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Tagged with normal-distribution stochastic-calculus
5 questions
7
votes
1
answer
598
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Show that $(W_t, \int_0^t W_s ds)$ has a normal joint distribution
I have to show that, if $W_t$ is a 1-d Brownian motion then
$\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution.
Hint: apply Ito formula to this bivariate process.
Any idea or suggestion on ...
- 73
2
votes
1
answer
380
views
Steven Shreve: Stochastic Calculus and Finance
The lecture notes have the following theorem:
Let $\theta\in \mathbb{R}$ be given and $B(t)$ stands for the Brownian motion which is a martingale, then $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
- 183
2
votes
1
answer
220
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Showing that the shortfall-to-quantile ratio of a normal distribution goes to one
I dont get why $$\lim_{x \to \infty}
\frac{\mu \{1 - \Phi(x)\} + \sigma \phi(x)}{(\mu + \sigma x) \{1 - \Phi(x)\} }
= \lim_{x \to \infty}
\frac{1}{1 - \sigma \frac{1 - \Phi(x)}{(\mu + \...
- 123
2
votes
1
answer
2k
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Show that the Ito integral is Gaussian
Let $f(t), 0 \leq t \leq T$ be a deterministic function with $f(t) = \sum_{i=1}^na_{i-1}1_[t_{i=1}, t_i)(t)$ with $0 \leq t_0<t_1<...<t_{n-1} = T$. Show that the stochastic integral $I_t(f) ...
- 121
0
votes
1
answer
303
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Necessary conditions to ensure that stochastic integral is a normal variable
Let $\left(W_t\right)_{t\geq 0}$ be a Brownian motion with respect to filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\geq 0}$. Let $\left(\alpha_t\right)_{t\geq 0}$ be an $\mathbb{F}$-adapted ...
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