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For a standard Brownian motion $W_t$ and a given crossing probability $\alpha < 1$, I want to have a boundary function $f(t) > 0$, such that the probability that $W_t$ ever crosses the boundary is bounded by the given probability, i.e. $$ P(|W_t| > f(t), \exists t > 0) \le \alpha $$ By Law of the iterated logarithm, any $f(t) = O(\sqrt{t\log\log t})$ does not meet this condition because $$ \lim\sup_{n\rightarrow \infty} \frac{W_t}{\sqrt{t\log\log t}} = \sqrt{2} $$ if I get it right. For my case, I would like an $f(t) = o(t)$. In other words, I want to have a function such that $$ \lim_{t\rightarrow \infty}\frac{f(t)}{t} = 0 $$.

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If $f(t)=b+t$, where $b>0$, then $ P(W_t > f(t)$ for some $t > 0)=e^{-2b}$, so if you choose $b=-\log\sqrt{\alpha/2}$, then $$ P(|W_t| > f(t)\hbox{ for some }t > 0)\le 2P(W_t > f(t)\hbox{ for some }t > 0)=2e^{-2b}=\alpha. $$

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  • $\begingroup$ Can you prove that exponential bound? (I believe it, I'm just not sure of the proof.) $\endgroup$ – Ian Jul 6 '16 at 16:42
  • $\begingroup$ Just use the "gambler's ruin" formula for $X_t:=W_t-t$ (a Brownian motion with unit negative drift): $\Bbb P[T_a<T_b]={S(0)-S(a)\over S(b)-S(a)}$, where $T_a:=\inf\{t:X_t=a\}$ is the hitting time of $a$, etc., and $S(x):=e^{2x}$ is a scale function for $X$. In the limit $a\to-\infty$ you get $\Bbb P[T_b<\infty]=\lim_{a\to-\infty}\Bbb P[T_b<T_a]=S(0)/S(b)=e^{-2b}$. $\endgroup$ – John Dawkins Jul 6 '16 at 19:40
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Fix $\alpha \in (0,1)$. Let $M_t := \sup_{s \leq t} W_s$ and $m_t := \inf_{s \leq t} W_s$ be the running maximum and mimimum, respectively. Then

$$\mathbb{P}(\exists t \in [0,n]: |W_t|>c_n) = \mathbb{P}(\{M_n>c_n\} \cup \{m_n < -c_n\})$$

for any constant $c_n>0$ and $n \in \mathbb{N}$. Since both $M_n$ and $m_n$ are real-valued, we can choose $c_n>0$ sufficiently large such that

$$\mathbb{P}(\exists t \in [0,n]: |W_t|>c_n) = \mathbb{P}(\{M_n>c_n\} \cup \{m_n < -c_n\}) \leq (1-\alpha) \alpha^n.$$

If we define

$$f(t) := \sum_{n=1}^{\infty} c_n 1_{(n-1,n]}(t)$$

then

$$\mathbb{P}(\exists t>0: |W_t|>f(t)) \leq \sum_{n \in \mathbb{N}} \mathbb{P}(\exists t \in [0,n]: |W_t|>c_n) \leq (1-\alpha) \sum_{n \in \mathbb{N}} \alpha^n = \alpha$$

which shows that $f$ does the job.

Remarks:

  • The distribution of $M_n$ and $m_n$ is known; in fact, $M_n \sim -m_n \sim |W_n|$ (se e.g. Brownian Motion - An Introduction to Stochastic Processes by Schilling & Partzsch, Chapter 6, for a proof). This allows us to compute bounds for the constants $c_n$.

  • This construction of $f$ works, more generally, for any stochastic process $(X_t)_{t \geq 0}$ with continuous sample paths.

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  • $\begingroup$ While perfectly reasonable, I think without an explicit characterization of or at least estimate for the distribution of $M_n,m_n$, it is not really what the OP was going for. $\endgroup$ – Ian Jul 6 '16 at 16:41
  • $\begingroup$ @Ian The distribution of $M_n$ and $m_n$ is known, that's hardly a problem. (The OP hasn't specified what he's looking for so....) $\endgroup$ – saz Jul 6 '16 at 16:48
  • $\begingroup$ It's known, sure, but is it known to the OP? Perhaps a reference to that effect would be useful, so that the OP can get a feel for the scale of $c_n$. $\endgroup$ – Ian Jul 6 '16 at 16:50
  • $\begingroup$ @Ian I have added a reference. Thanks for your feedback. $\endgroup$ – saz Jul 7 '16 at 5:51

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