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Results tagged with brownian-motion
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user 29212
In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.
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Write expectation of brownian motion conditional on filtration as an integral?
Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is
$f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\i …
- 14.8k
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Proof that integral of Brownian motion wrt time is not a martingale
Let $X_t=\int_0^t W_s ds$ where $W_s$ is Brownian motion, so $E[W_s]=0$.
Then $E[X_t]=\int_0^t E[W_s] ds=\int_0^t 0 ds=0$.
So $E[X_t|{\cal F}_s]=0\neq X_s$, almost everywhere. So by previous senten …
- 187