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Results tagged with wienerprocess
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user 11678
1
vote
Determine $E[W_p W_q W_r]$
Note that $\{W_t \mid t \geq 0\}$ is a martingale. Then, for $0<p<q<r$,
\begin{align*}
E(W_pW_qW_r) &= E\Big( E(W_pW_qW_r \mid \mathcal{F}_q)\Big)\\
&=E\Big(W_pW_q E(W_r \mid \mathcal{F}_q)\Big)\\
&=E …
- 21.3k
3
votes
Accepted
Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$
\begin{align*}
E\Big(W_t^3-3tW_t \mid \mathcal{F}_s\Big) &= E\Big((W_t-W_s+W_s)^3-3t(W_t-W_s+W_s) \mid \mathcal{F}_s\Big) \\
&=E\Big((W_t-W_s)^3+W_s^3+3(W_t-W_s)^2W_s + 3 (W_t-W_s)W_s^2\\
&\qquad \qqu …
- 21.3k
3
votes
Accepted
Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
Note that, for $t>s>0$,
\begin{align*}
X_t-X_s &= \frac{1}{t}\int_0^t udW_u - \frac{1}{s}\int_0^s udW_u\\
&=\frac{1}{t}\bigg(\int_s^t u dW_u + \int_0^s udW_u \bigg)- \frac{1}{s}\int_0^s udW_u\\
&=\fra …
- 21.3k
1
vote
Accepted
Solving a backwards heat equation using stochastic calculus
Based on the form of your equation, we can consider the SDE
\begin{align*}
dX_t = \sigma dW_t,
\end{align*}
where $W$ is a standard Brownian motion. Since, for $0 \leq t \leq T$,
\begin{align*}
X_T …
- 21.3k