Schilder's theorem

Let C₀ = C₀([0, T]; R^d) be the Banach space of continuous functions ${\displaystyle f:[0,T]\longrightarrow \mathbf {R} ^{d}}$ such that ${\displaystyle f(0)=0}$, equipped with the supremum norm ||·||_∞ and ${\displaystyle C_{0}^{\ast }}$ be the subspace of absolutely continuous functions whose derivative is in ${\displaystyle L^{2}}$ (the so-called Cameron-Martin space). Define the rate function

${\displaystyle I(\omega )={\frac {1}{2}}\int _{0}^{T}\|{\dot {\omega }}(t)\|^{2}\,\mathrm {d} t}$

on ${\displaystyle C_{0}^{\ast }}$ and let ${\displaystyle F:C_{0}\to \mathbb {R} ,G:C_{0}\to \mathbb {C} }$ be two given functions, such that ${\displaystyle S:=I+F}$ (the "action") has a unique minimum ${\displaystyle \Omega \in C_{0}^{\ast }}$.

Then under some differentiability and growth assumptions on ${\displaystyle F,G}$ which are detailed in Schilder 1966, one has

${\displaystyle \lim _{\lambda \to \infty }{\frac {\mathbb {E} \left[\exp \left(-\lambda F(\lambda ^{-1/2}\omega )\right)G(\lambda ^{-1/2}\omega )\right]}{\exp \left(-\lambda S(\Omega )\right)}}=G(\Omega )\mathbb {E} \left[\exp \left(-{\frac {1}{2}}\langle \omega ,D(\Omega )\omega \rangle \right)\right]}$

where ${\displaystyle \mathbb {E} }$ denotes expectation with respect to the Wiener measure ${\displaystyle \mathbb {P} }$ on ${\displaystyle C_{0}}$ and ${\displaystyle D(\Omega )}$ is the Hessian of ${\displaystyle F}$ at the minimum ${\displaystyle \Omega }$; ${\displaystyle \langle \omega ,D(\Omega )\omega \rangle }$ is meant in the sense of an ${\displaystyle L^{2}([0,T])}$ inner product.

Let B be a standard Brownian motion in d-dimensional Euclidean space R^d starting at the origin, 0 ∈ R^d; let W denote the law of B, i.e. classical Wiener measure. For ε > 0, let W_ε denote the law of the rescaled process √εB. Then, on the Banach space C₀ = C₀([0, T]; R^d) of continuous functions ${\displaystyle f:[0,T]\longrightarrow \mathbf {R} ^{d}}$ such that ${\displaystyle f(0)=0}$, equipped with the supremum norm ||·||_∞, the probability measures W_ε satisfy the large deviations principle with good rate function I : C₀ → R ∪ {+∞} given by

${\displaystyle I(\omega )={\frac {1}{2}}\int _{0}^{T}|{\dot {\omega }}(t)|^{2}\,\mathrm {d} t}$

if ω is absolutely continuous, and I(ω) = +∞ otherwise. In other words, for every open set G ⊆ C₀ and every closed set F ⊆ C₀,

${\displaystyle \limsup _{\varepsilon \downarrow 0}\varepsilon \log \mathbf {W} _{\varepsilon }(F)\leq -\inf _{\omega \in F}I(\omega )}$

and

${\displaystyle \liminf _{\varepsilon \downarrow 0}\varepsilon \log \mathbf {W} _{\varepsilon }(G)\geq -\inf _{\omega \in G}I(\omega ).}$

Taking ε = 1/c², one can use Schilder's theorem to obtain estimates for the probability that a standard Brownian motion B strays further than c from its starting point over the time interval [0, T], i.e. the probability

${\displaystyle \mathbf {W} (C_{0}\smallsetminus \mathbf {B} _{c}(0;\|\cdot \|_{\infty }))\equiv \mathbf {P} {\big [}\|B\|_{\infty }>c{\big ]},}$

as c tends to infinity. Here B_c(0; ||·||_∞) denotes the open ball of radius c about the zero function in C₀, taken with respect to the supremum norm. First note that

${\displaystyle \|B\|_{\infty }>c\iff {\sqrt {\varepsilon }}B\in A:=\left\{\omega \in C_{0}\mid |\omega (t)|>1{\text{ for some }}t\in [0,T]\right\}.}$

Since the rate function is continuous on A, Schilder's theorem yields

${\displaystyle {\begin{aligned}\lim _{c\to \infty }{\frac {\log \left(\mathbf {P} \left[\|B\|_{\infty }>c\right]\right)}{c^{2}}}&=\lim _{\varepsilon \to 0}\varepsilon \log \left(\mathbf {P} \left[{\sqrt {\varepsilon }}B\in A\right]\right)\\[6pt]&=-\inf \left\{\left.{\frac {1}{2}}\int _{0}^{T}|{\dot {\omega }}(t)|^{2}\,\mathrm {d} t\,\right|\,\omega \in A\right\}\\[6pt]&=-{\frac {1}{2}}\int _{0}^{T}{\frac {1}{T^{2}}}\,\mathrm {d} t\\[6pt]&=-{\frac {1}{2T}},\end{aligned}}}$

making use of the fact that the infimum over paths in the collection A is attained for ω(t) = t/T . This result can be heuristically interpreted as saying that, for large c and/or large T

${\displaystyle {\frac {\log \left(\mathbf {P} \left[\|B\|_{\infty }>c\right]\right)}{c^{2}}}\approx -{\frac {1}{2T}}\qquad {\text{or}}\qquad \mathbf {P} \left[\|B\|_{\infty }>c\right]\approx \exp \left(-{\frac {c^{2}}{2T}}\right).}$

In fact, the above probability can be estimated more precisely: for B a standard Brownian motion in Rⁿ, and any T, c and ε > 0, we have:

${\displaystyle \mathbf {P} \left[\sup _{0\leq t\leq T}\left|{\sqrt {\varepsilon }}B_{t}\right|\geq c\right]\leq 4n\exp \left(-{\frac {c^{2}}{2nT\varepsilon }}\right).}$

• Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications. Applications of Mathematics (New York) 38 (Second ed.). New York: Springer-Verlag. pp. xvi+396. ISBN 0-387-98406-2. MR 1619036. (See theorem 5.2)