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Baldi, Paolo

Large deviations and functional iterated logarithm law for diffusion processes. (English) Zbl 0587.60074

Probab. Theory Relat. Fields 71, 435-453 (1986).
Let y be the diffusion process on \(U\subset R^ m\) satisfying the equation \(dy_ t=b(y_ t)+\sigma (y_ t)dB_ t,\) \(y_ 0=x\), where \(B_ t\) is the Brownian motion in \(R^ k\), \(\sigma\) is an \(m\times k\) matrix field and b is a vector field on U. Let for every \(u>0\) \(z_ u(t)=\Gamma_{\sqrt{u \log \log u}}(y_{ut})\), \(0\leq t\leq 1\), where \(\Gamma_{\sqrt{u \log \log u}}(\cdot)\) is a certain family of contractions from U into U \((\Gamma_{\alpha}y=\alpha^{-1}y\) is a special case). Using a large deviation principle, it is shown that under some assumptions \(\{z_ u\}\) is relatively compact a.s. as \(u\to \infty\) in the set of continuous functions \(\{f: [0,1]\to U| \quad f(0)=x\}\) with the uniform convergence, and the limit set of \(\{z_ u\}\) is explicitly given. The above is a generalization of Strassen’s law of the iterated logarithm for Brownian motion. Applications for iterated integrals and diffusions on Lie groups are discussed.
Reviewer: A.Korzeniowski

Cited in 7 Reviews
Cited in 26 Documents

MSC:

60J60 Diffusion processes
60F10 Large deviations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F15 Strong limit theorems

Keywords:

Brownian motion; family of contractions; Strassen’s law of the iterated logarithm; iterated integrals; diffusions on Lie groups
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\textit{P. Baldi}, Probab. Theory Relat. Fields 71, 435--453 (1986; Zbl 0587.60074)
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References:

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