I was wondering if someone could recommend a reference that deals with time integrals of diffusion processes.
Suppose $X$ is an Ito diffusion process with dynamics
$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$. The process I'm interested in is $Y_t = \int_0^t X_s ds$. I haven't seen any treatment of the properties of $Y$ in the better-known texts on stochastic analysis – perhaps someone on MO can help.
I'll give a simple example to try to explain part of the reason I'm interested. Suppose $dX^{(1)} = dW_t^{(1)}$ and $dX^{(2)} = \sigma dW_t^{(2)}$, where $W^{(1)}$ and $W^{(2)}$ are independent Brownian motions. $X^{(1)}$ has quadratic variation $t$ almost surely, and $X^{(2)}$ has quadratic variation $\sigma t$. Thus, for $\sigma \neq 1$ the process laws are not equivalent.
I'm wondering what this implies for the laws of $\int^t X^{(1)}_s ds$ and $\int^t X^{(2)}_s ds$. Intuitively, integration should "hide" the small oscillations of the sample paths. Is it possible that the integrated processes have equivalent laws?
Best Answer
One can adapt the argument used to show that, for a standard Brownian motion $W$, the laws of $W$ and $\sigma W$ on any interval $[0,t]$ with $t > 0$ and $\sigma^2\ne1$ are singular.
For every positive $v$, let $E_v$ denote the space of $C^1$ real valued functions defined on $[0,t]$ such that the quadratic variation of their first derivative on $[0,t]$ exists and equals $v$. Let $X=(X_s)_{0\le s\le t}$ with $X_s=\displaystyle\int_0^sW_u\mathrm{d}u$. Then $[X\in E_{t}]$ and $[\sigma X\in E_{\sigma^2t}]$ are both almost sure events but $E_t$ and $E_{\sigma^2t}$ are disjoint hence the laws of $X$ and $\sigma X$ are singular.