On the Lyapunov Exponent of a Multidimensional Stochastic Flow

Michele Baldini ()
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Michele Baldini: Merrill Lynch, Global Equity Linked Products

Journal of Theoretical Probability, 2007, vol. 20, issue 2, 327-337

Abstract: Abstract Let X t be a reversible and positive recurrent diffusion in ℝd described by $$X_{t}=x+\sigma\,b(t)+\int_{0}^{t}m(X_{s})\mathrm {d}s,$$ where the diffusion coefficient σ is a positive-definite matrix and the drift m is a smooth function. Let X t (A) denote the image of a compact set A⊂ℝ d under the stochastic flow generated by X t . If the divergence of the drift is strictly negative, there exists a set of functions u such that $$\lim_{t\to\infty}\int_{\ensuremath {X_{t}}(A)}\ensuremath {u}(x)\mathrm {d}x=0\quad\mbox{a.s.}$$ A characterization of the functions u is provided, as well as lower and upper bounds for the exponential rate of convergence.

Keywords: Diffusion; Stochastic flow; Recurrence; Superharmonic function; Elliptic operator (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10959-007-0071-4

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