Tagged Questions

13 views

Showing $E[e^{-\lambda \tau_{a}\wedge\tau_{-a}}]=sech(a\sqrt{2\lambda})$

This is homework so no answers please. For $\tau_{a}=inf_{t}(B_{t}=a)$ , we already know $E[e^{-\lambda \tau_{a}}]=e^{-\sqrt{2\lambda}a}$. By $B_{t}$ I mean Brownian motion. The question is to show: ...
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weiner process expected max given current state? [on hold]

Suppose we have weiner process over interval t=0 to t=1, with std. 1 over interval and mean 0. If time now is r where r on interval 0 to 1 and x is current value of process and m is current maximum ...
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mean hitting time of a level and growth rate of maximum process

Let $X_t$ be the absolute value of Brownian motion starting at $0$, let $\tau_x$ be it's first hitting time of the level $x>0$, and let $M_t$ be it's running maximum up to time $t$. Suppose we knew ...
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Probability Brownian Motion doesn't hit a point in the limit.

This is a question from Revuz and Yor (exercise 3.18) for which I seem to get a different answer. Show that $\lim_{t \to \infty}\,t^{1/2}\,\mathbb{P}\{B_s\leq1\,\forall\, s\in[0,t]\}=\sqrt{2/\pi}$. ...
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two brownian motions in $\mathbb{Z}^2$

I was wondering what is the probability for 2 brownian walkers coming from 2 different initial positions to be at the same position at time t. I consider that at each step, each point can ...
57 views

Probability that the value at time T from one geometric Brownian motion process is greater than the value from another GBM

I am having a competition between $n$ people (starts at time $t$=0), each who accumulates points on a daily basis, which I assume is a geometric Brownian motion process with parameters $\mu_i$, ...
Let $B=\{B_t:t\geq 0\}$ be a standard Brownian motion. Define the Brownian brige $X=\{X_t:t\geq0\}$ as $$X_t=B_t-tB_1\quad t\in[0,1]$$ Show that $X$ is (i) Gaussian and find its (ii) mean and (iii) ...
While reading a paper related to functional of brownian motion I came across the following notation $1(B_t \in dx)$, where $1(A)$ is the indicator function of the set A, and $B_t$ is a standard ...