The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes
The authors study continuous local martingales. First they introduce a default function \(\gamma_M(t)\) of a local martingale \(M\) by \(\gamma_M (t)= EM_0-EM_t\) and note that a local martingale is a martingale if and only if \(\gamma_M(T)=0\) for every bounded stopping time \(T\). Assume now that the local martingale \(M\) is continuous and \(T\) is a finite stopping time such that the process \((M^T)^-\) belongs to class \(D\). Then they show that \[ E\bigl[ M_T\mathbf{1}_{\{\sup_{t\leq T}M_t< x\}}\bigr] +xP\left(\sup_{t\leq T}M_t\geq x\right)+ E(M_0-x)^+ =EM_0, \] and the following default formula holds \[ \lim_{x\to \infty} xP \left(\sup_{t\leq T}M_t\geq x\right)= \gamma_M(T). \] The authors give analogous results for continuous semi-martingales and apply these results to study the integrability of functionals of local martingales. They also describe the default functions of Bessel processes and radial Ornstein-Uhlenbeck processes in relation to their first hitting and last exit times.
scientific article; zbMATH DE number 1389147