Abstract
We consider controlled random walks that are martingales with uniformly bounded increments and nontrivial jump probabilities and show that such walks can be constructed so that $P(S_n^u=0)$ decays at polynomial rate $n^{-\alpha}$ where $\alpha>0$ can be arbitrarily small. We also show, by means of a general delocalization lemma for martingales, which is of independent interest, that slower than polynomial decay is not possible.
Highlights
Introduction and statement of resultsConsider a discrete time martingale {Mi}i≥0 adapted to a filtration Fi whose increments are uniformly bounded by 1, i.e. |Mi+1 − Mi| ≤ 1, and such thatP (|Mi+1 − Mi| = 1 | Fi) > c > 0.It is folklore that in many respects, such a martingale should be well approximated by Brownian motion
By means of a general delocalization lemma for martingales, which is of independent interest, that slower than polynomial decay is not possible
Charlie Smart kindly pointed out [11] that the continuous time results in [2] and [4] concerning the viscosity solution of certain optimal control problems could be adapted to the discrete time setting in order to show an integrated version of Corollary 1.1, namely that for any fixed β, γ > 0 a martingale {Mi} as in the corollary could be constructed so that for all δ small
Summary
Consider a discrete time martingale {Mi}i≥0 adapted to a filtration Fi whose increments are uniformly bounded by 1, i.e. |Mi+1 − Mi| ≤ 1, and such that. Our goal in this paper is to point out that this naive expectation is completely wrong We will frame this in the language of controlled processes below, but a corollary of our main result, Theorem 1.3 below, is the following. Charlie Smart kindly pointed out [11] that the continuous time results in [2] and [4] concerning the viscosity solution of certain optimal control problems could be adapted to the discrete time setting (using [5]) in order to show an integrated version of Corollary 1.1, namely that for any fixed β, γ > 0 a martingale {Mi} as in the corollary could be constructed so that for all δ small,.
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