Conditioning a reflected one-dimensional diffusion via its canonical decomposition
The author considers a reflected one-dimensional diffusion process \(X\) starting from \(x\geq 0\), the point 0 being an instantaneously reflecting boundary. If the scale function is convex and if appropriate conditions on the speed measure are fulfilled, the diffusion \(X\) can be written as \(X=N-B\), where \(B\) is a Brownian motion starting from \(-x\) and \(X\) a locally zero-energy additive functional. The author studies the diffusion conditioned by the event ``\(N\leq 0\)''. The conditional law \(P(\cdot/N\leq 0)\) is obtained as the limit of the conditional laws \(P(\cdot/vert N\leq y)\) when \(y\) goes to zero. The proof of this result uses a path decomposition analogous to the one of D. Williams. Then, the author gives a simple pathwise construction of the conditioned process from the original diffusion \(X\).
scientific article; zbMATH DE number 179066
scientific article