Construction and sample path properties of diffusion house-moving between two curves
Kensuke Ishitani, Soma Nishino
TL;DR
This work constructs diffusion house-moving $H_{\mu}^{g^-\to g^+}$ as the weak limit of diffusion bridges constrained between two curves, extending Brownian-house-moving concepts to general one-dimensional diffusions with drift $\mu$. It develops a density-based, path-decomposed description via $h_{\mu}$, proves a decomposition formula, and introduces a diffusion meander $X_{[0,T]}^{0,(g^-,g^+)}$ with explicit $k_{\mu}$-type densities. The authors show absolute continuity relations between the house-moving and the meander, providing Radon–Nikodym derivatives and measurable structure to compare these conditioned processes. They also establish sample-path regularity, proving local Hölder continuity, and lay groundwork for applications to higher-order Greeks and barrier-option analysis through rigorous weak-convergence results for conditioned diffusions.
Abstract
The purpose of this paper is to introduce the construction of a stochastic process called ``diffusion house-moving'' and to explore its properties. We study the weak convergence of diffusion bridges conditioned to stay between two curves, and we refer to this limit as diffusion house-moving. Applying this weak convergence result, we give the sample path properties of diffusion house-moving.