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Results tagged with stochastic-processes
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user 340165
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
1
vote
Accepted
How to understand the transition density of reflected Brownian motion
The solid line in the figure that represents $p(t;-x,y)+p(t;x,y)$ looks qualitatively OK but quantitatively wrong.
Let $y=|x_0|$ and $E\subset[0,\infty)\,.$ Then $p_t(y;E)=P(|x_t|\in E)=P(x_t\in E)+ …
- 233
2
votes
Accepted
Where does the extra term in the density of a diffusion with respect to $c B(t)$ come from?
In a nutshell you are asking the following:
When $F(x)=\int_0^xf(y)\,dy$ we get from Ito
$$
F(cB(t))=\int_0^Tf(cB(t))\,d(cB(t))+\frac{1}{2}\int_0^Tf'(cB(t))\,c^2\,dt\,
$$
so that the expressions
\begi …
- 233
1
vote
Probability that a geometric Brownian motion with additional determinstic drift ever hits zero
Clearly, the component $Y_t=e^{\mu t +\sigma W_t-\sigma^2 t/2}$ of the explicit solution never hits zero. This boils down the problem to the question if
$$
Z_t:=\int_0^t\frac{Cke^{-ks}}{Y_s}\,ds
$$
ev …
- 233
1
vote
Correlated Brownian motions across different times and representation with independent proce...
Take $W^1,W^2$ to be two independent Brownian motions and let $\varrho(t)$ be a function with values in $[-1,1]$. Set
$$
B^1_t=W^1_t\quad\text{ and }\quad B^2_t=\int_0^t\varrho(s)\,dW^1_s+\int_0^t\sqr …
- 233