On the Large Deviations for Stochatic Differential Equations Driven by a

Fractional Brownian Motion

Constantin Tudor Department of Mathematics University of Bucharest 70109 Bucharest, Romania ctudor@pro.math.unibuc.ro and CIMAT Apdo Postal 402, Guanajuato, Gto. 36000 Mexico tudor@fractal.cimat.mx

Abstract

In this paper we prove large deviations for stochastic differential equations driven by a fractiona Brownian motion of any Hurst parameter H ∈ (0, 1).

Key Words: fractional brownian motion, stochastic differential equations, large deviations AMS Classification: 60H10

1. Introduction

In recent years many phenomena in telecommunication network, mathematical finance, filtering theory, biology, etc., are modeled by fractional Brownian motion (fBm for short). It is well known that the fBm is not Markovian, nor a semimartingale, but has a long-range dependence and it is self-similar. There are several attempts to construct a stochastic calculus with respect to fBm, such as: sample paths theory, semimartingale approach and the Skorohod integral. SDE with respect to fBm are considered in [6], [3]. Such stochastic equations are of Volterra type with a weakly regular kernel and are not covered by standard stochastic Volterra equations as treated for example in Berger and Mizel [4], Protter [18]. Large deviations for stochastic equations driven by semimartingales are by now well known (see Azencot [1], Baldi and Chaleyat-Maurel [2], Cutland [7], Dozzi [10], Freidlin and Wentzell [11], Liptser and Pukhalskii [14], Nualart and Rovira [15], Pérez-Abreu and Tudor [16], Priouret [17], Rovira and Sanz-Solé [19]). In the present paper we consider Itô equations driven by a fBm and prove large deviations for them. Xiong [22] (see also Kallianpur and Xiong [13]) obtained a large deviations criterium for Banach space valued random variables which come from random transformations of the Wiener process and for which, since the transformation is not pointwise, the classical contraction principle does not work In the paper we use such an alternative for SDEs driven by fBm, together with an exponential inequality for a type of stochastic integrals that are not martingales.

2. Preliminaries about large deviations

Let (S, d) be a Polish space, {μ_{ε}}_{ε>0} a sequence of probability measures on the Borel σ-algebra of S and Λ : S → [0, +∞] be a function. Recall that the sequence {μ_{ε}}_{ε>0} is said to satisfy the large deviations principle with the rate function Λ if :

(a) Λ is lower semicontinuous.

(b) (Lower estimate). For every nonempty open set G ⊂ S

−Λ(G) ≤\underline{lim}_{ε→0} ε log μ_{ε}(G).  (2.1)

(c) (Upper estimate). For every closed set F ⊂ S

\overline{lim_{ε→0}} ε log μ_{ε}(F) ≤ −Λ(F),  (2.2)

where, for a set A ⊂ S, Λ(A) = inf_{x∈A} Λ(x).

The rate function Λ is said to be good if the level set Ψ_{Λ}(r) = {x ∈ S : Λ(x) ≤ r} is compact for each r ≥ 0. A useful criterion for proving large deviations is due to Xiong [22] (see also Kallianpur and Xiong [13]). We state it here part of it which is enough for our purpose.

Theorem 2.1 Let B be a real separable Banach space, A_{B} be a linear space of B-valued random variables defined on a probability space (Ω, F, P) . Let Λ : B → [0, ∞] be a good rate function with the skeleton K_{Λ} = ⋃_{r>0} Ψ_{Λ}(r). Let A : B → B, G : A_{B} → A_{B}, F : B × K_{Λ} → B and {X^{ε}}_{ε>0} be a family of B-valued random variables. Assume that:

(a) B ⊂ A_{B}.

(b) F (x, .) |_{Ψ_{Λ}(r)}is continuous for all x ∈ B, r > 0 and A, F are Lipschitz continuous, i.e., there are positive constants K_{1}, {K(r)}_{r>0} such that

∥A(x) − A(y)∥ ≤ K_{1} ∥x − y∥ , ∀x, y ∈ B,
∥F (x, φ) − F (y, φ)∥ ≤ K(r) ∥x − y∥ , ∀x, y ∈ B, φ ∈ Ψ_{Λ}(r).

