Georgian Mathematical Journal Volume 10 (2003), Number 2, 381–399

ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION

A. YU. VERETENNIKOV

In memory of Rezo Chitashvili

Abstract. We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

2000 Mathematics Subject Classification: 60F10, 60H10, 60H35, 65C30. Key words and phrases: Itô equation, Euler scheme, large deviations.

1. Introduction

Let X_{t} satisfy a SDE

X_{t} = x + ∫_{0}^{t} σ(X_{s}) dB_{s} (mod 1), t ≥ 0, X_{0} = x,

on the torus T^{1} = [0, 1], where B_{t} is a standard Wiener process, σ is bounded, Borel measurable, and non-degenerate. Another way to understand solutions (mod (1)) and the problem setting on T^{1} is to say that functions σ and f (below) are periodic with period 1, while X_{t} is a solution on R^{1}. Consider the problem of large deviations for the marginal distribution of the functional

F_{t}[X] := ∫_{0}^{t} f(X_{s}) ds, t → ∞.

Often, the answer is described via the rate function L which in “good cases” is a Fenchel–Legendre transformation of the (convex) function

L(α) = sup_{β} (αβ − H(β)) ,

where

H(β) := lim_{t→∞} t^{−1} log E exp (β ∫_{0}^{t} f(X_{s}) ds) ,  (1)

see [2], [10] et al. Suppose we use an approximate solution of the SDE instead of X_{t} itself (see [8]) with an approximate function H̃ (if any). The problem is whether H̃ is close to H; then corresponding L̃ should be also close to L.

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

For SDEs with a unit diffusion and drift, under appropriate conditions the answer turned out to be positive, see [11] (1D) and [12] (multi-dimensional case). The case with a variable diffusion is more technically involved. This is the reason why we do not consider drift here, just to simplify the calculus though a bit, and concentrate on the main new difficulty. This difficulty, which did not appear in the unit diffusion case, is a possible degeneration of the first ‘Malliavin derivative’ of the approximation process. To keep it positive, a stopping rule is applied. While differentiating the Girsanov identity, however, this stopping rule turns out to be a disadvantage, because of non-differentiability of indicator functions. Hence a good deal of the proof in the paper consists of explanation why Girsanov’s formula still can be differentiated.

Assumptions for the main result are: σ is bounded, non-degenerate, periodic and of the class C_{b}^{3}, f ∈ C_{b}^{1} and also periodic. Perhaps these rather strong assumptions are due to the method, however, they may also reflect the vulnerability of large deviations to small disturbances of the system under consideration.

We consider a standard Euler scheme,

X_{t}^{h} = x + ∫_{0}^{t} σ(X_{[s/h]h}^{h}) dB_{s},  (2)

where [a] denotes the integer part of the value a. However, for the purpose of the analysis of semigroup operators, we propose a minor modification of this scheme on a time interval [0, 1] which includes a stopping rule. Some reason for this will be presented shortly. Notice that the quasi-generator of X_{t}^{h} is σ^{2}(X_{[t/h]h}^{h})∂_{x}^{2}/2. The approximate value of H is

H^{h}(β) := lim_{t→∞} t^{−1} log E exp (β ∫_{0}^{t} f(X_{s}^{h}) ds) .  (3)

The main question is whether H^{h}, – suppose it is well-defined, – approximates H. If it does, then the approximate rate function

L^{h}(α) = sup_{β} (αβ − H^{h}(β))

will be also close to L(α), see [11]. Later on we shall state conditions which ensure the existence of both limits (1) and (3). For the sake of simplicity, we consider h’s such that n = 1/h is an integer.

The approach is based on the analysis of semigroup operators A^{β} and A^{h,β} (see below) which correspond to the processes X_{t} and X_{t}^{h} for 0 ≤ t ≤ 1. Due to assertions (8) and (9) in Lemma 1, the double inequality (10) is crucial in establishing an approximate equality of H^{h} and H. Hence we present this inequality as a main result of the paper, see below Theorem 1.

We propose the following modification of the algorithm (2) as 0 ≤ t ≤ 1: to investigate the properties of both operators; emphasize that this does not mean any improvement in the definition of the function H^{h}, but only concerns an analysis of operators at t = 1. Let X_{t}^{h} satisfy (2), ρ := h^{2/5}, γ_{n}(k) := inf(t ∈ [kh, (k + 1)h] : |B_{t} − B_{kh}| > ρ), γ_{n} := inf_{k≤n−1} γ_{n}(k), with a standard notation inf ∅ = +∞. Notice that P (γ_{n} = 1) = 0. Now we consider the process X_{min(t,γ_{n})}^{h}, or, in the other words, X_{t}^{h} as t ≤ γ_{n}. This changes the process X_{t}^{h} on [0, 1] if and only if γ_{n} < 1, that is, if the increments of B_{t} become extremely large. The probability of such an event possesses a sub-exponential bound

P (γ_{n} < 1) ≤ exp(−n^{c}), c > 0.  (4)

Indeed, as n → ∞,

P (γ_{n} < 1) ≤ nP (γ_{n}(1) ≤ h) = nP (sup_{t≤h} |B_{t}| > ρ) = nP (sup_{t≤1} |B_{t}| > ρ/\sqrt{h})

= 2nP (|B_{1}| > ρ\sqrt{n}) ∼ 4n\frac{1}{(ρ\sqrt{n})}e^{−ρ^{2}n/2} = 4n^{9/10} exp(−n^{1/5}/2).

