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Results tagged with brownian-motion
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user 493556
4
votes
1
answer
251
views
Explicit expression for the expected number of up-crossings of Brownian motion
Let $W$ be a standard Brownian motion starting at $1/2$, i.e. $W_0=1/2$. Set
$$\tau := \inf\big\{t>0: W_t\notin (0,1)\big\}.$$
As $(W_t^2-t)_t$ is a martingale, one has $\mathbb P[W_\tau =0]=1/2 = \ma …
- 1,409
2
votes
0
answers
277
views
Identify two continuous martingales in law as time-changed Brownian motions
Let $W$ be a Brownian motion and $\alpha$ be a progressively measurable process taking values in $\mathbb R_+$. Set $\beta_t:=\max(\alpha_t, 1)$ for all $t\ge 0$. Define respectively $X$, $Y$ by
$$X_t …
- 1,409