The Williams decomposition is
Let $(B_t-\nu t)_{t\geq 0}$ be a Brownian motion with negative drift $\nu>0$ and let $M_\infty^{-\nu}:=\sup_{t\in [0,\infty]}(B_t-\nu t)$. Then conditionally on $M_\infty^{-\nu}=M$, the law of the drifted Brownian motion $(B_t-\nu t)_{t\geq 0}$ is given by the joining of two independent conditioned Brownian paths i.e. two Bessel processes.
Now suppose we have two 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
$$E[(B_t^1-B_t^2)^2]=2\ln 2.$$
We see that as $t\to +\infty$, the correlation goes to one.
I am wondering if there is any modification/generalization possible of the above Williams result for increasingly correlated Brownian motions i.e. conditioning on the supremum of the first BM $M_{1,\infty}:=\sup_{t\in [0,\infty]}(B_t^1-\nu t)=m$ and then saying something about the second one $B_t^2-\nu t$.
In particular, I am studying the joint event
$$P \left[ \sup_{t\geq \ln 2} B^1_t-\nu t\geq u_1, \int_{\ln 2}^L e^{B^2_s-\nu s} \, d\mu(s)\leq u_2 \right],$$
(for some deterministic measure $d\mu=f(s) \, ds$ with $f>0$ and fixed constant $L>0$.).
So it would be great if I could condition on $M_{1,\infty}=m$ and then be able to at least upper bound using some process $Y^{2}_{m,s}$ (eg. some modification of Bessel in the spirit of William's result).
$$\leq c_1 \int_{u_1}^\infty P\left[c_2 \int_{\ln 2}^L e^{Y^2_{m,s}} \, d\mu(s)\leq u_2 \right] g(m) \, dm$$
for some density $g(m)$ and constants $c_1,c_2>0$. (I don't expect an equality result as in Williams because $B^1,B^2$ are related only asymptotically.)
Q1: Any suggestions or references would be helpful.
- Attempts
1)If the correlation was constant then we can write $B_t^1=\rho B^2_t+\sqrt{1-\rho^2}\tilde{B}^1_t$ where $\tilde{B}^1_t$ is independent of $B^2$ and so I can condition on the $\tilde{B}^1$ part and apply William's result.
See also the work in "A class of copulae associated with Brownian motion processes and their maxima" where they study the joint law of $B^1_T, M_{2,T}$.
Even if the correlation is not constant, there is such such a decomposition (see Common Decomposition of Correlated Brownian Motions and its Financial Applications)
The problem here is that the correlation happens also across different $t\neq w$ and so we don't have this decomposition.
Q2: Is there any way to re-express the pair $(B^1,B^2)$ as a more tractable transformation of an independent pair? (Linear won't work due to the correlation for $t\neq w$).
2)Another route is to instead condition ,as per usual in William's result, by $M^2=\sup_{t\in [0,\infty]}(B_t^2-\nu t)=m$ and then see if that gives me anything for $M^1=\sup_{t\in [0,\infty]}(B_t^1-\nu t)$.
For large $t$ we have that $B^1_t,B^2_t$ are close, so the same is true for their tail-suprema $\sup_{t\geq B}$ for large $B$. So the issue here is what happens for $t\leq B$.
3)Another possible route is as follows. First we condition by $M_{1,\infty}$ and so apply Williams to get different $\tilde{B}^1$ process. Lets assume that we also get some concrete $\tilde{B}^2$ process.
If we could prove that
- they are both stochastically dominated by Gaussians eg. drifted Brownian motions,
- the covariance of their differences is at least uppper bounded by the previous ones $E[(B^1_t-B^2_t)(B^1_s-B^2_s)],E[(B^1_t-B^2_t)B^1_s]$,
then we could upper bound by the negative moment $$E\left[\left(\int_{\ln 2}^L e^{\tilde B^2_s-\nu s} \, d\mu(s)\right)^{-p}\right]=E\left[\left(\int_{\ln 2}^L e^{\tilde{B}^2_s-\tilde{B}^1_s+\tilde{B}^1_s-\nu s} \, d\mu(s)\right)^{-p}\right]$$
apply Kahane's inequality for the convex function $x^{-p}$, to dominate $\tilde{B}^2$ by $N(0,2\ln 2) + \tilde{B}^1$ and so upper bound it by
$$cE\left[\left(\int_{\ln 2}^L e^{\tilde{B}^1_s-\nu s} \, d\mu(s)\right)^{-p}\right].$$
The challenge here is that I don't know how exactly the William's result changes the law of $\tilde{B}^2$ i.e. whether the above two bullet points are true.