Possibly, the result and technique can be also helpful in investigating moderate deviations of approximate solutions of SDEs.

2. Auxiliary and Main Results

In this section we consider the process and operator on the torus T^{1} and C(T^{1}) respectively. It is known [2] that the limit H does exist and is equal to

H(β) = log r(A^{β}),

where r(A^{β}) is the spectral radius of the semigroup operator on the function space C(T^{1}),

A^{β}ϕ(x) = E_{x}ϕ(X_{1}) exp (β ∫_{0}^{1} f(X_{s}) ds) .

We will state this assertion in Lemma 1 for the reader’s convenience. We also need an approximate semigroup operator A^{h,β} on C(T^{1}),

A^{h,β}ϕ(x) = E_{x}ϕ(X_{1}) exp (β ∫_{0}^{1} f(X_{s}^{h}) ds) ,

and r(A^{β,h}) denotes its spectral radius.

Recall [5] that the operator A is called positive if f ≥ 0 implies Af ≥ 0. The operator A is called 1-bounded if for any g : g ∈ C(T^{1}), g ≥ 0, g ≠ 0, with some c = c(g),

0 < c^{−1} ≤ Ag(x) ≤ c.  (5)

Notice that for any such g, \sqrt{g} ∈ L_{1}(T^{1}), and ∥\sqrt{g}∥_{L_{1}(T^{1})} > 0. Denote

q(A) = inf(q : |λ| ≤ q r(A), λ ∈ spectrum of A);

this value is called the spectral gap of the operator A.

Lemma 1.

1. (Frobenius–Krasnosel’skii: [5], Theorem 11.5) Let A be a positive compact 1-bounded operator on C(T^{1}). Then the spectral radius r(A^{β}) is an isolated simple spectrum point of the operator A, its eigenfunction is positive, and there exists q < 1 such that all other spectrum points λ satisfy the bound |λ| ≤ q r(A^{β}).

For the next assertions, 2 to 6, it is assumed that σ, σ^{−1}, f are bounded and Borel, and σ > 0.

2. (Krylov and Safonov [6]) Two operators A^{β} and A^{h,β} are compact in C(T^{1}).

3. For any b > 0, there exists C such that for any |β| < b,

C^{−1} ≤ r(A^{β}) ≤ C,  (6)

and uniformly in h > 0,

C^{−1} ≤ r(A^{h,β}) ≤ C.  (7)

4. For any b > 0, there exists C such that for any |β| < b and all h small enough,

| log r(A^{h,β}) − log r(A^{β})| ≤ C∥A^{h,β} − A^{β}∥_{C(T^{1})}.  (8)

5. (Freidlin–Gärtner, see [2]) The limit (1) does exist, and H(β) coincides with λ(A^{β}) := log r(A^{β}). Moreover, the limit in (3) exists, too, and coincides with the spectral radius of the approximate semigroup operator A^{h,β},

H^{h}(β) = log r(A^{h,β}).  (9)

6. For any b > 0 there exist q̄ < 1 and C > 0 such that for any |β| < b,

q(A^{β}) ≤ q̄, and Δ_{0} ≥ C^{−1}.

For additional details of the proof of this lemma, which just combines several important technical results from various areas, see [11]. Now we formulate the main result of the paper.

Theorem 1. Let σ ∈ C_{b}^{3}, σ^{−1} > 0, and f ∈ C_{b}^{1}. Then for any b > 0 there exists C such that for any |β| < b,

∥A^{β} − A^{h,β}∥_{C(T^{1})} ≤ ∥A^{β} − A^{h,β}∥_{C(R^{1})} ≤ Ch^{1/2}.  (10)

Notice that all X_{t}^{h}’s and X_{t} are defined on a common probability space with a Wiener process.

Corollary 1. Under the assumptions of Theorem 1, for any inf f < α < sup f, there exists β(α) such that L(α) = αβ(α) − H(β(α)), and

L(α) ≤ L^{h}(α) + C_{β(α)}h^{1/2}Δ_{0}^{−1};

if, in addition, H is strictly convex at β(α), then

L(α) ≥ L^{h}(α) − C_{β(α)}h^{1/2}Δ_{0}^{−1};

if α ∉ [inf f, sup f] , and f ≠ const, then

L(α) = +∞ and L^{h}(α) = +∞.

3. Proof of Theorem 1

1. We use the idea from [1]. Let ∥ϕ∥_{C(R^{1})} ≤ 1. We are going to compare the two expressions,

E_{x}ϕ(X_{1}) exp ( ∫_{0}^{1} βf(X_{s})ds) and E_{x}ϕ(X_{1}^{h}) exp ( ∫_{0}^{1} βf(X_{s}^{h})ds).

The goal is to establish a bound for their difference which may only depend on ∥ϕ∥_{C}, but not on the modulus of continuity of ϕ. We will use the representation E_{x}ϕ(X_{1}) exp(β ∫_{0}^{1} f(X_{s})ds) = u(0, x), where u(t, x) is an appropriate solution of the problem

∂_{t}u + Lu + βfu = 0, u(1, x) = ϕ(x),

and L denotes the generator of the process X_{t},

L = a(x)∂_{x}^{2}, a(x) = σ^{2}(x)/2.

