Directory UMM :Journals:Journal_of_mathematics:GMJ:
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: Itˆo equation, Euler scheme, large deviations.
1. Introduction
Let Xt satisfy a SDE
Zt
Xt = x + σ(Xs ) dBs
(mod 1),
t ≥ 0,
X0 = x,
0
1
on the torus T = [0, 1], where Bt 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 Xt is a solution on R1 . Consider the problem
of large deviations for the marginal distribution of the functional
Zt
Ft [X] := f (Xs ) ds, t → ∞.
0
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 (αβ − H(β)) ,
β
where
H(β) := lim t−1 log E exp β
t→∞
Zt
0
f (Xs ) ds ,
(1)
see [2], [10] et al. Suppose we use an approximate solution of the SDE instead
˜ (if any). The problem
of Xt itself (see [8]) with an approximate function H
˜ is close to H; then corresponding L
˜ should be also close to L.
is whether H
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
382
A. YU. VERETENNIKOV
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 Cb3 , f ∈ Cb1 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,
Zt
h
h
Xt = x + σ(X[s/h]h
) dBs ,
(2)
0
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 Xth is
h
σ 2 (X[t/h]h
)∂x2 /2. The approximate value of H is
t
Z
H h (β) := lim t−1 log E exp β f (Xsh ) ds .
(3)
t→∞
0
h
The main question is whether H , – suppose it is well-defined, – approximates
H. If it does, then the approximate rate function
¡
¢
Lh (α) = sup αβ − H h (β)
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 Aβ and Ah,β
(see below) which correspond to the processes Xt and Xth 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
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
383
an analysis of operators at t = 1. Let Xth satisfy (2), ρ := h2/5 , γn (k) :=
inf(t ∈ [kh, (k + 1)h] : |Bt − Bkh | > ρ), γn := inf k≤n−1 γn (k), with a standard
notation inf ∅ = +∞. Notice that P (γn = 1) = 0. Now we consider the process
h
Xmin(t,γ
, or, in the other words, Xth as t ≤ γn . This changes the process Xth
n)
on [0, 1] if and only if γn < 1, that is, if the increments of Bt become extremely
large. The probability of such an event possesses a sub-exponential bound
P (γn < 1) ≤ exp(−nc ),
c > 0.
(4)
Indeed, as n → ∞,
√
P (γn < 1) ≤ nP (γn (1) ≤ h) = nP (sup |Bt | > ρ) = nP (sup |Bt | > ρ/ h)
t≤1
t≤h
√
1
2
= 2nP (|B1 | > ρ n) ∼ 4n √ e−ρ n/2 = 4n9/10 exp(−n1/5 /2).
(ρ n)
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(Aβ ),
where r(Aβ ) is the spectral radius of the semigroup operator on the function
space C(T 1 ),
1
Z
Aβ φ(x) = Ex φ(X1 ) exp β f (Xs ) ds .
0
We will state this assertion in Lemma 1 for the reader’s convenience. We also
need an approximate semigroup operator Ah,β on C(T 1 ),
1
Z
Ah,β φ(x) = Ex φ(X1 ) exp β f (Xsh ) ds ,
0
and r(Aβ,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 6= 0, with
some c = c(g),
0 < c−1 ≤ Ag(x) ≤ c.
(5)
√
√
Notice that for any such g, g ∈ L1 (T 1 ), and k gkL1 (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.
384
A. YU. VERETENNIKOV
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(Aβ ) 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(Aβ ).
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 Aβ and Ah,β are compact in
C(T 1 ).
3. For any b > 0, there exists C such that for any |β| < b,
C −1 ≤ r(Aβ ) ≤ C,
(6)
C −1 ≤ r(Ah,β ) ≤ C.
(7)
| log r(Ah,β ) − log r(Aβ )| ≤ CkAh,β − Aβ kC(T 1 ) .
(8)
and uniformly in h > 0,
4. For any b > 0, there exists C such that for any |β| < b and all h small
enough,
5. (Freidlin–G¨artner, see [2]) The limit (1) does exist, and H(β) coincides
with λ(Aβ ) := log r(Aβ ). Moreover, the limit in (3) exists, too, and
coincides with the spectral radius of the approximate semigroup operator
Ah,β ,
H h (β) = log r(Ah,β ).
