On some integral estimates for solutions of stochastic equations driven by symmetric stable processes

Let X be a solution of stochastic differential equation dX_t = b(X_t-)dZ_t + γ|b|α(X_t)dt, t ≥ 0 where Z is a one-dimensional symmetric stable process of index 0 < α ≤ 2 and let τ_m(X) = inf[t ≥ 0: |X_t| ≥ m], m ∈ ℤ₊. We prove various Lp-estimates for processes X for p = 1, 2. In particular, it is shown that, if γ ≠ 0 and 0 < α ≤ 2, then for all t > 0 and a measurable function f: ℝ → [0,∞], it holds E ∫₀ ^tΛτm(X)|b|α(X_s)f(X_s)ds ≤ N||f||_2,m where ||f||_2,m is the L₂-norm of the function f on the interval [-m,m] and the constant N depends on α,m, γ, and t only. For γ = 0 and 1/2 < αa ≤ 2, similar L_p-estimates with p = 1, 2 are proven. As an application, we use obtained estimates to prove the existence of (weak) solutions for corresponding stochastic differential equations with γ ≠ 0 and γ = 0.