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Convergent martingales of asymptotically minimal fluctuation
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  • Published: August 1987

Convergent martingales of asymptotically minimal fluctuation

  • J. M. C. Clark1 

Probability Theory and Related Fields volume 75, pages 531–543 (1987)Cite this article

Summary

This paper establishes that certain martingales converge to their limits as fast as any other similarly adapted process, not just in a least squares sense, but also in a strong “almost sure” sense. In particular, a Brownian motion (W t), stopped at a type of “differentiably predictable” stopping time T, possesses a minimal fluctuation property of the following sort: for any process (Y t) adapted to the filtration generated by (W t),

$$\overline {\mathop {lim}\limits_{h \downarrow 0} } [(2h log log h^{ - 1} )^{ - 1/2} |W_T - Y_{T - h} |] \underline \geqslant 1$$

a.s., with equality if Y t=Wt. A random time-change argument extends this result to a class of continuous martingales; for instance, if M t is an Ito integral \(\int\limits_0^t {g_s d W_s }\) and Y t is a similarly adapted process, then

$$\overline {\mathop {\lim }\limits_{t \uparrow 1} } [(2(1 - t) log log(1 - t)^{ - 1} )^{ - 1/2} |M_1 - Y_t |] \underline \geqslant |g_1 |$$

a.s. with equality if Y t=Mt, provided that g t2 is continuous and positive at t=1. Finally, discrete parameter martingales, obeying conditions required in Heyde's law of the iterated logarithm for the tails of convergent martingales possess a minimal fluctuation property that can be interpreted as showing that Heyde's law gives a ‘best’ rate of convergence.

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References

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Authors and Affiliations

  1. Department of Electrical Engineering, Imperial College, SW7 2BT, London, England

    J. M. C. Clark

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  1. J. M. C. Clark
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Clark, J.M.C. Convergent martingales of asymptotically minimal fluctuation. Probab. Th. Rel. Fields 75, 531–543 (1987). https://doi.org/10.1007/BF00320332

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  • Received: 22 October 1986

  • Issue Date: August 1987

  • DOI: https://doi.org/10.1007/BF00320332

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Keywords

  • Filtration
  • Stochastic Process
  • Brownian Motion
  • Probability Theory
  • Mathematical Biology
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