Example of a bi dimensional process with Brownian marginals which is not a bi dimensional Brownian Motion

Question

I am looking for an example (if it exists) of a bi-dimensional process say $(W^1_t,W^2_t)$ where the margins are plain uni-dimensional Brownian motions but for which we can assert that it is not a 2 dimensional Brownian motion.

I mean this in the extended sense by allowing to enter the definition of two dimensional BM any processes the margin of which are correlated Brownian motions or otherwise said that there exists a $\rho \in (-1,1)$ such that $<W^1_t,W^2_t>\not= \rho.t$.

So for this it would suffice to prove that $<W^1_t,W^2_t>\not= \rho.t$ for all $\rho \in (-1,1)$ or alternatively that the bivariate process is not a local martingale (by Lévy's characterization). The thing is that I am looking for a constructive example.

Best regards

Answer 1

For a one-dimensional Brownian motion $(W_t)_{t \geq 0}$, the reflected process

$$B_t := \begin{cases} W_t, & t \leq 1, \\ W_1-(W_t-W_1), & t>1 \end{cases} \tag{1}$$

defines a one-dimensional Brownian motion.

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Since $\mathbb{E}(W_s W_t) = \min\{s,t\}$, we have

$$\mathbb{E}(W_t B_t) = \mathbb{E}(W_t^2)=t \quad \text{for all $t \leq 1$}$$

and

$$\mathbb{E}(W_t B_t) =2 \mathbb{E}(W_1 W_t) - \mathbb{E}(W_t^2) = 2-t \quad \text{for all $t >1$}.$$

This means, in particular, that there does not exist $\varrho \in [-1,1]$ such that $\langle W,B \rangle_t = \varrho t$, and so the two-dimensional process $(W_t,B_t)_{t \geq 0}$ is not a two-dimensional Brownian motion (in the sense of the definition given in the OP).

More generally, consider

$$B_t := \int_0^t f(s) \, dW_s$$

for some Borel-measurable function $f:[0,\infty) \to \mathbb{R}$ such that $|f(s)|=1$ for all $s \geq 0$. Then

$$\langle W,B \rangle_t = \int_0^t f(s) \, ds.$$

If we choose $f(s) := 1_{[0,1]}(s)-1_{(1,\infty)}(s)$, then we get the reflected Brownian motion $(1)$.