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Results tagged with wienerprocess
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user 41821
7
votes
Accepted
Probability Density Function of a Wiener Process Minimum
Firstly, $m_T=\min\limits_{t\in[0,T]} B_t = -\max\limits_{t\in[0,T]} -B_t \overset{Law}{=} -\max\limits_{t\in[0,T]} B_t = -M_T$. So, you can either consider the running maximum or minimum.
Let $\tau$ …
- 16.4k
5
votes
Accepted
Differentiating Wiener process
Let $\text dX_t=\mu_t\text dt+\sigma_t\text dW_t$ be an Itô process. Itô's Lemma tells us $$\text df(t,X_t)=\left(f_t+\mu_tf_x+\frac{1}{2}\sigma_t^2f_{xx}\right)\text dt+\sigma_tf_x\text dW_t.$$
You'r …
- 16.4k
8
votes
Accepted
Proof that $\exp(aW(t)-0.5a^2t)$ is a martingale
Let $(W_t)$ be a standard Brownian motion and $a>0$. We define $X_t=e^{aW_t-\frac{1}{2}a^2t}$. Then, the process $(X_t)$ is adapted and integrable which are the first two conditions of being a marting …
- 16.4k