(c) X^{ε} ∈ A_{B} and

X^{ε} = A(X^{ε}) + ε^{\frac{1}{2}} G(X^{ε}), a.s.in B.

(d) G is exponentially continuous in the following sense: for any M > 0 there exist δ > 0 such that for any {X_{1}^{ε}} , {X_{2}^{ε}} ⊂ A_{B} we have

\overline{lim_{ε→0}} ε log P (ε^{\frac{1}{2}} ∥G(X_{1}^{ε}) − G(X_{2}^{ε})∥ > \sqrt{δ}, ∥X_{1}^{ε} − X_{2}^{ε}∥ < δ) ≤ −M.

(e) {X^{ε}}_{ε>0} is exponentially tight, i.e., for any M > 0 there exist a compact C_{M} ⊂ B such that

\overline{lim_{ε→0}} ε log P (X^{ε} ∉ C_{M}) ≤ −M.

(f) For any x ∈ B,the family B-valued random variables

X^{ε,x} = A(x) + ε^{\frac{1}{2}} G(x), ε > 0,

satisfies the LDP ion B with the good rate function

I^{x}(y) = inf {Λ(φ) : φ ∈ K_{Λ}, F (x, φ) = y} , y ∈ B.  (2.3)

Then the function I : B → [0, ∞] defined by

I(x) = inf {Λ(φ) : φ ∈ K_{Λ}, F (x, φ) = x} , x ∈ B,  (2.4)

is a rate function (not necessarily good!) and the upper large deviations bound hold for the sequence μ_{ε} = P ∘ (X^{ε})^{−1} .

We shall need the following theorem due to Kallianpur and Odaira (see [12], [13] or [21]). Theorem 2.2 Let μ be a centered Gaussian measure on a separable Banach space E. Let S be a continuous linear map from a separable Hilbert space H to E such that

∥S′l∥_{H}^{2} = ∫_{E} l(x)^{2}dμ(x), ∀l ∈ E′,  (2.5)

where S′ : E′ → H′ = H is the dual of S. For ε > 0, l let μ_{ε} be the probability measure on E defined by

μ_{ε}(C) = μ(x ∈ E : ε^{\frac{1}{2}} x ∈ C), ∀C ∈ B(E).

Then {μ_{ε}}_{ε>0} satisfies the LDP in E with the good rate function

I_{μ}(x) = inf \{\frac{1}{2} ∥h∥_{H}^{2} : h ∈ H, S(h) = x\} , x ∈ E.  (2.6)

3. Stochastic Differential Equations Driven by Fractional Brownian Motion

We consider T = [0, 1] and W = {x ∈ C(T, R) : x(0) = 0} equipped with the supremum norm. We denote by W′ the strong topological dual of W. For H ∈ (0, 1) we denote by P_{H} the unique probability measure on W such that the canonical process {W_{t}}_{t∈T} is a fBm with Hurst parameter H. Recall that the covariance R_{H} of W is given by

R_{H}(s, t) = \frac{C_{H}}{2} (s^{2H} + t^{2H} − |t − s|^{2H}) ,  (3.1)

C_{H} = \frac{Γ(2 − 2H) cos πH}{πH(1 − 2H)} .  (3.2)

In order to represent the fBm in terms of standard Brownian motion (obtained for H = \frac{1}{2}) we need hypergeometric functions. We consider the Gauss hypergeometric function F (α, β, γ, z) which is the analytic continuation on C × C × C \ {0, −1, −2, ....} × {z ∈ C : Arg(1 − z) < π} of the power series

F (α, β, γ, z) = ∑_{k=0}^{∞} \frac{(α)_{k} (β)_{k}}{(γ)_{k} k!} z^{k},  (3.3)

(a)_{k} = a(a + 1)....(a + k − 1). Define

K_{H}(t, r) = \frac{(t − r)^{H− \frac{1}{2}}}{Γ(H + \frac{1}{2})} F (\frac{1}{2} − H, H − \frac{1}{2}, H + \frac{1}{2}, 1 − \frac{t}{r})1_{[0,t)}(r).  (3.4)

It is known that (see [9])

R_{H}(t, s) = ∫_{0}^{t∧s} K_{H}(t, r)K_{H}(s, r)dr.  (3.5)

Next we shall denote by ∇_{H} the Gross-Sobolev derivative operator, δ_{H} its dual (divergence operator) and we define the stochastic integral with respect the fBm by

∫_{0}^{1} u_{s}δ_{H}W_{s} = δ_{H}(K_{H}u),  (3.6)

for processes u for which K_{H}u is in the domain of δ_{H}.