It is well-known that u(0, x) = E_{x}ϕ(X_{1}) exp(β ∫_{0}^{1} f(X(s))ds). The solution exists and is unique in the class of functions

C([0, 1] × R^{1}) ⋂_{t_{0}<1} ⋂_{p>1} W_{p,loc}^{1,2}([0, t_{0}] × R^{1}),

see [7] (also see [9] for an exact reference); we do not use Hölder classes (which would be appropriate) because of the lack of references concerning equations with only continuous initial data (i.e. ϕ). Embedding theorems and a priori inequalities in Sobolev spaces (cf. [7], ch. 3 and 5) imply the following for any t_{0} < 1 (in the next three formulas C = C(t_{0}, β)):

∥u∥_{C^{0,1}([0,t_{0}]×R^{1})} ≤ C∥ϕ∥_{C}.

Moreover, since the coefficients σ ∈ C_{b}^{3} and f ∈ C_{b}^{1}, we can differentiate the equation with respect to x, and by virtue of the same a priori estimates in Sobolev spaces, embedding theorems and the equation,

∥u∥_{C^{1,2}([0,t_{0}]×R^{1})} ≤ C∥ϕ∥_{C(R^{1})}.  (11)

Also, in the maximal cylinder, [0, 1] × R^{1},

∥u∥_{C([0,1]×R^{1})} ≤ C∥ϕ∥_{C(R^{1})}.  (12)

Finally, we will use the bound (e.g., it is a straightforward consequence from [3], Ch. 9, Theorem 7),

∥u(t, ·)∥_{C^{1}(R^{d})} ≤ C(1 − t)^{−1/2}∥ϕ∥_{C}, 0 ≤ t < 1.  (13)

2. Now, with h = 1/n, we find

E_{x}ϕ(X_{1}^{h}) exp ( ∫_{0}^{1} βf(X_{s}^{h})ds) − E_{x}ϕ(X_{1}) exp ( ∫_{0}^{1} βf(X_{s})ds)

= E_{x}u(1, X_{1}^{h}) exp ( ∫_{0}^{1} βf(X_{s}^{h})ds) − u(0, x)

= E_{x} (u(1, X_{1}^{h}) exp ( ∫_{0}^{1} βf(X_{s}^{h})ds) − u(0, x)) 1(γ_{n} < 1)

+ E_{x} (u(1, X_{1}^{h}) exp ( ∫_{0}^{1} βf(X_{s}^{h})ds) − u(0, x)) 1(γ_{n} ≥ 1).

Here the first expectation is estimated due to the bound (4),

|E_{x} (u(1, X_{1}^{h}) exp ( ∫_{0}^{1} βf(X_{s}^{h})ds) − u(0, x)) 1(γ_{n} < 1)|

≤ C∥ϕ∥_{C(R^{1})}P (γ_{n} < 1) ≤ C∥ϕ∥_{C(R^{1})} exp(−n^{c}).

Hence it remains to estimate the second term with 1(γ_{n} > 1) (recall that P (γ_{n} = 1) = 0). Denote Λ_{n} = 1(γ_{n} > 1), and let us use the identity,

E_{x} (u(1, X_{1}^{h}) exp ( ∫_{0}^{1} βf(X_{s}^{h})ds) − u(0, x)) Λ_{n}

= ∑_{k=0}^{n−1} E_{x} [u ((k + 1/n), X_{(k+1)/n}^{h}) exp ( ∫_{0}^{(k+1)/n}βf(X_{s}^{h})ds)

−u (k/n, X_{k/n}^{h}) exp ( ∫_{0}^{k/n}βf(X_{s}^{h}) ds)] Λ_{n}.

Denote

L_{z} = a(z)∂_{x}^{2}.

The operator L_{X_{k/n}^{h}} is the generator of the process X_{s}^{h} on k/n ≤ s ≤ (k + 1)/n. Due to the Itô formula on the set (k + 1)/n ≤ γ_{n} which holds true for any k including k = n−1 (see below – at the end of the paper – about the latter case)

E_{x} [u ((k + 1)/n, X_{(k+1)/n}^{h}) exp ( ∫_{0}^{(k+1)/n}βf(X_{s}^{h})ds)

ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION 387

−u (k/n, Xh_{\frac{k}{n}} ) exp ( ∫_{0}^{k/n}βf(X_{s}^{h})ds)] Λ_{n}

= E_{x} Λ_{n} ∫_{k/n}^{(k+1)/n}exp ( ∫_{0}^{t} βf(X_{s}^{h})ds) [∂_{t} + L_{Xh ; \frac{k}{n}} + f(X_{t}^{h})] u(t, X_{t}^{h}) dt

= E_{x} Λ_{n} ∫_{k/n}^{(k+1)/n}exp ( ∫_{0}^{t} βf(X_{s}^{h})ds) [−Lu(t, X_{t}^{h}) + L_{Xh ; \frac{k}{n}} u(t, X_{t}^{h})] dt

= ∫_{k/n}^{(k+1)/n}E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h})ds) [−a(X_{t}^{h}) + a(Xh_{\frac{k}{n}})] u_{xx}(t, X_{t}^{h}) dt

= − ∫_{k/n}^{(k+1)/n}E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(Xh_{\frac{k}{n}})a_{xx}(X_{s}^{h}) ds) u_{xx}(t, X_{t}^{h}) dt

− ∫_{k/n}^{(k+1)/n}E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h})ds) ( ∫_{k/n}^{t} a_{x}(X_{s}^{h})σ(Xh_{\frac{k}{n}}) dB_{s}) u_{xx}(t, X_{t}^{h}) dt

≡ −I_{1} − I_{2}.

