On distribution of functionals of a Brownian motion stopped at the inverse range time (Q1603210)

Consider a real Brownian motion \(W\), its local time \(l(t,z)\), its inverse range process \[ \theta_v:=\text{Inf}\left\{t\left|\sup_{0\leq s\leq t}W_s\right. -\underset{0\leq s\leq t} {\text{Inf}} W_s=v\right\}, \] \(f:\mathbb{R}\to\mathbb{R}_+\), \(x\in \mathbb{R}^d\), \(\gamma_1,\dots,\gamma_k\geq 0\), and the two functionals \[ A_t:=\int^t_0f(W_s)ds+\sum^k_{j=1}\gamma_jl(t,x_j),\quad B_t:=\int^\infty_{-\infty}f(l(t,z))dz. \] Then it is proved that \[ \frac{\partial}{\partial y}\mathbb{E}_x\left(e^{-A_{\theta_v}}\times 1_{\{W_{\theta_v}<y\}}\right)=\frac{\partial}{\partial v}\mathbb{E}_x\left(e^{-A_{H_{y,y+v}}}\times 1_{W_{\{H_{y,y+v}}=y\}}\right) \] for \(x-v<y<x\), where \(H_{a,b}\) denotes the exit time of \(]a,b[\) for \(W\). This allows to deduce formulae relative to \(\theta_v\) from known formulae relative to \(H_{a,b}\). For example it is proved that the process \(z\mapsto l(t,z)\) is Markovian, and its law is precisely calculated. Then the law of \(\text{Sup}_{z\in\mathbb{R}} l(\theta_v,z)\) is computed, and so is also the Laplace transform of its joint law with \(B_{\theta_v}\). This is finally extended to the case of \(W\) having a constant drift.