Correlated Brownian motions across different times and representation with independent processes

This is a more wide-net question of Two increasingly correlated Brownian motions and Williams decomposition.

In our problem we have two correlated Brownian motions $$B^{1},B^{2}$$ (starting at time $$t=0$$ from zero) with covariance

$$E[B_{t}^{1}B_{w}^{2}]=\left\{\begin{matrix}0,&e^{-t}+e^{-w}\geq 1 \\ \ln\frac{1}{e^{-t}+e^{-w}},& e^{-t}+e^{-w}\leq 1\end{matrix}\right.$$

,correlation $$\rho_{t}=1-\frac{\ln 2}{t}, t\geq \ln 2$$ and zero-otherwise and their difference for $$t\geq ln2$$ is

$$E[(B_{t}^{1}-B_{t}^{2})^{2}]=2\ln 2.$$

So as $$t\to +\infty$$, the correlation goes to one.

Q: Is there a way to express the correlated $$(B^{1},B^{2})$$ in terms of some concrete function of independent $$(W^{1},W^{2})$$ (not necessarily Brownian motions)?

In the case of Martingale pair $$(B^1_{t},B^{2}_{t})$$ with some cross-variation $$=H_{t}$$, then in Revuz-Yor V.thm.39 and "Correlated Coalescing Brownian Flows on R and the Circle" have a nice formula in terms of independent BMs $$X=(X^1,X^2)$$

$$(B^1_{t},B^{2}_{t})=\int_{0}^{t}a_{r} dX_{r}$$ for some matrix $$a_{r}$$ containing $$H'_{r}$$.

In the above problem, the pair has correlations across different times and so we don't have Martingale structure.

So we will need some nonlinear functiona eg. "Large Deviations For Sticky Brownian Motions" they study Brownian motions interacting in non-linear ways.

Q2: Also, can you suggest in the literature some cases of Brownian motions correlated for different t,s? That way I can try to transfer their techniques to the above problem. So far I only came across articles studying BMs correlated for same $$t=s$$: $$E[B^1_{t}B^{2}_t]=\rho_{t}$$.

Take $$W^1,W^2$$ to be two independent Brownian motions and let $$\varrho(t)$$ be a function with values in $$[-1,1]$$. Set $$B^1_t=W^1_t\quad\text{ and }\quad B^2_t=\int_0^t\varrho(s)\,dW^1_s+\int_0^t\sqrt{1-\varrho^2(s)}\,dW^2_s\,.$$
• $$B^2$$ is a Brownian motion because it is a continuous martingale with quadratic variation $$t\,.$$
• The correlation between $$B^1$$ and $$B^2$$ is $$\mathbb E[B^1_tB^2_w]=\mathbb E[B^1_tB^2_t]=\int_0^t\varrho(s)\,ds\text{ for }w\ge t\,.$$
Apparently you are looking for two Brownian motions were this correlation depends on $$t$$ and $$w$$. When they share a common filtration I think this is not possible because $$B^2_w-B^2_t$$ should then be independent of $$B^1_t$$. Therefore, $$\mathbb E[B^1_tB^2_w]=\underbrace{\mathbb E[B^1_t(B^2_w-B^2_t)]}_{=0}+\mathbb E[B^1_tB^2_t]$$ must always hold, i.e., the correlation can only depend on the minimum of $$t$$ and $$w$$.