We have used the identity due to Itô’s formula on the set t ≤ γ_{n} (in the sequel, k/n ≤ t ≤ (k + 1)/n),

a(X_{t}^{h}) − a(X_{k/n}^{h}) = ∫_{k/n}^{t} L_{X_{k/n}^{h}}a(X_{s}^{h}) ds + ∫_{k/n}^{t} a_{x}(X_{s}^{h})σ(X_{k/n}^{h}) dB_{s}.

3. \underline{Consider first the case k/n ≤ 1/2.} We have

|exp ( ∫_{0}^{t} βf(X_{s}^{h})ds) ( ∫_{k/n}^{t} L_{X_{k/n}^{h}}a(X_{s}^{h}) ds) u_{xx}(t, X_{t}^{h})| ≤ Ch.

Hence, \underline{the integral I_{1}} possesses the bound

|I_{1}| = | ∫_{k/n}^{(k+1)/n}E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h})ds) ( ∫_{k/n}^{t} L_{X_{k/n}^{h}}a(X_{s}^{h}) ds) u_{xx}(t, X_{t}^{h}) dt|

≤ C h^{2}.

Let us estimate I_{2}:

| ∫_{k/n}^{(k+1)/n}E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h})ds) ( ∫_{k/n}^{t} a_{x}(X_{s}^{h})σ(Xh_{\frac{k}{n}}) dB_{s}) u_{xx}(t, X_{t}^{h}) dt|

≤ C ∫_{k/n}^{(k+1)/n}E_{x} | ∫_{k/n}^{t} a_{x}(X_{s}^{h})σ(Xh_{\frac{k}{n}}) dB_{s}| dt ≤ Ch^{3/2}.

Overall, one half of the whole sum is estimated by

∑_{k=0}^{n/2} |E_{x} [u ((k + 1/n), X_{(k+1)/n}^{h}) exp ( ∫_{0}^{(k+1)/n}βf(X_{s}^{h})ds)

−u (k/n, X_{k/n}^{h}) exp ( ∫_{0}^{k/n}βf(X_{s}^{h}) ds)] Λ_{n}| ≤ Ch^{1/2}.

4. \underline{Consider the case k/n > 1/2, the term I_{1}}. Note that generally speaking u_{xx} is not bounded as t → 1, therefore, we must get rid of terms like this. We will replace it by u_{x} which has an integrable singularity at zero; ideally, it would be nice to fulfill a second differentiation and replace this term by the bounded u, but there are technical reasons which prevent us from this second differentiation. We use the Bismut approach to stochastic calculus, based on Girsanov’s formula. Denote by X_{t}^{h,ϵ} the solution of the SDE

X_{t}^{h,ϵ} = x + ∫_{0}^{t} σ(X_{[s/h]h}^{h,ϵ}) (dB_{s} + ϵ ds),  (14)

Y_{t}^{h,ϵ} = \frac{∂X_{t}^{h,ϵ}}{∂ϵ} , Z_{t}^{h,ϵ} = \frac{∂Y_{t}^{h,ϵ}}{∂ϵ}

(both derivative processes are understood in the classical sense, see [2]), and

Y_{t}^{h} = Y_{t}^{h,0}, Z_{t}^{h} = Z_{t}^{h,0}.

In this stage we use essentially the factor Λ_{n} in all expectations, and hence can restrict ourselves to considering stopped processes, X_{t∧γ_{n}}^{h}, etc. We are going to show the following properties and bounds: for some h_{0} > 0,

•

Y_{t∧γ_{n}}^{h} > 0 for any t > 0, h < h_{0};  (15)
•
lim sup_{n→∞} sup_{h<h_{0}} sup_{0≤t≤1} EΛ_{n} ((Y_{t}^{h})^{p}) < ∞, ∀p > 0;  (16)
•
lim sup_{n→∞} sup_{h<h_{0}} sup_{0≤t≤1} EΛ_{n}(Y_{t}^{h})^{−p} < ∞, ∀p > 0;  (17)

•

lim sup_{n→∞} sup_{h<h_{0}} sup_{0≤t≤1} EΛ_{n} (|Z_{t∧γ_{n}}^{h}|^{p}) < ∞, ∀p > 0.  (18)

We always have for t ≤ γ_{n},

Y_{t}^{h,ϵ} = ∫_{0}^{t} σ(X_{[s/h]h}^{h,ϵ}) ds + ∫_{0}^{t} σ_{x}(X_{[s/h]h}^{h,ϵ})Y_{[s/h]h}^{h,ϵ} (dB_{s} + ϵ ds).

So, for ϵ = 0,

Y_{t}^{h} = ∫_{0}^{t} σ(X_{[s/h]h}^{h}) ds + ∫_{0}^{t} σ_{x}(X_{[s/h]h}^{h})Y_{[s/h]h}^{h} dB_{s}.  (19)

All the properties (16)–(18) mentioned above follow from this representation even though we cannot assert that Y_{t}^{h} nor (Y_{t}^{h})^{−1} is bounded. The first consequence of (19) is the bound (16). It follows from the Burkholder – Davies – Gundy inequality. (A similar bound holds true without γ_{n}, too; but we will not use it here.) Next, denote b_{k} = σ(X_{kh}^{h}), and d_{k} = σ_{x}(X_{kh}^{h}). Both random sequences are bounded, and inf_{k} inf_{ω} b_{k} > 0. From (19) we conclude,

Y_{(k+1)h}^{h} = Y_{kh}^{h}(1 + d_{k}ΔB_{(k+1)h}) + b_{k}h, ΔB_{(k+1)h} := B_{(k+1)h} − B_{kh}.

