Onsager-Machlup functional for the fractional Brownian motion
Let \(B^H_t\) \((0\leq t\leq 1)\) be a fractional Brownian motion with Hurst parameter \(0<H<1\). For any \(b\in {\mathcal C}^2_b(R)\) consider the stochastic differential equation \[ X_t=x+B^H_t+ \int^t_0b(X_sds). \tag{*} \] The purpose of this paper is to study the limit behaviour of \[ \gamma_\varepsilon (\varphi)= {P\bigl(\|X-\varphi \|\leq \varepsilon\bigr) \over P\bigl(\|B^H\|\leq\varepsilon \bigr)} \quad\text{as }\varepsilon \downarrow 0, \] where \(\varphi\) is a function such that \(\varphi-x\) belongs to the Cameron-Martin space associated with the fractional Brownian motion, and \(\|\cdot\|\) is a suitable norm. If the limit is of the form \(\exp(J (\varphi))\), the functional \(J\) is called the Onsager-Machlup functional associated with (*) and the norm \(\|\cdot \|\). The authors compute the Onsager-Machlup functional of \(X\), e.g. for the Hölder norms of order \(\beta\) with \(0<\beta <H-1/4\) in the case \(1/4<H <1/2\) (singular case) and for the Hölder norms of order \(\beta\) with \(H-1/2< \beta< H-1/4\) when \(H>1/2\) (regular case).
scientific article; zbMATH DE number 1837374