Remark 3.1 It is known that (see [9])

(i) {∫_{0}^{t} K_{H}(t, s)δ_{H}W_{s}}_{t∈T} = {W_{t}}_{t∈T} .

(ii) {∫_{0}^{t} 1_{[0,t]}δ_{H}W_{s}}_{t∈T} := {B_{t}}_{t∈T} is a standard Brownian motion on (W, B(W), P_{H}) .

(iii) For every u ∈ L^{2}(W × T), u adapted,

∫_{0}^{t} u_{s}δ_{H}W_{s} = ∫_{0}^{t} u_{s}dB_{s}, t ∈ T,

(the last integral is the usual Itô integral). (iv) We have the following representation for the fBm

W_{t} = ∫_{0}^{t} K_{H}(t, s)dB_{s}, t ∈ T.

By a stochastic differential equation driven by a fBm we mean an equation of the form

X_{t} = ξ + ∫_{0}^{t} K_{H}(t, s)b(s, X_{s})ds + ∫_{0}^{t} K_{H}(t, s)σ(s, X_{s})dB_{s}, t ∈ T,  (3.7)

where ξ ∈ R and σ(t, x), b(t, x) : T × R → R are measurable. Next we denote

H_{0} = |H − \frac{1}{2}| , Δ_{H} = [0, \frac{1}{H_{0}} ) ,

and for p ∈ Δ_{H} we put

α_{p} = (1 − pH_{0})^{−1} , L_{+}^{α_{p}} = ∪_{α>α_{p}}L^{α}(T).

Definition 3.2 By a solution of (3.7) we mean an nonanticipating process {X_{t}}_{t∈T} such that t ⟶ E [|X_{t}|^{2}] ∈ L_{+}^{α_{2}} and (3.7) is satisfied dP_{H} ⊗ dt a.s.. The following result is due tu Coutin and Decreusefond [6] Theorem 3.3 If b, σ are Lipschitz in the second variable, uniformly with respect to the first variable, and there exist x_{0}, y_{0} ∈ R, α_{b} > α_{1}, α_{σ} > 2α_{2} such that

b(., x_{0}) ∈ L^{α_{b}}(T), σ(., x_{0}) ∈ L^{α_{σ}}(T),

then (3.7) has a unique sulution {X_{t}}_{t∈T} . Moreover for every p ∈ Δ_{H} ∩ [2, ∞)

sup_{t∈T} E [|X_{t}|^{p}] < ∞.  (3.8)

If σ is bounded then X is continuous.

4. Main Result

X_{t}^{ε} = ξ + ∫_{0}^{t} K_{H}(t, s)b(s, X_{s}^{ε})ds + ε^{\frac{1}{2}} ∫_{0}^{t} K_{H}(t, s)σ(s, X_{s}^{ε})dB_{s}, t ∈ T, (4.1)

Let H_{\frac{1}{2}} be the Cameron-Martin space associated to the standard Brownian motion, that is

H_{\frac{1}{2}} = \{h ∈ L^{2}(T) : h(t) = ∫_{0}^{t} ḣ(s)ds, ḣ ∈ L^{2}(T)\} .