(9)
6. For any b > 0 there exist q¯ < 1 and C > 0 such that for any |β| < b,
q(Aβ ) ≤ q¯,
and
∆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 ∈ Cb3 , σ −1 > 0, and f ∈ Cb1 . Then for any b > 0 there
exists C such that for any |β| < b,
kAβ − Ah,β kC(T 1 ) ≤ kAβ − Ah,β kC(R1 ) ≤ Ch1/2 .
(10)
Notice that all Xth ’s and Xt 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) ≤ Lh (α) + Cβ(α) h1/2 ∆−1
0 ;
if, in addition, H is strictly convex at β(α), then
L(α) ≥ Lh (α) − Cβ(α) h1/2 ∆−1
0 ;
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
385
if α 6∈ [inf f, sup f ] , and f 6= const, then
L(α) = +∞
and
Lh (α) = +∞.
3. Proof of Theorem 1
1. We use the idea from [1]. Let kφkC(R1 ) ≤ 1. We are going to compare the
two expressions,
Ex φ(X1 ) exp
µ Z1
0
¶
µ Z1
¶
h
h
βf (Xs )ds
and Ex φ(X1 ) exp
βf (Xs )ds .
0
The goal is to establish a bound for their difference which may only depend on
kφkC , but not on
R 1the modulus of continuity of φ. We will use the representation
Ex φ(X1 ) exp(β 0 f (Xs )ds) = u(0, x), where u(t, x) is an appropriate solution
of the problem
∂t u + Lu + βf u = 0, u(1, x) = φ(x),
and L denotes the generator of the process Xt ,
L = a(x)∂x2 ,
a(x) = σ 2 (x)/2.
R1
It is well-known that u(0, x) = Ex φ(X1 ) exp(β 0 f (X(s))ds). The solution
exists and is unique in the class of functions
\ \ 1,2
Wp,loc ([0, t0 ] × R1 ),
C([0, 1] × R1 )
t0 1
see [7] (also see [9] for an exact reference); we do not use H¨older 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
t0 < 1 (in the next three formulas C = C(t0 , β)):
kukC 0,1 ([0,t0 ]×R1 ) ≤ CkφkC .
Moreover, since the coefficients σ ∈ Cb3 and f ∈ Cb1 , 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,
kukC 1,2 ([0,t0 ]×R1 ) ≤ CkφkC(R1 ) .
(11)
Also, in the maximal cylinder, [0, 1] × R1 ,
kukC([0,1]×R1 ) ≤ CkφkC(R1 ) .
(12)
Finally, we will use the bound (e.g., it is a straightforward consequence from
[3], Ch. 9, Theorem 7),
ku(t, ·)kC 1 (Rd ) ≤ C(1 − t)−1/2 kφkC ,
0 ≤ t < 1.
(13)
386
A. YU. VERETENNIKOV
2. Now, with h = 1/n, we find
1
1
Z
Z
Ex φ(X1h ) exp βf (Xsh )ds − Ex φ(X1 ) exp βf (Xs )ds
0
0
= Ex u(1, X1h ) exp
Z1
0
= Ex u(1, X1h ) exp
βf (Xsh )ds − u(0, x)
Z1
0
+ Ex u(1, X1h ) exp
βf (Xsh )ds − u(0, x) 1(γn < 1)
Z1
0
βf (Xsh )ds − u(0, x) 1(γn ≥ 1).
Here the first expectation is estimated due to the bound (4),
¯
¯
1
¯
¯
Z
¯
¯
h
h
¯Ex u(1, X1 ) exp βf (Xs )ds − u(0, x) 1(γn < 1)¯
¯
¯
¯
¯
0
≤ CkφkC(R1 ) P (γn < 1) ≤ CkφkC(R1 ) exp(−nc ).
Hence it remains to estimate the second term with 1(γn > 1) (recall that P (γn =
1) = 0). Denote Λn = 1(γn > 1), and let us use the identity,
1
Z
Ex u(1, X1h ) exp βf (Xsh )ds − u(0, x) Λn
0
=
n−1
X
k=0
¢
¡
h
exp
Ex u (k + 1/n), X(k+1)/n
Denote
¢
¡
h
exp
−u k/n, Xk/n
Zk/n
0
(k+1)/n
Z
0
βf (Xsh )ds
βf (Xsh ) ds Λn .
