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### 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.
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### 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{...
• 169k
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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 $0<h&... • 15.6k 1 vote ### Geometric Brownian Motion problem - Compute$\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} = ...

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