We have the following main result. Theorem 4.1 Assume that σ, b are bounded and Lipschitz in the second variable, uniformly with respect to the first variable. Let X^{ε} be the unique continuous solution of (4.1). Then the family {X^{ε}}_{ε>0} satisfies a LDP in W with the good rate function

I(f) = inf \{\frac{1}{2} ∥ḣ∥_{L^{2}(T)}^{2} : h ∈ H_{\frac{1}{2}} , f = S(h)\} ,  (4.2)

where S(h) is the unique continuous solution of

S(h)_{t} = ξ+∫_{0}^{t} K_{H}(t, s)b(s, S(h)_{s})ds+∫_{0}^{t} K_{H}(t, s)σ(s, S(h)_{s})ḣ(s)ds, t ∈ T. (4.3)

For the proof of Theorem 4.1 we need the following lemmas. Next a process {U_{t}} and α > 0 we define

[U]_{α} = sup_{0≤s<t≤T} \frac{|U_{t} − U_{s}|}{|t − s|^{α}}.

We have the following exponential estimate for stochastic Volterra integrals with singular kernel(see [15] for the case of regular coefficients). Lemma 4.2 Let {f(t)}_{t∈T} be a bounded nonanticipating real process and 0 < α < H. Then, for any a > 0,

P ([∫_{0}^{.} K_{H}(., s)f(s)dB_{s}]_{α} > a) ≤ C_{1} exp {− (aC_{2} − 1)^{2}} ,  (4.4)

where 0 < C_{i} = C_{i}(α, H, ∥f∥_{∞}) < ∞. In particular

P ( sup_{0≤t≤1} |∫_{0}^{t} K_{H}(t, s)f(s)dB_{s}| > a) ≤ C_{1} exp {− (aC_{2} − 1)^{2}} .  (4.5)

Proof. By the Kolmogorov criterium the process t ⟶ I(t) := ∫_{0}^{t} K_{H}(t, s)f(s)dB_{s} is α-Hölder. Fix 0 ≤ s < t ≤ 1 and define

g_{t,s}(r) = 1_{s≤r≤t}K_{H}(t, r) + 1_{r≤s} [K_{H}(t, r) − K_{H}(s, r)] , r ∈ T.

Then we have

I(t) − I(s) = ∫_{0}^{1} g_{t,s}(r)f(r)dB_{r}.  (4.6)

Define the martingales

M_{u} = C_{H}^{− \frac{1}{2}} (t − s)^{−α} ∫_{0}^{u} g_{t,s}(r)f(r)dB_{r},

M̃_{u} = C_{H}^{− \frac{1}{2}} (t − s)^{−α} ∫_{0}^{u} \frac{M_{r}g_{t,s}(r)f(r)}{(1 + M_{r}^{2})^{\frac{1}{2}}}dB_{r}, u ∈ T.

By Itô’s formula we have

d (1 + M_{u}^{2})^{\frac{1}{2}} = \frac{1}{2} (1 + M_{u}^{2})^{− \frac{1}{2}} g_{t,s}^{2}(r)f^{2}(r)dr−

\frac{1}{4} (1 + M_{u}^{2})^{− \frac{3}{2}} g_{t,s}^{2}(r)f^{2}(r)dr + M̃_{u}^{2}.  (4.7)

By the time change in stochastic integral we choose a Brownian motion \left\{{ \atop W̃_{u}}\right\}_{u∈T} such that

M̃_{u} = W̃_{τ(u)}, τ(u) = C_{H}^{−1} (t − s)^{−2α} ∫_{0}^{u} \frac{M_{r}^{2}g_{t,s}^{2}(r)f^{2}(r)}{1 + M_{r}^{2}}dr.

Next we have

τ(u) ≤ C_{H}^{−1} (t − s)^{−2α} ∥f∥_{∞}^{2} ∫_{0}^{u} g_{t,s}^{2}(r)dr =

C_{H}^{−1} (t − s)^{−2α} ∥f∥_{∞}^{2} [∫_{s}^{t} K_{H}^{2}(t, r)dr + ∫_{0}^{s} (K_{H}(t, r) − K_{H}(s, r))^{2} dr]

≤ (t − s)^{2(H−α)} ≤ 1,

where from by using this in (4.7) we obtain

(1 + M_{1}^{2})^{\frac{1}{2}} ≤ 1 + sup_{t≤1} |W̃_{t}| + C_{1},

and thus

M_{1}^{2} ≤ C_{1} (1 + sup_{t≤1} |W̃_{t}|^{2})^{2} .

