Systems of equations driven by stable processes |
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Authors: | Richard F. Bass Zhen-Qing Chen |
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Affiliation: | (1) Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA;(2) Department of Mathematics, University of Washington, 354350 Seattle, WA 98195-4350, USA |
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Abstract: | Let Zjt, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations where the matrix A(x)=(Aij(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∈ ℝd with |a|, |b| ∈ (λ1, λ2) and p sufficiently large, the Lp-norm of the operator whose Fourier multiplier is (|u · a|α - |u · b|α) / ∑j=1d |ui|α is bounded by a constant multiple of |a−b|θ for some θ > 0, where u=(u1 , . . . , ud) ∈ ℝd. We deduce from this the Lp-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|α / ∑j=1d |ui|α, where u=(u1,...,ud) and a is a continuous function on ℝd with |a(x)|∈ (λ1, λ2) for all x∈ ℝd. Research partially supported by NSF grant DMS-0244737. Research partially supported by NSF grant DMS-0303310. |
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Keywords: | Primary: 60H10 Secondary: 60G52 60J75 42B20 35S05 |
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