I have come across a stochastic process on which I have two questions.
Let $W(r)$ and $B(r)$ denote two independent Wiener processes, $r \in [0,1]$, and let $-\infty<x<\infty$.
Define the process $P(x)$, such that
- $P(0)=0$;
- $P(x)=\int_0^1W^2(r)dB(xr)$ for each $x$
- $\operatorname{Cov}(P(x),P(y))=(\min\{x,y\})^{1/2}\int_0^1W^2(r)dB(r)$
- $P(x)$ and $P(x+\delta)-P(x)$ are independent conditional on the sigma-field associated with $W(r)$.
My first question is: has this process been studied already, and does it have continuous sample paths? It is a mixture Gaussian process, so far as I can understand, which is why I have studied the four points above only.
The process $P(x)$ is part of the following functional, which I am studying in the context of an M-estimator - in essence, it characterises the limit of the loss function whose minimum is the estimator I am studying:
$-|x|\int^1_0W^2(r)dr+P(x)$
defined, as mentioned above, over $-\infty<x<\infty$.
My second question is: does the functional above have a unique maximum? I am trying to use the arg-max continuous mapping theorem, hence my question.