Now by Fernique’s inequality we obtain that

E [exp {C_{2}M_{1}^{2}}] = C_{3} < ∞.  (4.8)

If we denote

Ψ(x) = e^{C_{2}x^{2}} − 1, p(x) = x^{α},

η = ∫_{0}^{1} ∫_{0}^{1} exp \{C_{2} \begin{matrix}|I(t) − I(s)| & 2 \\ |t − s| & 2α \end{matrix} \} dtds,

then (4.6) and (4.8) imply

E(η) ≤ ∫_{0}^{1} ∫_{0}^{1} E [exp(C_{2}M_{1}^{2})] dtds ≤ C_{3}.  (4.9)

Also we have

η − 1 = ∫_{0}^{1} ∫_{0}^{1} Ψ (\frac{|I(t) − I(s)|}{p(|t − s|)}) dtds

From Kallianpur-Xiong [13]-pp. 313-314 and (4.9) it follows that

|I_{1}^{t}(f) − I_{1}^{s}(f)| ≤ C_{4} (\sqrt{|log η|} + 1) |t − s|^{α} ,  (4.10)

and then

P ([I(.)]_{α} > a) ≤ P [η > exp {(aC_{4}^{−1} − 1)^{2}}] ≤
≤ C_{3} exp {(aC_{4}^{−1} − 1)^{2}} ,

i.e., the estimate (4.4). ■ Lemma 4.3 Under the assumptions of Theorem 4.1 the equation (4.3) has a unique continuous solution.

Proof. If f, g are continuous solutions of (4.3) then it follows

|f_{t} − g_{t}| ≤ C ∫_{0}^{t} |K_{H}(t, s)| φ_{s}ds, ∀t ∈ T,  (4.11)

where

φ_{t} = |f_{t} − g_{t}| (1 + |ḣ_{t}|) ∈ L^{2}(T).

Of course

∫_{0}^{t} |K_{H}(t, s)| φ_{s}ds ≤ [∫_{0}^{t} |K_{H}(t, s)|^{2} ds]^{\frac{1}{2}} [∫_{0}^{1} φ_{s}^{2}ds]^{\frac{1}{2}} ≤ D < ∞.  (4.12)

If we define for r ≥ 1, K_{n}^{(r)}(t) = ∫_{0}^{t} |K_{H}(t, s_{1})|^{r} ds_{1} ∫_{0}^{s_{1}} |K_{H}(s_{1}, s_{2})|^{r} ds_{2}... ∫_{0}^{s_{n−1}} |K_{H}(s_{n−1}, s_{n})|^{r} ds_{n}, then iterating (4.11) and using Lemma 3.3 of [6] we obtain

|f_{t} − g_{t}| ≤ DC^{n}K_{n−1}^{(1)}(t) ⟶ 0.

For existence we use the successive approximations. Define f_{t}^{0} = ξ, f_{t}^{n} = ξ + ∫_{0}^{t} K_{H}(t, s)b(s, f_{s}^{n−1})ds + ∫_{0}^{t} K_{H}(t, s)σ(s, f_{s}^{n−1})ḣ(s)ds, t ∈ T. The the successive approximations f_{t}^{n} are continuous since K_{H}(L^{2}(T)) ⊂ W (see [20]-pag. 187). If we denote

α_{n}(t) = |f_{t}^{n+1} − f_{t}^{n}, |

then we have

α_{0}(t) ≤ C ∫_{0}^{t} |K_{H}(t, s)| (1 + |ḣ_{s}|) ds ≤ D,
α_{n}(t) ≤ C ∫_{0}^{t} |K_{H}(t, s)| α_{n−1}(s) (1 + |ḣ_{s}|) ds,

and by iteration

α_{n}(t) ≤ C^{n}DK_{n}^{(1)}(t),

and by Lemma 3.3 of [6], we deduce that α_{n}(t) ⟶ α(t) uniformly in t and α is a solution of (4.3). ■ Lemma 4.4 For f ∈ W consider the gaussian process

