Systems of equations driven by stable processes

Let Z ^j _t , j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α (0,2). We consider the system of stochastic differential equations [InlineMediaObject not available: see fulltext.] where the matrix A(x)=(A _ij (x))1≤i, j≤d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let λ₂ > λ₁ > 0. We show that for any two vectors a, b ∈ ℝ^d with |a|, |b| (λ₁, λ₂) and p sufficiently large, the L ^p -norm of the operator whose Fourier multiplier is (|u • a| ^α - |u • b| ^α) / Σ_j=1^d |u _i|^α is bounded by a constant multiple of |a-b| ^θ for some θ > 0, where u=(u₁ , . . . , u_d) ∈ ℝ^d. We deduce from this the L ^p -boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u • a(x)| ^α / Σ_j=1 ^d |u_i|^α, where u=(u ₁,...,u _d ) and a is a continuous function on ℝ^d with |a(x)| (λ₁, λ₂) for all x ∈ ℝ^d.