Hence, by induction,

Y_{kh}^{h} = ∑_{j=0}^{k−1} hb_{j} ∏_{i=j+1}^{k−1}(1 + d_{i}ΔB_{(i+1)h}).

A similar representation holds true for kh ≤ t < (k + 1)h:

Y_{t}^{h} = (1 + d_{k}(B_{t} − B_{kh})) ∑_{j=0}^{k−1} hb_{j} ∏_{i=j+1}^{k−1}(1 + d_{i}ΔB_{(i+1)h}) + b_{k}(t − kh).

Now choose h > 0 so small that ρ sup_{k} |d_{k}| < 1; it suffices that ρ ∥σ_{x}∥_{C} < 1. Then all values Y_{t}^{h} (as t ≤ γ_{n}) are positive.

5. Let us show the bound

sup_{h<h_{0}} EΛ_{n}|Y_{t}^{h}|^{−p} < ∞.

We have

EΛ_{n}|Y_{t}^{h}|^{−p} ≤ ∑_{m>0}(m + 1)^{p}P (|Y_{t∧γ_{n}}^{h}| ≤ 1/m; γ_{n} > 1),

so that we only need to estimate the probabilities under the sum. For the sake of simplicity, we restrict the calculus to the case 0 < t = kh ≤ 1; the case 0 < (k − 1)h < t < kh ≤ 1 is considered similarly although it requires more new notations. Notice that on {(i + 1)/n ≤ γ_{n}}, ln(1 + d_{i}ΔB_{(i+1)h}) =

390 A. YU. VERETENNIKOV

d_{i}ΔB_{(i+1)h} − d_{i}^{2}(ΔB_{(i+1)h})^{2}/2 + o_{i}(h), where sup_{i} |o_{i}(h)|/h → 0. We can choose h > 0 so small that inf_{i}(n o_{i}(h)) ≥ − ln 2. Let κ^{−1} ∈ (0, inf_{i} b_{i}). Then we get

P (|Y_{kh}^{h}| ≤ 1/m; γ_{n} > 1) = P (∑_{j=0}^{k−1} hb_{j} ∏_{i=j+1}^{k−1}(1 + d_{i}ΔB_{(i+1)h}) ≤ 1/m; γ_{n} > 1)

= P (∑_{j=0}^{k−1} hb_{j} exp ( ∑_{i=j+1}^{k−1} ln(1 + d_{i}ΔB_{(i+1)h})) ≤ 1/m; γ_{n} > 1)

≤ P (∑_{j=0}^{k−1} hb_{j} exp ( ∑_{i=j+1}^{k−1} [d_{i}ΔB_{(i+1)h} − (d_{i}ΔB_{(i+1)h})^{2}/2 + o_{i}(h)])

≤ 1/m; γ_{n} > 1)

≤ P ( inf_{0≤j≤k−1} exp ( ∑_{i=j+1}^{k−1} [d_{i}ΔB_{(i+1)h} − (d_{i}ΔB_{(i+1)h})^{2}/2]) ≤ 2/(mt); γ_{n} > 1)

= P ( inf_{0≤j≤k−1} ( ∑_{i=j+1}^{k−1} [d_{i}ΔB_{(i+1)h} − (d_{i}ΔB_{(i+1)h})^{2}/2])

≤ − log(mt/2); γ_{n} > 1))

≤ P ( sup_{0≤j≤k−1} (∑_{i=0}^{j} [d_{i}ΔB_{(i+1)h} − (d_{i}ΔB_{(i+1)h})^{2}/2])

≥ log(mt/2)/(2κ); γ_{n} > 1)

+ P (− ∑_{i=0}^{k−1} [d_{i}ΔB_{(i+1)h} − (d_{i}ΔB_{(i+1)h})^{2}/2] ≥ log(mt/2)/(2κ); γ_{n} > 1)

:= P_{1} + P_{2}.

6. Let us estimate P_{2} (we use a constant λ > 0 to be fixed later):

P_{2} ≤ P (exp (−λ ∑_{i=0}^{k−1} [d_{i}ΔB_{(i+1)h} − (d_{i}ΔB_{(i+1)h})^{2}/2])
≥ exp ((λ/(2κ)) log(mt/2)) )

≤ exp (−(λ/(2κ)) log(mt/2))

× E exp (− ∑_{i=0}^{k−1} [λd_{i}ΔB_{(i+1)h} ∓ λ^{2}d_{i}^{2}h − λ(d_{i}ΔB_{(i+1)h})^{2}/2])

≤ exp (−(λ/(2κ)) log(mt/2)) (E exp (−2λ ∑_{i=0}^{k−1} d_{i}ΔB_{(i+1)h} − 2λ^{2}d_{i}^{2}h))^{1/2}

× exp(Cλ^{2}) (E exp (C ∑_{i=0}^{k−1} λ(ΔB_{(i+1)h})^{2}))^{1/2} .

Since E exp (−2λ ∑_{i=0}^{k−1} d_{i}ΔB_{(i+1)h} − 2λ^{2}d_{i}^{2}h) = 1, and for h > 0 small enough

E exp (Cλ ∑_{i=0}^{k−1}(ΔB_{(i+1)h})^{2}) = (\frac{1}{\sqrt{1 − λCh}})^{k} ≤ (1 + \frac{λC}{2n} )^{n} ≤ e^{λC/2},

we get

P_{2} ≤ C_{λ,t}m^{−λ/2}, C_{λ,t} < ∞.