Lz = a(z)∂x2 .
is the generator of the process Xsh on k/n ≤ s ≤ (k + 1)/n.
The operator LXk/n
h
Due to the Itˆo formula on the set (k + 1)/n ≤ γn which holds true for any k
including k = n − 1 (see below – at the end of the paper – about the latter case)
(k+1)/n
Z
¢
¡
h
exp
βf (Xsh )ds
Ex u (k + 1)/n, X(k+1)/n
0
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
³
−u k/n, X hk
= Ex Λn
(k+1)/n
Z
(k+1)/n
Z
k/n
= E x Λn
k/n
=
(k+1)/n
Z
k/n
=−
(k+1)/n
Z
k/n
−
(k+1)/n
Z
k/n
n
Zt
exp
0
exp
Ex Λn exp
Ex Λn exp
Ex Λn exp
Zt
0
Zt
Zt
0
Zt
0
0
´
Zk/n
βf (Xsh )ds Λn
exp
387
0
βf (Xsh )ds
βf (Xsh )ds
βf (Xsh )ds
·
∂t + LX hk +
·
¸
f (Xth )
n
−Lu(t, Xth )
+ LX hk
u(t, Xth ) dt
¸
u(t, Xth )
n
dt
i
h
h
h
−a(Xt ) + a(X k ) uxx (t, Xth ) dt
n
t
Z
βf (Xsh ) ds
a(X hk )axx (Xsh ) ds uxx (t, Xth ) dt
n
k/n
t
Z
βf (Xsh )ds
ax (Xsh )σ(X hk ) dBs uxx (t, Xth ) dt
n
k/n
≡ −I1 − I2 .
We have used the identity due to Itˆo’s formula on the set t ≤ γn (in the
sequel, k/n ≤ t ≤ (k + 1)/n),
h
a(Xth ) − a(Xk/n
)=
Zt
h
LXk/n
h a(Xs ) ds +
k/n
Zt
h
ax (Xsh )σ(Xk/n
) dBs .
k/n
3. Consider first the case k/n ≤ 1/2. We have
¯
¯
t
t
¯
¯
Z
Z
¯
¯
¯
h ¯
h
LXk/n
h a(Xs ) ds uxx (t, Xt )¯ ≤ Ch.
¯exp βf (Xsh )ds
¯
¯
¯
¯
0
k/n
Hence, the integral I1 possesses the bound
¯
¯
t
t
¯ (k+1)/n
¯
Z
Z
¯ Z
¯
¯
¯
h
h
h
|I1 | = ¯
Ex Λn exp
LXk/n
βf (Xs )ds
h a(Xs ) ds uxx (t, Xt ) dt¯
¯
¯
¯
¯ k/n
0
k/n
≤ C h2 .
388
A. YU. VERETENNIKOV
Let us estimate I2 :
¯
¯
t
t
¯
¯ (k+1)/n
Z
Z
Z
¯
¯
¯
¯
ax (Xsh )σ(X hk ) dBs uxx (t, Xth ) dt¯
Ex Λn exp βf (Xsh )ds
¯
n
¯
¯
¯
¯ k/n
0
k/n
¯
¯
¯
¯ Zt
(k+1)/n
Z
¯
¯
¯
¯
h
h
≤C
Ex ¯
ax (Xs )σ(X k ) dBs ¯ dt ≤ Ch3/2 .
n
¯
¯
¯
¯ k/n
k/n
Overall, one half of the whole sum is estimated by
¯
(k+1)/n
Z
n/2 ¯
X
¯
¢
¡
h
¯Ex u (k + 1/n), X(k+1)/n
exp
βf (Xsh )ds
¯
k=0 ¯
0
k/n
¯
¯
Z
¯
¢
¡
h
h
−u k/n, Xk/n exp
βf (Xs ) ds Λn ¯¯ ≤ Ch1/2 .
¯
0
4. Consider the case k/n > 1/2, the term I1 . Note that generally speaking
uxx is not bounded as t → 1, therefore, we must get rid of terms like this.
We will replace it by ux 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 Xth,ǫ the solution of the SDE
Zt
h,ǫ
h,ǫ
Xt = x + σ(X[s/h]h
) (dBs + ǫ ds),
(14)
0
∂Xth,ǫ
∂Yth,ǫ
, Zth,ǫ =
∂ǫ
∂ǫ
(both derivative processes are understood in the classical sense, see [2]), and
Yth,ǫ =
Yth = Yth,0 , Zth = Zth,0 .
