the existence of solutions for both stochastic equations for the values 0 < α < 1 and only measurable coefficients a and b satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.