Large Deviation Principle for Diffusions

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 X_ε (•) = ε^1/2X(•) , then X (•) = F(ε^1/2 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

$$ \begin{gathered} \inf \left\{ {{I_{{{}_wT}}}(\phi ):F(\phi ) \in C} \right\} \hfill \\ = \inf \left\{ {{I_{{{}_wT}}} \circ {F^{{ - 1}}}(\phi ):\phi \in C} \right\} \hfill \\ \end{gathered} $$

.