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),
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
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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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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DOI: https://doi.org/10.1007/BF00320332
Keywords
- Filtration
- Stochastic Process
- Brownian Motion
- Probability Theory
- Mathematical Biology