Math Problems Vault

Problem 12.1

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ . Use the Borel - Cantelli lemma to show that the running maximum $M_{n}$ : = $\sup_{0 \leqslant t \leqslant n} B_{t}$ cannot grow faster than $C \sqrt{n \log n}$ for any $C > 2$ . Use this to show that

\varlimsup _{t \rightarrow \infty } \frac{M_{t}}{C \sqrt{t \log t}} \leqslant 1 \quad \text{ a.s. }

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Remark.

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We can obtain the exceptional set uniformly: on $\Omega_{0}: = \bigcap_{\mathbb {Q}\ni C>2}\Omega_{C}$ , $\mathbb {P}\left(\Omega_{0}\right)=1$ and $\forall \omega \in \Omega_{0}: \varlimsup_{n\rightarrow \infty } \frac{M_{n}}{\sqrt{n\log n}} \leqslant 2$ .

Problem 12.2

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ . Apply Doob's maximal inequality (A.13) to the exponential martingale $M_{t}^{\xi }: = \exp \left(\xi B_{t} - \frac{1}{2} \xi^{2} t\right)$ to show that for all $x, \xi > 0$

\mathbb {P}\left(\sup _{s \leqslant t}\left(B_{s} - \frac{1}{2} \xi s\right) > x\right) \leqslant e^{ - x \xi }

(This inequality can be used for the upper bound in the proof of Theorem 12.1, avoiding the combination of (12.2) and the upper bound in (12.1).)

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Problem 12.3

Show that the proof of Khinchine's LIL, Theorem 12.1, can be modified to give

\varlimsup _{t \rightarrow \infty } \frac{\sup _{s \leqslant t}\left|B(s)\right|}{\sqrt{2 t \log \log t}} \leqslant 1

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Remark.

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Dupuis's paper [4] shows LILs for processes $(X_{t})_{t\geqslant 0}$ with stationary independent increments, using estimates of the type $\mathbb {P}\left(X_{t}>x\right)$ . If $\mathbb {P}\left(X_{t}>x\right) \asymp \mathbb {P}\left(\sup_{s\leqslant t}X_{s}>x\right)$ , an LIL for $X_{t}$ holds if, and only if, an LIL for $\sup_{s\leqslant t}X_{s}$ holds.

Problem 12.4

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ . Use Theorem 12.5 to show that $\kappa (t) = (1+\epsilon ) \sqrt{2 t \log \left|\log t\right|}$ is an upper function for $t \rightarrow 0$ .

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Problem 12.5

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ . Deduce from Theorem 12.5 the following test for upper functions in large time. Assume that $\kappa \in \mathcal{C}[1, \infty )$ is a positive function such that $\kappa (t) / t$ is decreasing and $\kappa (t) / \sqrt{t}$ is increasing. Then

\mathbb {P}\left(B(t) < \kappa (t) \text{ as } t \rightarrow \infty \right) = \begin{cases} 0 \\ 1 \end{cases} \text{ according to } \int _{1}^{\infty } \frac{\kappa (s)}{s^{3 / 2}} \exp \left(-\frac{\kappa ^{2}(s)}{2 s}\right) d s \begin{cases} = \infty \\ < \infty \end{cases} .

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Problem 12.6

(a)(b)(c)

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a one - dimensional Brownian motion, $a, b > 0$ and $\tau : = \inf \left\{ t \geqslant 0:\left|B_{t}\right| = b \sqrt{a+t}\right\}$ . Show that

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(a)

$\mathbb {P}\left(\tau < \infty \right) = 1$ ;

(b)

$\mathbb {E} \tau = \infty \quad (b \geqslant 1)$ ;

(c)

$\mathbb {E} \tau < \infty \quad (b < 1)$ .

Problem 12.7

Let $X = \left(X_{t}\right)_{t \geqslant 0}$ be a one - dimensional process satisfying (B0), (B1), (B2). Assume, in addition, that $X$ is continuous in probability, i.e. $\lim_{t \rightarrow 0} \mathbb {P}\left(\left|X_{t} - X_{0}\right| > \epsilon \right) = 0$ for all $\epsilon > 0$ . A function $g:[0, \infty ) \rightarrow [0, \infty )$ is said to be of regular growth, if there exist functions $\kappa_{j}(\lambda ), j = 1,2$ , such that $\lim_{\lambda \rightarrow 1} \kappa_{2}(\lambda ) = 1, \lim_{\lambda \rightarrow \infty } \kappa_{1}(\lambda ) = \infty$ and

\kappa _{1}(\lambda ) g(t) \leqslant g(\lambda t) \leqslant \kappa _{2}(\lambda ) g(t) \text{ for all } \lambda \geqslant 0, t > 0 \text{. }

Show that for every symmetric (i.e. $X_{t} \sim - X_{t}$ ) process $X$ and increasing $g$ of regular growth

\int _{1}^{\infty } \frac{1}{t} \mathbb {P}\left(X_{t} > \frac{1}{3} g(t)\right) d t < \infty \Longrightarrow g \text{ is an upper function as } t \rightarrow \infty .

Remark. $X$ is a so - called Lévy process. Its paths $t \mapsto X_{t}$ are not continuous, but they may have jump discontinuities which happen, however, never at fixed times. This is due to the continuity "in probability". See [231] and [235, Chapter X] for more on this topic.

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Remark.

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With minor variations, this proof also works for $\left|X(t)\right|$ ; the main change needed is in the step using $2\mathbb {P}\left(X(t)-X(a^{k})\geqslant 0\right)\geqslant 1$ , which must be replaced by an Ottaviani--Skorokhod-type bound — see [9, p.295] for details.