In this stage we use essentially the factor Λn in all expectations, and hence can
h
, etc. We are going to
restrict ourselves to considering stopped processes, Xt∧γ
n
show the following properties and bounds: for some h0 > 0,
•
h
> 0 for any t > 0, h < h0 ;
(15)
Yt∧γ
n
•
¡
¢
(16)
lim sup sup sup EΛn (Yth )p < ∞, ∀p > 0;
n→∞
•
h 0;
n→∞
h 0.
lim sup sup sup EΛn |Zt∧γ
n
n→∞
=
Zt
(18)
h 0. From (19) we conclude,
h
h
Y(k+1)h
= Ykh
(1 + dk ∆B(k+1)h ) + bk h,
∆B(k+1)h := B(k+1)h − Bkh .
Hence, by induction,
h
Ykh
=
k−1
X
hbj
j=0
k−1
Y
(1 + di ∆B(i+1)h ).
i=j+1
A similar representation holds true for kh ≤ t < (k + 1)h:
Yth
= (1 + dk (Bt − Bkh ))
k−1
X
j=0
hbj
k−1
Y
i=j+1
(1 + di ∆B(i+1)h ) + bk (t − kh).
Now choose h > 0 so small that ρ supk |dk | < 1; it suffices that ρ kσx kC < 1.
Then all values Yth (as t ≤ γn ) are positive.
5. Let us show the bound
sup EΛn |Yth |−p < ∞.
h 1),
n
m>0
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 ≤ γn }, ln(1 + di ∆B(i+1)h ) =
390
A. YU. VERETENNIKOV
di ∆B(i+1)h − d2i (∆B(i+1)h )2 /2 + oi (h), where supi |oi (h)|/h → 0. We can choose
h > 0 so small that inf i (n oi (h)) ≥ − ln 2. Let κ−1 ∈ (0, inf i bi ). Then we get
h
P (|Ykh
| ≤ 1/m; γn > 1) = P
=P
à k−1
X
hbj exp
≤P
k−1
X
j=0
hbj exp
j=0
Ã
hbj
k−1
Y
i=j+1
!
(1 + di ∆B(i+1)h ) ≤ 1/m; γn > 1
!
ln(1 + di ∆B(i+1)h )
i=j+1
j=0
à k−1
X
Ã
à k−1
X
!
≤ 1/m; γn > 1
!
k−1
X
£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2 + oi (h)
i=j+1
!
≤ 1/m; γn > 1
≤P
Ã
=P
Ã
inf
exp
inf
Ã
0≤j≤k−1
0≤j≤k−1
Ã
!
!
k−1
X
£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2 ≤ 2/(mt); γn > 1
i=j+1
k−1
X
i=j+1
!
£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2
!
≤ − log(mt/2); γn > 1)
≤P
Ã
sup
0≤j≤k−1
Ã
!
j
X
£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2
i=0
!
≥ log(mt/2)/(2κ); γn > 1
+P
Ã
!
k−1
X
£
¤
−
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2 ≥ log(mt/2)/(2κ); γn > 1
i=0
:= P1 + P2 .
6. Let us estimate P2 (we use a constant λ > 0 to be fixed later):
P2 ≤ P
Ã
Ã
!
k−1
X
£
¤
exp −λ
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2
i=0
≥ exp ((λ/(2κ)) log(mt/2))
≤ exp (−(λ/(2κ)) log(mt/2))
!
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
Ã
!
k−1
X
£
¤
× E exp −
λdi ∆B(i+1)h ∓ λ2 d2i h − λ(di ∆B(i+1)h )2 /2
i=0
Ã
Ã
≤ exp (−(λ/(2κ)) log(mt/2)) E exp −2λ
Ã
Ã
× exp(Cλ2 ) E exp C
³
Since E exp −2λ
Ã
E exp Cλ
we get
k−1
X
391
Pk−1
i=0
k−1
X
λ(∆B(i+1)h )2
i=0
k−1
X
i=0
di ∆B(i+1)h − 2λ2 d2i h
!!1/2
!!1/2
.
