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Systems of equations driven by stable processes
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  • Published: 10 February 2005

Systems of equations driven by stable processes

  • Richard F. Bass1 &
  • Zhen-Qing Chen2 

Probability Theory and Related Fields volume 134, pages 175–214 (2006)Cite this article

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Abstract

Let Zj t , 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)=(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 Lp-norm of the operator whose Fourier multiplier is (|u · a|α - |u · b|α) / ∑ 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 ) ∈ ℝd. We deduce from this the Lp-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|α / ∑ j=1 d |u i |α, where u=(u1,...,u d ) and a is a continuous function on ℝd with |a(x)|∈ (λ1, λ2) for all x∈ ℝd.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

    Richard F. Bass

  2. Department of Mathematics, University of Washington, 354350, Seattle, WA 98195-4350, USA

    Zhen-Qing Chen

Authors
  1. Richard F. Bass
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  2. Zhen-Qing Chen
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Corresponding author

Correspondence to Richard F. Bass.

Additional information

Research partially supported by NSF grant DMS-0244737.

Research partially supported by NSF grant DMS-0303310.

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Cite this article

Bass, R., Chen, ZQ. Systems of equations driven by stable processes. Probab. Theory Relat. Fields 134, 175–214 (2006). https://doi.org/10.1007/s00440-004-0426-z

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  • Received: 16 October 2003

  • Revised: 08 July 2004

  • Published: 10 February 2005

  • Issue Date: February 2006

  • DOI: https://doi.org/10.1007/s00440-004-0426-z

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Mathematics Subject Classifications (2000)

  • Primary: 60H10
  • Secondary: 60G52
  • 60J75
  • 42B20
  • 35S05

Key words or phrases

  • Stable processes
  • Stochastic differential equations
  • Martingale problem
  • Weak solution
  • Weak uniqueness
  • Pseudodifferential operators
  • Method of rotations
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