Y_{t}^{ε} = ξ + ∫_{0}^{t} K_{H}(t, s)b(s, f_{s})ds + ε^{\frac{1}{2}} ∫_{0}^{t} K_{H}(t, s)σ(s, f_{s})dB_{s}, t ∈ T. (4.13)

Then the family {Y^{ε}}_{ε>0} satisfies a LDP in W with the good rate function

I^{f}(h) = inf \{\frac{1}{2} \left∥{ \atop φ̇}\right∥_{L^{2}(T)}^{2} : φ ∈ H_{\frac{1}{2}} , h = S_{f}(φ)\} ,  (4.14)

where

S_{f}(φ)_{t} = ξ + ∫_{0}^{t} K_{H}(t, s)b(s, f_{s})ds + ∫_{0}^{t} K_{H}(t, s)σ(s, f_{s})φ̇(s)ds, t ∈ T. (4.15)

From the above lemma and Theorem 3.3 of [9] we obtain in particular the standard result (see for example [5]-Corollary 4.9.3 or [21]-Theorems 3.45 and 3.48)

Corollary 4.5 The the family \{ε^{\frac{1}{2}} W \}_{ε>0} satisfies a LDP in W with the good rate function

I(h) = inf \{\frac{1}{2} ∥ḣ∥_{L^{2}(T)}^{2} : ḣ ∈ L^{2} (T) , h(.) = ∫_{0}^{.} K_{H}(., s)ḣ(s)ds\} .  (4.16)

Proof of Lemma 4.4 Without loss of generality we may assume that ξ = b = 0. We shall apply Theorem 2.2 to the operator S_{f} : H_{\frac{1}{2}} ⟶ W. Consider the Gaussian measure μ on W

μ = P_{H} ∘ (∫_{0}^{.} K_{H}(., r)σ(r, f_{r})dB_{r})^{−1} .

For η ∈ W′ we have

∫ η^{2}(w)dμ(w) = ∫_{0}^{1} ∫_{0}^{1} E {[∫_{0}^{t} K_{H}(t, r)σ(r, f_{r})dB_{r}] ×

× [∫_{0}^{s} K_{H}(s, r)σ(r, f_{r})dB_{r}]} dη(t)dη(s) =

∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} K_{H}(t, r)K_{H}(s, r)σ^{2}(r, f_{r})drdη(t)dη(s) =

∫_{0}^{1} [∫_{0}^{1} K_{H}(t, r)dη(t)]^{2} σ^{2}(r, f_{r})dr.

Next for each φ ∈ H_{\frac{1}{2}} we have

⟨S′_{f}η, φ⟩_{H_{\frac{1}{2}}} = ∫_{0}^{1} S_{f}φ(t)dη(t) =

∫_{0}^{1} [∫_{0}^{1} K_{H}(t, r)dη(t)] σ(r, f_{r})φ̇(r)dr,

which implies that

∥S′_{f}η∥_{H_{\frac{1}{2}}}^{2} = ∫_{0}^{1} [∫_{0}^{1} K_{H}(t, r)dη(t)]^{2} σ^{2}(r, f_{r})dr. ■

Proof of Theorem 4.1 Next we shall denote: B = C (T) , A_{B} be the family of all real continuous adapted processes {f_{t}}_{t∈T} ,

Λ : B → [0, ∞] , Λ(φ) = \{ \begin{matrix}\frac{1}{2} \left∥{ \atop φ̇}\right∥_{L^{2}(T)}^{2} , & φ ∈ H_{\frac{1}{2}} \\ ∞, & φ ∉ H_{\frac{1}{2}} , \end{matrix}

A : B → B, A(f)_{t} = ξ + ∫_{0}^{t} K_{H}(t, s)b(s, f_{s})ds,

G : A_{B} → A_{B}, G(X)_{t} = ∫_{0}^{t} K_{H}(t, s)σ(s, X_{s})dB_{s},

F : B × H_{\frac{1}{2}} → B, F (f, φ)_{t} = S_{f}(φ).

We check the hypotheses of Theorem 2.1 in the above setting. The conditions (a), (b) are easily seen. In order to see (d), we define

f(t) = δ^{−1} [σ(s, X_{ε}^{1}(s)) − σ(s, X_{ε}^{2}(s))] 1_{{|X_{ε}^{1}(t)−X_{ε}^{2}(t)|<δ}}, t ∈ T.

