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2 votes

Sum of hitting times and hitting times of sum of Brownian Motion

Let $\tau_a=\inf\{t:B_t=a\}$, $a>0$. We have $P(\tau_a\leq t)=P(\sup_{s\leq t}B_s\geq a)$. Let $\Phi,\phi$ be respectively the standard normal cdf and pdf. The reflection principle yields $P(\tau_a\...
Snoop's user avatar
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1 vote

Simulate a Brownian motion by exponential time stepping

1. The formula in the paper is correct, and here is a more detailed derivation: If $T$ is an exponential random variable with rate $\alpha$, and if $B_t$ is the standard Wiener process independent of $...
Sangchul Lee's user avatar
1 vote
Accepted

Proving the Conditional Expectation of a Uniformly Distributed Random Variable Given the Sum of Two i.i.d. Uniform Random Variables

$E(X|Z)=E(Y|Z)$ and $E(X|Z)+E(Y|Z)=E(X+Y|Z)=E(Z|Z)=Z$ implies the result.
Letac Gérard's user avatar
1 vote
Accepted

May the sum of Wiener processes be a Wiener process?

Let $B_t$ and $\tilde{B}_t$ be two independent standard Wiener processes, and define $W_t$ and $\tilde{W}_t$ by \begin{align*} W_t &= \frac{1}{2}B_t + \frac{\sqrt{3}}{2} \tilde{B}_t, & \tilde{...
Sangchul Lee's user avatar
1 vote

Showing the stationary distribution of Langevin Diffusion without Fokker Plank (Rosenthal 15.6.9)

The author seems to call for a heuristic argument (as is (15.6.3)), so here is one; all feedback appreciated. By Ex. 15.3.7(b) we would need to show that $\int P_h(z,.)\pi(dz)=\pi(.)$ for all $0<h&...
Snoop's user avatar
  • 15.6k
1 vote

Geometric Brownian Motion problem - Compute $\mathbb{P}(X_3 < 3)$

Whereas $X_{t} \sim GBM(\mu, \sigma^2)$, another way to solve it is to remember that $$ X_{t+s} = X_{t} \ e^{\left( \mu-\frac{\sigma^2}{2} \right)s \ + \ \sigma(W_{t+s} - W_{t})}, $$ then $$X_{1+2} = ...
Victor Nunes's user avatar

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