Ray-Knight theorems related to a stochastic flow
The stochastic flow of homeomorphisms of \(\mathbb R\) associated with a stochastic differential equation \[ X_{t}(x) = x + \beta _1\int ^{t}_{0} \mathbf 1_{\{X_{s}(x)\leq 0\}} ds+ \beta _2\int ^{t}_{0} \mathbf 1_{\{X_{s}(x)> 0\}} ds+ B_{t}, \quad x\in \mathbb R, \;t\geq 0, \] is studied, \(B\) being a one-dimensional Wiener process and \(\beta _1, \beta _2\in \mathbb R\) fixed constants. First, it is shown that there exists an almost surely bicontinuous process \((L^{x}_{t}, t\geq 0,x\in \mathbb R)\) such that \((L^{x}_{t}, t\geq 0)\) is the semimartingale local time at 0 for the process \(X_{t}(x)\) for each \(x\in \mathbb R\). Consider stopping times \[ T(a) = \inf \{t>0; X_{t}(a) = 0\}, \quad \tau _{r}(a) = \inf \{ t>0; L^{a}_{t} >r\}, \;a\in \mathbb R, \;r\geq 0. \] Under suitable assumptions on \(\beta _1\), \(\beta _2\) it is proven that processes like \((L^{a+x}_{T(a)}, x\geq 0)\) or \((L^{b+x}_{\tau _{r}(b)}, x\geq 0)\) are inhomogeneous diffusions whose generators are computed explicitly. The derivative \(DX_{t}(x)\) of the (increasing) function \(x\mapsto X_{t}(x)\) satisfies \(DX_{t}(x) = \exp ((\beta _2-\beta _1)L^{x}_{t})\), therefore, \((DX_{T}(x), x\in \mathbb R)\) is also a diffusion for certain stopping times \(T\). These results complement and extend those obtained by \textit{R. F. Bass} and \textit{K. Burdzy} [Ann. Probab. 27, 50-108 (1999; Zbl 0943.60087)].
scientific article; zbMATH DE number 2000173
scientific article