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A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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 …
Kurt G.'s user avatar
  • 233
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)+ …
Kurt G.'s user avatar
  • 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 …
Kurt G.'s user avatar
  • 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 …
Kurt G.'s user avatar
  • 233