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2 Fixed several trivial errors

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)} $W^{(1)}$ and W^{(2)} $W^{(2)}$ 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?

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# Time integrals of diffusion processes

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?