Then by Lemma 4.2 we have

ε log P_{H} (ε^{\frac{1}{2}} sup_{0≤t≤1} |G(X_{ε}^{1})_{t} − G(X_{ε}^{2})_{t}| > \sqrt{δ}, sup_{0≤t≤1} |X_{ε}^{1}(t) − X_{ε}^{2}(t)| < δ)

εP_{H}([∫_{0}^{.} K_{H}(., s)f(s)dB_{s}]_{α} > \frac{1}{(εδ)^{\frac{1}{2}}} ) → −∞ as ε → 0.

For the condition (e), define the compact set

C_{M} = {x ∈ C (T) : [x]_{α} ≤ M} , M > 0, 0 < α < H.

It is easily seen that for s ≤ t,

|∫_{0}^{t} K_{H}(t, r)b(r, X_{r}^{ε})dr − ∫_{0}^{s} K_{H}(s, r)b(r, X_{r}^{ε})dr| ≤ D_{1} (t − s)^{H} .  (4.17)

Then by Lemma 4.2 and (4.17),

lim_{M→∞} \overline{lim_{ε→0}} ε log P_{H}(X^{ε} ∉ C_{M}) ≤

lim_{M→∞} max \{\overline{lim_{ε→0}} ε log P_{H}([∫_{0}^{.} K_{H}(., r)b(r, X_{r}^{ε})dr]_{α} > \frac{M}{2} ),

\overline{lim_{ε→0}} ε log P_{H}([∫_{0}^{.} K_{H}(., r)σ(r, X_{r}^{ε})dr]_{α} > \frac{M}{2} )\} = −∞.

Next if f ∈ C (T) then the family

X^{ε,f} = ξ + ε^{\frac{1}{2}} B(f),

satisfies, by Lemma 4.4, the LDP in C (T) with the good rate function

I^{f}(h) = inf \{\frac{1}{2} \left∥{ \atop φ̇}\right∥_{L^{2}(T)}^{2} : φ ∈ H_{\frac{1}{2}} , h = F (f, φ)\} ,

which is the assertion in the condition (f). Therefore by Theorem 2.1 the upper estimate of large deviations hold. For the lower estimate all we need to show is that for every f ∈ C (T) with I(f) < ∞ and for all δ, η > 0 we have

\underline{lim}_{ε→0} ε log P ( sup_{0≤t≤1} |X_{t}^{ε} − f_{t}| < δ) ≥ −I(f) − η.

Choose φ ∈ H_{\frac{1}{2}} such that f = S_{f}(φ) and \frac{1}{2} \left∥{ \atop φ̇}\right∥_{L^{2}(T)}^{2} ≤ I(f) + η.

By Girsanov’s theorem the process

B̃_{t} = B_{t} − \frac{1}{\sqrt{ε}}φ_{t}, t ∈ T,

is a Brownian Motion with respect to the probability measure

P̃ = exp \{\frac{1}{\sqrt{ε}} ∫_{0}^{1} φ̇_{t}dB_{t} − \frac{1}{2ε} ∫_{0}^{1} \left|{ \atop φ̇_{t}}\right|^{2} dt\} .P.

Consider the process X̃_{t}^{ε} = X_{t}^{ε} − f_{t}, t ∈ T. It is clear that X̃_{t}^{ε} satisfies under P̃ the stochastic equation

X̃_{t}^{ε} = ε^{\frac{1}{2}} ∫_{0}^{t} K_{H}(t, s)σ(s, X̃_{s}^{ε} + f_{s})dB̃_{s}+

∫_{0}^{t} K_{H}(t, s) [b(s, X̃_{s}^{ε} + f_{s}) − b(s, f_{s})] ds+
∫_{0}^{t} K_{H}(t, s) [σ(s, X̃_{s}^{ε} + f_{s}) − σ(s, f_{s})] φ̇ds.  (4.18)