´
di ∆B(i+1)h − 2λ2 d2i h = 1, and for h > 0 small enough
2
(∆B(i+1)h )
i=0
!
=
µ
1
√
1 − λCh
P2 ≤ Cλ,t m−λ/2 ,
¶k
≤
µ
λC
1+
2n
¶n
≤ eλC/2 ,
Cλ,t < ∞.
7. Now let us estimate P1 . Using the Kolmogorov–Doob inequality, we get a
similar bound:
!
Ã
à j
X£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2
P1 ≤ P
sup exp
0≤j≤k−1
i=0
≥ exp ((λ/(2κ)) log(mt/2))
!
≤ exp(−(λ/(2κ)) log(mt/2))
à k−1
!
X£
¤
λdi ∆B(i+1)h ∓ λ2 d2i h − λ(di ∆B(i+1)h )2 /2
× E exp
i=0
¡
¢
≤ exp −(λ/(2κ)) log(mt/2) + Cλ2
Ã
à k−1
!!1/2
k−1
X
X
× E exp 2λ
di ∆B(i+1)h − (4λ2 /2)
d2i h
i=0
≤ Cλ,t m
−λ/2
,
i=0
Cλ,t < ∞.
Hence, choosing λ > 2p + 2, we estimate,
X
h −p
h
EΛn |Ykh
| ≤
(m + 1)p P (|Ykh
| ≤ 1/m; γn > 1)
m>0
≤
X
(m + 1)p Cλ,t m−λ/2 < ∞.
m>0
Thus (17) is established; we remind that the general case t 6= kh is considered
similarly.
392
A. YU. VERETENNIKOV
8. Now (18) follows from (17) and a representation
Zth,ǫ
=2
Zt
h,ǫ
h,ǫ
σx (X[s/h]h
)Y[s/h]h
0
+
ds +
Zt
0
Zt
´2
³
h,ǫ
h,ǫ
(dBs + ǫ ds)
σxx (X[s/h]h
) Y[s/h]h
h,ǫ
h,ǫ
σx (X[s/h]h
)Z[s/h]h
(dBs + ǫ ds).
0
At ǫ = 0 this reads as
Zth = 2
+
Zt
h
h
σx (X[s/h]h
)Y[s/h]h
ds +
0
t
Z
Zt
0
¡ h ¢2
h
σxx (X[s/h]h
) Y[s/h]h
dBs
h
h
σx (X[s/h]h
)Z[s/h]h
dBs .
0
By virtue of (16) and (17) this implies (18).
9. Let Λǫn = 1(γnǫ > 1), where γnǫ is defined as γn for the process Bt +ǫt, t ≥ 0.
To get rid of uxx in our integrals, consider Girsanov’s transformation,
t
t
Z
Z
h,ǫ
a(Xk/n
)axx (Xsh,ǫ ) ds ux (t, Xth,ǫ )
Ex Λǫn exp βf (Xsh,ǫ ) ds
0
k/n
¶
µ
´−1
³
ǫ2
h,ǫ
= const.
exp −ǫB1 −
× Yt
2
We differentiate this identity with respect to ǫ at ǫ = 0 to get the following:
t
t
Z
Z
Λǫ − Λn
h
exp βf (Xsh ) ds
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
0 = limEx n
ǫ→0
ǫ
0
k/n
t
Z
Zt
h
a(Xk/n
)axx (Xsh ) ds uxx (t, Xth )
+ Ex Λn exp βf (Xsh ) ds
0
k/n
t
Zt
Z
h
+ Ex Λn exp βf (Xsh ) ds
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
0
×
¡
¢−1
Yth
Zt
0
βfx (Xsh )Ysh ds
k/n
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
+ Ex Λn exp
¡ ¢−1
× Yth
Zt
Zt
0
βf (Xsh ) ds
Zt
k/n
393
h
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
h
h
ax (Xk/n
)Yk/n
axx (Xsh ) ds
k/n
+ Ex Λn exp
×
¡
¢−1
Yth
Zt
Zt
0
t
Z
h
βf (Xsh ) ds
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
k/n
h
a(Xk/n
)axxx (Xsh )Ysh ds
k/n
+ Ex Λn exp
¡
¢ ³
h −1
¡
¢
h −1
Zt
0
t
Z
h
βf (Xsh ) ds
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
k/n
¡
h
¢ ´
h −2
−Zt Yt
t
t
Z
Z
h
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
+ Ex Λn exp βf (Xsh,ǫ ) ds
× Yt
0
× Yt
k/n
(−B1 ).
