Systems of equations driven by stable processes

Let Z(t)(j), j = 1,..., d, be independent one-dimensional symmetric stable processes of index alpha is an element of (0, 2). We consider the system of stochastic differential equations dX(t)(i) = Sigma(d)(j=1) A(ij)(Xt-)dZ(t)(j), i = 1,...,d, 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 lambda(2) > lambda(1) > 0. We show that for any two vectors a, b is an element of R-d with |a|, |b| is an element of (lambda(1), lambda(2)) and p sufficiently large, the L-p-norm of the operator whose Fourier multiplier is (| u center dot a|(alpha) - | u center dot b|(alpha))/Sigma(d)(j=1) |u(i) |(alpha) is bounded by a constant multiple of |a - b|(theta) for some theta > 0, where u = (u(1),..., u(d)) is an element of R-d. We deduce from this the L-p-boundedness of pseudodifferential operators with symbols of the form psi(x, u) = | u center dot a(x)|(alpha)/Sigma(d)(j=1)|u(i) |(alpha), where u = (u(1),..., u(d)) and a is a continuous function on R-d with |a(x)| is an element of(lambda(1), lambda(2)) for all x is an element of R-d.