7. Now let us estimate P_{1}. Using the Kolmogorov–Doob inequality, we get a similar bound:

P_{1} ≤ P ( sup_{0≤j≤k−1} exp (∑_{i=0}^{j} [d_{i}ΔB_{(i+1)h} − (d_{i}ΔB_{(i+1)h})^{2}/2])

≥ exp ((λ/(2κ)) log(mt/2)) )

≤ exp(−(λ/(2κ)) log(mt/2))

× E exp (∑_{i=0}^{k−1} [λd_{i}ΔB_{(i+1)h} ∓ λ^{2}d_{i}^{2}h − λ(d_{i}ΔB_{(i+1)h})^{2}/2])

≤ exp (−(λ/(2κ)) log(mt/2) + Cλ^{2})

× (E exp (2λ ∑_{i=0}^{k−1} d_{i}ΔB_{(i+1)h} − (4λ^{2}/2) ∑_{i=0}^{k−1} d_{i}^{2}h))^{1/2}

≤ C_{λ,t}m^{−λ/2}, C_{λ,t} < ∞.

Hence, choosing λ > 2p + 2, we estimate,

EΛ_{n}|Y_{kh}^{h}|^{−p} ≤ ∑_{m>0}(m + 1)^{p}P (|Y_{kh}^{h}| ≤ 1/m; γ_{n} > 1)

≤ ∑_{m>0}(m + 1)^{p}C_{λ,t}m^{−λ/2} < ∞.

Thus (17) is established; we remind that the general case t ≠ kh is considered similarly.

8. Now (18) follows from (17) and a representation

Z_{t}^{h,ϵ} = 2 ∫_{0}^{t} σ_{x}(X_{[s/h]h}^{h,ϵ})Y_{[s/h]h}^{h,ϵ} ds + ∫_{0}^{t} σ_{xx}(X_{[s/h]h}^{h,ϵ}) (Y_{[s/h]h}^{h,ϵ})^{2} (dB_{s} + ϵ ds)

+ ∫_{0}^{t} σ_{x}(X_{[s/h]h}^{h,ϵ})Z_{[s/h]h}^{h,ϵ}(dB_{s} + ϵ ds).

At ϵ = 0 this reads as

Z_{t}^{h} = 2 ∫_{0}^{t} σ_{x}(X_{[s/h]h}^{h})Y_{[s/h]h}^{h} ds + ∫_{0}^{t} σ_{xx}(X_{[s/h]h}^{h}) (Y_{[s/h]h}^{h})^{2} dB_{s}
+ ∫_{0}^{t} σ_{x}(X_{[s/h]h}^{h})Z_{[s/h]h}^{h}dB_{s}.

By virtue of (16) and (17) this implies (18).

9. Let Λ_{n}^{ϵ} = 1(γ_{n}^{ϵ} > 1), where γ_{n}^{ϵ} is defined as γ_{n} for the process B_{t}+ϵt, t ≥ 0. To get rid of u_{xx} in our integrals, consider Girsanov’s transformation,

E_{x} Λ_{n}^{ϵ} exp ( ∫_{0}^{t} βf(X_{s}^{h,ϵ}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h,ϵ})a_{xx}(X_{s}^{h,ϵ}) ds) u_{x}(t, X_{t}^{h,ϵ})

× (Y_{t}^{h,ϵ})^{−1} exp (−ϵB_{1} − \frac{ϵ^{2}}{2} ) = const.

We differentiate this identity with respect to ϵ at ϵ = 0 to get the following:

0 = lim_{ϵ→0}E_{x} \frac{Λ_{n}^{ϵ} − Λ_{n}}{ϵ} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{xx}(t, X_{t}^{h})

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} ∫_{0}^{t} βf_{x}(X_{s}^{h})Y_{s}^{h} ds

ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION 393

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} ∫_{k/n}^{t} a_{x}(X_{k/n}^{h})Y_{k/n}^{h}a_{xx}(X_{s}^{h}) ds

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xxx}(X_{s}^{h})Y_{s}^{h} ds

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} (−Z_{t}^{h} (Y_{t}^{h})^{−2})

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h,ϵ}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} (−B_{1}).

Here all the terms with u_{x} but the first one possess the bound by absolute value, ≤ C(1 − t)^{−1/2}h.

10. Let us to show the bound

lim sup_{ϵ→0} |E_{x} \frac{Λ_{n}^{ϵ} − Λ_{n}}{ϵ} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})|
≤ Che^{−n^{c}}/\sqrt{1 − t}.

We have,

lim sup_{ϵ→0} |E_{x} \frac{Λ_{n}^{ϵ} − Λ_{n}}{ϵ} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})|

≤ Ch(1 − t)^{−1/2} lim sup_{ϵ→0} E_{x} |\frac{Λ_{n}^{ϵ} − Λ_{n}}{ϵ} | .

The random variables Λ_{n} and Λ_{n}^{ϵ} may be represented as follows:

Λ_{n} = ∏_{i=1}^{n} 1 ( sup_{ih<t<(i+1)h} |B_{t} − B_{ih}| ≤ ρ) ,

Λ_{n}^{ϵ} = ∏_{i=1}^{n} 1 ( sup_{ih<t<(i+1)h} |B_{t} − B_{ih} + ϵ(t − ih)| ≤ ρ) .

