By Brewer M. J.

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4 is devoted to the characterization of all such martingales; in particular, we show that Xγ = 0 if and only if E [X|Fγ ] = 0. 6, we investigate how, in general, the quantities Xγ and E [X|Fγ ] diﬀer. 1 Some Quantities Associated with γ Most of the results in this section may be read from Tables 1α and 2α; however, the following discussion is essentially self-contained. 1 This is diﬀerent from, but related to, Williams’ path decomposition [Wil74] before and after γT1 = sup {t < T1 ; Bt = 0} R. Mansuy and M.

We recall, using our terminology, that the classical BDG inequalities assert √ (1) (1) that (At = sups≤t |Bs |, t ≥ 0) and (Ct = t, t ≥ 0) are momentequivalent. (2) (2) a) Are (At = sups≤t Bs ; t ≥ 0) and (Ct = sups≤t |Bs |, t ≥ 0) momentequivalent? (3) (3) b) Are (At = Lt ; t ≥ 0), the local time of B at 0, and (Ct = sups≤t |Bs |, t ≥ 0) moment-equivalent? 10) 0 (n) with (βt ; t ≥ 0) a standard Brownian motion. Note that (nρt ; t ≥ 0) is a squared Bessel process of dimension n. e. 2 This exercise is closely related to the so-called Poincar´e lemma; see [DF87] and [Str93] for some historical comments about it.

A) Prove that l P (∀t ≤ τl , Rt ≤ ϕ(Lt )) = exp − dλ n max εu ≥ ϕ(λ) 0 u where ε, under the Itˆ o measure n of excursions of R away from 0, denotes the generic excursion and τ is the right-inverse of the local time process L. b) Show that n (maxu εu ≥ a) = a−2ν . 1). Hint: One may use the martingale property of (h(Lt )Rt2ν +1−H(Lt ), t ≥ 0) for a conveniently chosen measurable, positive function h such that ∞ x h(u)du = 1 and H(x) = 0 h(y)dy. 5 This result can also be deduced from the result of Exercise 14 about reﬂecting Brownian motion.

### A Bayesian model for local smoothing in kernel density estimation by Brewer M. J.

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