If we denote

A^{ε} = { sup_{0≤t≤1} |X̃_{t}^{ε}| < δ} ,

then we obtain

P (A^{ε}) = ∫_{A^{ε}} exp \{−\frac{1}{\sqrt{ε}} ∫_{0}^{1} φ̇_{t}dB_{t} + \frac{1}{2ε} ∫_{0}^{1} \left|{ \atop φ̇_{t}}\right|^{2} dt\} dP̃ =

exp \{−\frac{1}{2ε} ∫_{0}^{1} \left|{ \atop φ̇_{t}}\right|^{2} dt\} ∫_{A^{ε}} exp \{−\frac{1}{\sqrt{ε}} ∫_{0}^{1} φ̇_{t}dP̃B̃_{t}\} dP̃,

and from Hölder and Jensen’s inequalities,

ε log P (A^{ε}) ≥ −I(f) − η + ε log P̃(A^{ε}) − \sqrt{ε}E_{P̃|_{A^{ε}}} (∫_{0}^{1} φ̇_{t}dP̃B̃_{t}) ,

which shows that it is enough to prove that

log P̃(A^{ε}) and E_{P̃|_{A^{ε}}} (∫_{0}^{1} φ̇_{t}dP̃B̃_{t}) ,

are bounded for ε ≤ ε_{0}. Since

|E_{P̃|_{A^{ε}}} (∫_{0}^{1} φ̇_{t}dP̃B̃_{t})| = \frac{1}{P̃(A^{ε})} ∫_{A^{ε}} (∫_{0}^{1} φ̇_{t}dP̃B̃_{t}) dP̃

≤ \frac{1}{P̃(A^{ε})} [EP̃ (|∫_{0}^{1} φ̇_{t}dP̃B̃_{t}|^{2})]^{\frac{1}{2}} P̃(A^{ε})^{\frac{1}{2}} = \frac{[EP̃ (∫_{0}^{1} \left|{ \atop φ̇_{t}}\right|^{2} dt)]^{\frac{1}{2}}}{P̃(A^{ε})^{\frac{1}{2}}} ,

the assertion follows if we prove that P̃(A^{ε}) → 1 as ε → 0. In fact we show more, that is EP̃ [sup_{0≤t≤1} |X̃_{t}^{ε}|^{r}] → 0 as ε → 0 for r large enough. From (4.18), by using a standard procedure, we have for r large enough

EP̃ [|X̃_{t}^{ε}|^{r}] ≤ C_{1} \{ε^{\frac{r}{2}} + ∫_{0}^{t} K_{H}^{r}(t, s)EP̃ [|X̃_{s}^{ε}|^{r}] ds\} ,

where from

EP̃ [|X̃_{t}^{ε}|^{r}] ≤ C_{2} ∑_{l=1}^{∞} ε^{\frac{lr}{2}} (K_{l}^{(r)}1) (t),

for ε ≤ ε_{0}, with C_{1} independent of ε. Therefore

sup_{t≤1} EP̃ [|X̃_{t}^{ε}|^{r}] ≤ C_{3}ε^{\frac{r}{2}} → 0 as ε → 0.  (4.19)

From Lemma 3.2 of [6] we have

∫_{0}^{s} K_{H}^{r}(s, s_{1}) |X̃_{s_{1}}^{ε}|^{r} ds_{1} ≤ sup_{t≤1} ∥K_{H}^{r}(t, .)∥_{L^{r_{1}}} ∫_{0}^{1} |X̃_{s_{1}}^{ε}|^{r_{2}} ds_{1}.  (4.20)

Now from Remark 4.1 of [8],(4.19) and (4.20) it follows that

EP̃ [sup_{s≤t} |X̃_{s}^{ε}|^{r}] ≤ C_{4} \{ε^{\frac{r}{2}} + E [sup_{s≤t} ∫_{0}^{s} K_{H}^{r}(s, s_{1}) |X̃_{s_{1}}^{ε}|^{r} ds_{1}]\} ≤

C_{5} \{ε^{\frac{r}{2}} + sup_{s≤1} EP̃ (|X̃_{s}^{ε}|^{r}) ds\} → 0 as ε → 0.

Finally the exponentially tightness and the lower estimate imply that the rate function is good.

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