Hence

|Λ_{n}^{ϵ} − Λ_{n}| ≤ ∑_{i=1}^{n} |1 ( sup_{ih<t<(i+1)h} |B_{t} − B_{ih} + ϵ(t − ih)| ≤ ρ)

−1 ( sup_{ih<t<(i+1)h} |B_{t} − B_{ih}| ≤ ρ)|

≤ ∑_{i=1}^{n} 1 (ρ − ϵh ≤ sup_{ih<t<(i+1)h} |B_{t} − B_{ih}| ≤ ρ + ϵh) .

So, as ϵ ≪ h,

E_{x} |\frac{Λ_{n}^{ϵ} − Λ_{n}}{ϵ} | ≤ nϵ^{−1}E_{x}1 (ρ − ϵh ≤ sup_{0<t<h} |B_{t}| ≤ ρ + ϵh)

= nϵ^{−1}P_{x} (h^{−1/2}(ρ − ϵh) ≤ sup_{0<t<1} |B_{t}| ≤ h^{−1/2}(ρ + ϵh))

= 4nϵ^{−1} ∫_{h^{−1/2}(ρ−ϵh)}^{h^{−1/2}(ρ+ϵh)}(2π)^{−1/2} exp(−x^{2}/2) dx

≤ 4nϵ^{−1}2h^{−1/2}ϵh(2π)^{−1/2} exp(−(h^{−1/2}(ρ − ϵh))^{2}/2)

= \frac{8}{\sqrt{2π}} n^{1/2} exp (−(h^{−1/2}(h^{2/5} − ϵh))^{2}/2)

∼ \frac{8}{\sqrt{2π}} n^{1/2} exp (−n^{1/5}/2) = o(h), n → ∞.

Thus, in fact, all the terms with u_{x} including the first one possess the bound by absolute value, ≤ C(1 − t)^{−1/2}h. After integration with respect to t from kh to (k + 1)h, for k ≤ n − 2 this gives a bound

|E_{x} Λ_{n} ∫_{k/n}^{(k+1)/n}exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{xx}(t, X_{t}^{h}) dt|

≤ C(1 − (k + 1)h)^{−1/2}h^{2}.

After summation over n/2 < k ≤ n − 2, we get

∑_{n/2<k≤n−2} |E_{x} Λ_{n} ∫_{k/n}^{(k+1)/n}exp ( ∫_{0}^{(k+1)/n}βf(X_{s}^{h}) ds)

× ( ∫_{k/n}^{(k+1)/n}a(X_{k/n}^{h})a_{xx}(X_{s}^{h}) ds) u_{xx}(t, X_{t}^{h}) dt| ≤ Ch.

Notice that we have used a ∈ C_{b}^{3}, f ∈ C_{b}^{1}.

11. Let us consider the case k = n−1. The reason why it should be estimated separately is that we cannot apply the Itô formula directly up to t = 1 since u_{x} and u_{xx} may be unbounded.

For all terms but the first one, we get an upper bound, with 1 − h < T < 1,

∫_{1−h}^{T} Ch(1 − t)^{−1/2} dt ≤ Ch.

The first term is bounded by the value

Ce^{−n^{c}} ∫_{1−h}^{T} (1 − t)^{−1/2} dt ≤ Ch^{1/2}e^{−n^{c}} ≤ Ch.

The bound is, of course, rather rough, but we cannot make better the inequality in the step (10). Now we put T → 1 and apply the Fatou lemma to get the same bound with T = 1.

12. The \underline{integral I_{2} for k/n > 1/2} is estimated similarly. We have

E_{x} Λ_{n}^{ϵ} exp ( ∫_{0}^{t} βf(X_{s}^{h,ϵ}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h,ϵ})a_{x}(X_{s}^{h,ϵ}) dB_{s}^{ϵ}) u_{x}(t, X_{t}^{h,ϵ})

× (Y_{t}^{h,ϵ})^{−1} exp (−ϵB_{1} − \frac{ϵ^{2}}{2} ) = const.

Now differentiate this identity with respect to ϵ, which at ϵ = 0 reads,

0 = lim_{ϵ→0} E_{x} \frac{Λ_{n}^{ϵ} − Λ_{n}}{ϵ} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{x}(t, X_{t}^{h})

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{xx}(t, X_{t}^{h})

396 A. YU. VERETENNIKOV

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} ∫_{0}^{t} βf_{x}(X_{s}^{h})Y_{s}^{h} ds

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} ∫_{k/n}^{t} σ_{x}(X_{k/n}^{h})Y_{k/n}^{h}a_{x}(X_{s}^{h}) dB_{s}

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{xx}(X_{s}^{h})Y_{s}^{h} dB_{s}

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} (−Z_{t}^{h} (Y_{t}^{h})^{−2})

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h,ϵ}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{x}(t, X_{t}^{h})

× (Y_{t}^{h})^{−1} (−B_{1})

+ E_{x} Λ_{n} exp ( ∫_{0}^{t} βf(X_{s}^{h,ϵ}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h}) (Y_{t}^{h})^{−1} .

Here all the terms with u_{x} but the first one possess the bound by absolute value, ≤ C(1 − t)^{−1/2}h^{1/2}, or better. After integration with respect to t from kh to (k + 1)h, for k ≤ n − 2 this gives a bound ≤ C(1 − (k + 1)h)^{−1/2}h^{3/2}. After summation over n/2 < k ≤ n − 2, we therefore get a bound

∑_{n/2<k≤n−2} |E_{x} Λ_{n} ∫_{k/n}^{(k+1)/n}exp ( ∫_{0}^{(k+1)/n}βf(X_{s}^{h}) ds)

× ( ∫_{k/n}^{(k+1)/n}σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{xx}(t, X_{t}^{h}) dt| ≤ Ch^{1/2}.