Here all the terms with ux but the first one possess the bound by absolute value,
≤ C(1 − t)−1/2 h.
10. Let us to show the bound
¯
¯
t
t
¯
¯
Z
Z
¯ Λǫ − Λ
¯
¯
¯
n
h
exp βf (Xsh ) ds
lim sup ¯Ex n
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )¯
¯
ǫ
ǫ→0 ¯
¯
¯
0
k/n
√
c
≤ Che−n / 1 − t.
We have,
¯
¯
t
t
¯
¯
Z
Z
¯ Λǫ − Λ
¯
¯
¯
n
h
exp βf (Xsh ) ds
lim sup ¯Ex n
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )¯
¯
ǫ
ǫ→0 ¯
¯
¯
0
k/n
¯ ǫ
¯
¯ Λ − Λn ¯
¯.
≤ Ch(1 − t)−1/2 lim sup Ex ¯¯ n
¯
ǫ
ǫ→0
394
A. YU. VERETENNIKOV
The random variables Λn and Λǫn may be represented as follows:
Ã
!
n
Y
Λn =
1
sup |Bt − Bih | ≤ ρ ,
Λǫn =
i=1
n
Y
ih
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: Itˆo equation, Euler scheme, large deviations.
1. Introduction
Let Xt satisfy a SDE
Zt
Xt = x + σ(Xs ) dBs
(mod 1),
t ≥ 0,
X0 = x,
0
1
on the torus T = [0, 1], where Bt 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 Xt is a solution on R1 . Consider the problem
of large deviations for the marginal distribution of the functional
Zt
Ft [X] := f (Xs ) ds, t → ∞.
0
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 (αβ − H(β)) ,
β
where
H(β) := lim t−1 log E exp β
t→∞
Zt
0
f (Xs ) ds ,
(1)
see [2], [10] et al. Suppose we use an approximate solution of the SDE instead
˜ (if any). The problem
of Xt itself (see [8]) with an approximate function H
˜ is close to H; then corresponding L
˜ should be also close to L.
is whether H
c Heldermann Verlag www.heldermann.de
ISSN 1072-947X / $8.00 / °
382
A. YU. VERETENNIKOV
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 Cb3 , f ∈ Cb1 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,
Zt
h
h
Xt = x + σ(X[s/h]h
) dBs ,
(2)
0
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 Xth is
h
σ 2 (X[t/h]h
)∂x2 /2. The approximate value of H is
t
Z
H h (β) := lim t−1 log E exp β f (Xsh ) ds .
(3)
t→∞
0
h
The main question is whether H , – suppose it is well-defined, – approximates
H. If it does, then the approximate rate function
¡
¢
Lh (α) = sup αβ − H h (β)
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 Aβ and Ah,β
(see below) which correspond to the processes Xt and Xth 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
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
383
an analysis of operators at t = 1. Let Xth satisfy (2), ρ := h2/5 , γn (k) :=
inf(t ∈ [kh, (k + 1)h] : |Bt − Bkh | > ρ), γn := inf k≤n−1 γn (k), with a standard
notation inf ∅ = +∞. Notice that P (γn = 1) = 0. Now we consider the process
h
Xmin(t,γ
, or, in the other words, Xth as t ≤ γn . This changes the process Xth
n)
on [0, 1] if and only if γn < 1, that is, if the increments of Bt become extremely
large. The probability of such an event possesses a sub-exponential bound
P (γn < 1) ≤ exp(−nc ),
c > 0.
(4)
Indeed, as n → ∞,
√
P (γn < 1) ≤ nP (γn (1) ≤ h) = nP (sup |Bt | > ρ) = nP (sup |Bt | > ρ/ h)
t≤1
t≤h
√
1
2
= 2nP (|B1 | > ρ n) ∼ 4n √ e−ρ n/2 = 4n9/10 exp(−n1/5 /2).
(ρ n)
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(Aβ ),
where r(Aβ ) is the spectral radius of the semigroup operator on the function
space C(T 1 ),
1
Z
Aβ φ(x) = Ex φ(X1 ) exp β f (Xs ) ds .
