On the small deviation problem for some iterated processes

Abstract

We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for n-iterated Brownian motions and, more generally, for the iteration of n fractional Brownian motions. We also give a new and correct proof of some results in [21].• to show how the technique can be modified if Y has jumps. This is illustrated by several examples, among them the α-time Brownian motion, previously studied in [21]. Here, we give a correct proof of (a weaker version of) the results from [21].Small deviation problems (also called small ball problems or lower tail probability problems) were studied intensively during recent years, which is due to many connections to other subjects such as the functional law of the iterated logarithm of Chung type, strong limit laws in statistics, metric entropy properties of linear operators, quantization, and several other approximation quantities for stochastic processes. For a detailed account, we refer to the surveys [15] and [13] and to the literature compilation [16].The interest in iterated processes, in particular iterated Brownian motion, started with the works of Burdzy (cf.[6] and [7]). Iterated processes have interesting connections to higher order PDEs, cf. [1] and [22] for some recent results. Small deviations of iterated processes or the corresponding result for the law of the iterated logarithm are treated in [9] (X and Y Brownian motions), [10] (X Brownian motion, Y = |Y ′ | with Y ′ being Brownian motion), [21] (see Section 5 below), [18] (X fractional Brownian motion, Y a subordinator), and, most recently, [19] (X fractional Brownian motion, Y a subordinator, and the sup-norm is taken over a possibly fractal index set).In Section 2, we give general results under the assumption that the small deviation probabilities of X and Y , respectively, are known to some extent and that Y has a continuous modification. The proofs for these results are given in Section 3 and the results are illustrated with several examples in Section 4. In Section 5, we treat examples where Y has jumps, in particular, the so-called α-time Brownian motion, studied earlier in [21]. Finally, we mention some possible extensions and applications of our results and collect some open questions in Section 6. General resultsBefore we formulate our main results, let us define some notation. We write f g or g f if lim sup f /g < ∞, while the equivalence f ≈ g means that we have both f g and g f . Moreover, f g or g f say that lim sup f /g ≤ 1. Finally, the strong equivalence f ∼ g means that lim f /g = 1.We say that a process X is H-self-similar if (X(ct)) d = (c H X(t)) for all c > 0, where d = means that the finite-dimensional distributions coincide. Recall that, for example, fractional Brownian motion with Hurst parameter H is H-self-similar. However, there are many interesting self-similar processes outside the Gaussian framework, e.g. a strictly α-stable Lévy process is 1/α-self-similar ([24], [8], [23]). Let ...

On the asymptotics of supremum distribution for some iterated processes

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