13. Similarly to step (10), one gets a bound

lim sup_{ϵ→0} |E_{x} \frac{Λ_{n}^{ϵ} − Λ_{n}}{ϵ} exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) ds) u_{x}(t, X_{t}^{h})|

≤ Ch^{1/2}e^{−n^{c}}/\sqrt{1 − t}.

Hence all the terms with u_{x} including the first one possess the bound by absolute value, ≤ C(1−t)^{−1/2}h^{1/2}. After integration with respect to t from kh to (k+1)h, for k ≤ n − 2 this gives a bound

|E_{x} Λ_{n} ∫_{k/n}^{(k+1)/n}exp ( ∫_{0}^{t} βf(X_{s}^{h}) ds) ( ∫_{k/n}^{t} σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) dB_{s}) u_{xx}(t, X_{t}^{h}) dt|

≤ C(1 − (k + 1)h)^{−1/2}h^{3/2}.

After summation over n/2 < k ≤ n − 2, we get

∑_{n/2<k≤n−2} |E_{x} Λ_{n} ∫_{k/n}^{(k+1)/n}exp ( ∫_{0}^{(k+1)/n}βf(X_{s}^{h}) ds)

× ( ∫_{k/n}^{(k+1)/n}σ(X_{k/n}^{h})a_{x}(X_{s}^{h}) ds) u_{xx}(t, X_{t}^{h}) dt| ≤ Ch^{1/2}.

Here we have used σ ∈ C_{b}^{2}, f ∈ C_{b}^{1}.

14. Let us consider the case k = n − 1. For all terms but the first one, we get, with 1 − h < T < 1,

∫_{1−h}^{T} Ch^{1/2}(1 − t)^{−1/2} dt ≤ Ch.

The first term is bounded by the value

Ce^{−n^{c}} ∫_{1−h}^{T} (1 − t)^{−1/2} dt ≤ Ch^{1/2}e^{−n^{c}} ≤ Ch.

Now we put T → 1 and apply the Fatou lemma to get the same bound for integrals up to 1. Combining all the bounds obtained, we get

∥A^{h,β} − A^{β}∥ ≤ Ch^{1/2}.

The theorem is proved.

Acknowledgements

First attempts in studying the problem with unit diffusion and non-trivial drift were made jointly with Jean Memin who then had to leave this project for his other duties to the author’s regret as his interest and hints were very essential. This work could not have been done without discussions on positive operators with Alexander Sobolev and Mark Krasnosel’skii. The paper is written in memory of Rezo Chitashvili, a great enthusiast of stochastic analysis who shared generously his enthusiasm with others. The author is deeply thankful to all these colleagues and friends of his, including the anonymous referee.

This work was supported by grants EPSRC-GR/R40746/01, NFGRF 2301863, INTAS-99-0590 and RFBR-00-01-22000.

References

1. V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104(1996), No. 1, 43–60.

2. M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems. (Translated from the Russian) Fundamental Principles of Mathematical Sciences, 260. Springer-Verlag, New York, 1984.

3. A. Friedman, Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

4. T. Kato, Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, 1976.

5. M. A. Krasnosel’skii, E. A. Lifshitz, and A. V. Sobolev, Positive linear systems. The method of positive operators. (Translated from Russian) Sigma Series in Applied Mathematics, 5. Heldermann Verlag, Berlin, 1989.

6. N. V. Krylov and M. V. Safonov, A certain property of solutions of parabolic equations with measurable coefficients. Math. USSR, Izv. 16(1981), 151–164.

7. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural’ceva, Linear and quasilinear equations of parabolic type. (Translated from Russian) Translations of Mathematical Monographs. 23. American Mathematical Society, Providence, 1968.

8. G. N. Milstein, Numerical integration of stochastic differential equations. Translated and revised from the 1988 Russian original. Mathematics and its Applications, 313. Kluwer Academic Publishers Group, Dordrecht, 1995.

9. A. Yu. Veretennikov, Parabolic equations and Itô stochastic equations with coefficients discontinuous in the time variable. (Russian) Mat. Zametki 31(1982), No. 4, 549– 557; English transl.: Math. Notes 31(1982), 278–283.

10. A. Yu. Veretennikov, On large deviations for diffusion processes with measurable coefficients. (Russian) Uspekhi Mat. Nauk 50(1995), No. 5(305), 135–146; English transl.: Russian Math. Surveys 50(1995), No. 5, 977–987.

11. A. Yu. Veretennikov, On large deviations for approximations of SDEs. Probab. Theory Related Fields 125 (2003), No. 1, 135–152.

12. A. Yu. Veretennikov, On large deviations for approximations of SDEs on the torus. (Russian) Teoriya Veroyatnost. i Primenen. 47(2002), No. 4, 772–780; English translation: Theory Probab. Appl. 47(to appear).

(Received 19.11.2002; revised 23.02.2003)

Authors’s addresses:

School of Mathematics University of Leeds UK

Mathematics Department University of Kansas Lawrence, KS USA

Institute of Information Transmission Problems Moscow, Russia E-mail: veretenn@maths.leeds.ac.uk