Schilder's theorem

In mathematics, Schilder's theorem is a result in the large deviations theory of stochastic processes. Roughly speaking, Schilder's theorem gives an estimate for the probability that a (scaled-down) sample path of Brownian motion will stray far from the mean path (which is constant with value 0). This statement is made precise using rate functions. Schilder's theorem is generalized by the Freidlin–Wentzell theorem for Itō diffusions.

Statement of the theorem

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 (√ε)B. Then, on the Banach space C₀ = C₀([0, T]; R^d) of continuous functions $f:[0,T]\longrightarrow {\mathbf {R}}^{d}$ such that $f(0)=0$ , equipped with the supremum norm ||·||_∞, the probability measures W_ε satisfy the large deviations principle with good rate function I : C₀ → R ∪ {+∞} given by

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,

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

and

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

Example

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

{\mathbf {W}}(C_{{0}}\setminus {\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

\|B\|_{{\infty }}>c\iff {\sqrt {\varepsilon }}B\in A:={\big \{}\omega \in C_{{0}}{\big |}|\omega (t)|>1{\mbox{ for some }}t\in [0,T]{\big \}}.

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

\lim _{{c\to \infty }}{\frac {1}{c^{{2}}}}\log {\mathbf {P}}{\big [}\|B\|_{{\infty }}>c{\big ]}

=\lim _{\varepsilon \to 0}\epsilon \log \mathbf {P} {\big [}{\sqrt {\varepsilon }}B\in A{\big ]}

=-\inf \left\{\left.{\frac {1}{2}}\int _{{0}}^{{T}}|{\dot {\omega }}(t)|^{{2}}\,{\mathrm {d}}t\right|\omega \in A\right\}

=-{\frac {1}{2}}\int _{{0}}^{{T}}{\frac {1}{T^{{2}}}}\,{\mathrm {d}}t

=-{\frac {1}{2T}},

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

{\frac {1}{c^{{2}}}}\log {\mathbf {P}}{\big [}\|B\|_{{\infty }}>c{\big ]}\approx -{\frac {1}{2T}},

or, in other words,

{\mathbf {P}}{\big [}\|B\|_{{\infty }}>c{\big ]}\approx \exp \left(-{\frac {c^{{2}}}{2T}}\right).

In fact, the above probability can be estimated more precisely as follows: for B a standard Brownian motion in Rⁿ, and any T, c and ε > 0, it holds that

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

References

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)

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