Distribution of functionals of certain non-Markovian processes (Q2711135)

Let \((W(s))_{s\in[0,\infty)}\) be a Brownian motion on \({\mathbb R}\), started from \(x\) under \({\mathbb P}_x\), and let \((l(t,y))_{t\in[0,\infty], y\in {\mathbb R}}\) be its local time process. With arbitrary real numbers \(\alpha_1,\dots,\alpha_m\), \(r_1,\dots,r_m\), define \(\xi(s)=W(s)+\sum_{k=1}^m\alpha_kl(s, r_k)\). Since \((\xi(s))_{s\in[0,\infty)}\) is a semimartingale, it also possesses a local times process, which we denote \((L(t,y))_{t\in[0,\infty),y\in{\mathbb R}}\). The author calculates the (Laplace transforms of the) joint distributions of the five random variables \(\xi(t)\), \(I_{f}(t)=\int_0^t f(\xi(s)) ds\), \(\underline\xi(t)=\inf_{0\leq s\leq t} \xi(s)\), \(\overline \xi(t)=\sup_{0\leq s\leq t} \xi(s)\), and \(L(t,y)\). NEWLINENEWLINENEWLINEIn order to give a flavor, we state the result about the former four of these five random variables, in the special case where \(m=1\), i.e., we consider \(\xi(s)=W(s)+\alpha l(s,r)\). Let \(\tau\) be an exponential random variable, independent of the motion, such that \(\mathbb P(\tau>t)=e^{-\lambda t}\), \(t>0\), for some \(\lambda>0\). Fix \(a<b\) and fix piecewise continuous functions \(F\) and \(f\) on \([a,b]\), where \(F\) is bounded and \(f\geq 0\). Define NEWLINE\[NEWLINEQ_r(x)= \mathbb E_x\bigl\{F(\xi(\tau))\exp\{-I_f(\tau)\}1\{a<\underline\xi(\tau)\} 1\{\overline\xi(\tau)<b\}\bigr\}.NEWLINE\]NEWLINE Then \(r\mapsto Q_r(r)\) is differentiable on \((a,b)\), and for any \(r\in(a,b)\), the map \(x\mapsto Q_r(x)\) is the unique solution to the problem NEWLINE\[NEWLINE \begin{aligned} \frac 12 Q_r''(x)-(\lambda+f(x))Q_r(x)&=-\lambda F(x),\quad x\not=r,\tag{1}\\ Q_r'(r+)-Q_r'(r-)&=-2\alpha\frac d{dr} Q_r(r),\tag{2}\end{aligned} NEWLINE\]NEWLINE with zero boundary conditions at \(a\) and \(b\), and \(\lim_{r\downarrow a}Q_r(r)= \lim_{r\uparrow b}Q_r(r)=0\). (In the case \(a=-\infty, b=\infty\), the zero boundary conditions for \(Q_r(\cdot)\) and \(r\mapsto Q_r(r)\) have to be replaced by asymptotic boundedness.) For \(\alpha=0\), the result is well-known and is due to Kac (1949). For \(\alpha\not= 0\), (2) determines how to glue together the solutions of (1) on \((a,r)\) and \((r,b)\). NEWLINENEWLINENEWLINEIn the last section, the previous results are used to give explicit expressions for the density of the distribution of \(\sup_{0\leq s\leq \tau}(W(s)-l(s,\tau))\) and \(\xi(t)=W(t)+\alpha l(t,r)\) and for the Laplace transform of the local time \(L(\tau,y)\) of the process \(\xi(s)=W(s)-l(s,r)\).