Stochastic Equations Driven by a Cauchy Process

Abstract

Using the method of Krylov’s estimates, we prove the existence of (weak) solutions of the one-dimensional stochastic equation dX_t=b(X_t−)dZ_t+a(X_t)dt with arbitrary initial value x₀∈ℝ and the driven symmetric Cauchy process Z. The bounded coefficient b is assumed to be of non-degenerate form and the drift a to satisfy the condition |a(x)|≤(1/2)|b(x)| for all x∈ℝ.