# Large Deviation Principle for Diffusions

Chapter
Part of the Universitext book series (UTX)

## Abstract

In exercise (3.46) ii) we had an example of a diffusion, besides Brownian motion, for which we can derive a large deviations principle. Of course, our derivation in (3.46) ii) rested on the observation that if
$$X(t) = \beta (t) - \int_0^t X (s)ds\quad, \quad t \geqslant 0$$
, where β(•) is a Brownian motion starting at 0, then X(•) is a centered Gaussian process. A second approach (and an approach which has a chance of generalizing to non-Gaussian diffusions) to obtaining a large deviations principle for X(•) is the following. Define F: C([0, ∞); R1) → C([0,∞); R1) so that
$$(F(\phi ))(t) = \phi (t)\int_0^t {(F(\phi ))(s)ds\quad, \quad } t \geqslant 0$$
. Clearly, for each T > 0, F determines a continuous injective surjection from C([0,T];R1) onto C([0,T];R1) . Moreover, if Xε (•) = ε1/2X(•) , then X (•) = F(ε1/2 β(•)). Hence, if µ denotes the distribution of X(•)|[0,T], then for any closed C ⊆ C([0,T];R1):
$$\begin{gathered} \overline {\mathop{{\lim }}\limits_{{\varepsilon \downarrow 0}} \,} \varepsilon \log \,\mu \left( {\frac{1}{{{}_{\varepsilon }1/2}}C} \right) = \overline {\mathop{{\lim }}\limits_{{\varepsilon \downarrow 0}} \,} \varepsilon \,\log \,P({\varepsilon^{{1/2}}}\beta ( \cdot ) \in {F^{{ - 1}}}(C)), \hfill \\ \leqslant - \inf \left\{ {{I_{{{}_wT}}}(\phi ):F(\phi ) \in C} \right\} \hfill \\ \end{gathered}$$
$${I_{{{}_wT}}}$$ is the rate function for Wiener measure $${}_wT$$ on C([0,T];R1). Because F is 1-1 and onto, we see that
$$\begin{gathered} \inf \left\{ {{I_{{{}_wT}}}(\phi ):F(\phi ) \in C} \right\} \hfill \\ = \inf \left\{ {{I_{{{}_wT}}} \circ {F^{{ - 1}}}(\phi ):\phi \in C} \right\} \hfill \\ \end{gathered}$$
.