0
We will state this assertion in Lemma 1 for the reader’s convenience. We also
need an approximate semigroup operator Ah,β on C(T 1 ),
1
Z
Ah,β φ(x) = Ex φ(X1 ) exp β f (Xsh ) ds ,
0
and r(Aβ,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 6= 0, with
some c = c(g),
0 < c−1 ≤ Ag(x) ≤ c.
(5)
√
√
Notice that for any such g, g ∈ L1 (T 1 ), and k gkL1 (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.
384
A. YU. VERETENNIKOV
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(Aβ ) 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(Aβ ).
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 Aβ and Ah,β are compact in
C(T 1 ).
3. For any b > 0, there exists C such that for any |β| < b,
C −1 ≤ r(Aβ ) ≤ C,
(6)
C −1 ≤ r(Ah,β ) ≤ C.
(7)
| log r(Ah,β ) − log r(Aβ )| ≤ CkAh,β − Aβ kC(T 1 ) .
(8)
and uniformly in h > 0,
4. For any b > 0, there exists C such that for any |β| < b and all h small
enough,
5. (Freidlin–G¨artner, see [2]) The limit (1) does exist, and H(β) coincides
with λ(Aβ ) := log r(Aβ ). Moreover, the limit in (3) exists, too, and
coincides with the spectral radius of the approximate semigroup operator
Ah,β ,
H h (β) = log r(Ah,β ).
(9)
6. For any b > 0 there exist q¯ < 1 and C > 0 such that for any |β| < b,
q(Aβ ) ≤ q¯,
and
∆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 ∈ Cb3 , σ −1 > 0, and f ∈ Cb1 . Then for any b > 0 there
exists C such that for any |β| < b,
kAβ − Ah,β kC(T 1 ) ≤ kAβ − Ah,β kC(R1 ) ≤ Ch1/2 .
(10)
Notice that all Xth ’s and Xt 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) ≤ Lh (α) + Cβ(α) h1/2 ∆−1
0 ;
if, in addition, H is strictly convex at β(α), then
L(α) ≥ Lh (α) − Cβ(α) h1/2 ∆−1
0 ;
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
385
if α 6∈ [inf f, sup f ] , and f 6= const, then
L(α) = +∞
and
Lh (α) = +∞.
3. Proof of Theorem 1
1. We use the idea from [1]. Let kφkC(R1 ) ≤ 1. We are going to compare the
two expressions,
Ex φ(X1 ) exp
µ Z1
0
¶
µ Z1
¶
h
h
βf (Xs )ds
and Ex φ(X1 ) exp
βf (Xs )ds .
0
The goal is to establish a bound for their difference which may only depend on
kφkC , but not on
R 1the modulus of continuity of φ. We will use the representation
Ex φ(X1 ) exp(β 0 f (Xs )ds) = u(0, x), where u(t, x) is an appropriate solution
of the problem
∂t u + Lu + βf u = 0, u(1, x) = φ(x),
and L denotes the generator of the process Xt ,
L = a(x)∂x2 ,
a(x) = σ 2 (x)/2.
R1
It is well-known that u(0, x) = Ex φ(X1 ) exp(β 0 f (X(s))ds). The solution
exists and is unique in the class of functions
\ \ 1,2
Wp,loc ([0, t0 ] × R1 ),
C([0, 1] × R1 )
t0 1
see [7] (also see [9] for an exact reference); we do not use H¨older 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
t0 < 1 (in the next three formulas C = C(t0 , β)):
kukC 0,1 ([0,t0 ]×R1 ) ≤ CkφkC .
Moreover, since the coefficients σ ∈ Cb3 and f ∈ Cb1 , 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,
kukC 1,2 ([0,t0 ]×R1 ) ≤ CkφkC(R1 ) .
(11)
Also, in the maximal cylinder, [0, 1] × R1 ,
kukC([0,1]×R1 ) ≤ CkφkC(R1 ) .
(12)
Finally, we will use the bound (e.g., it is a straightforward consequence from
[3], Ch. 9, Theorem 7),
ku(t, ·)kC 1 (Rd ) ≤ C(1 − t)−1/2 kφkC ,
0 ≤ t < 1.
(13)
386
A. YU. VERETENNIKOV
2. Now, with h = 1/n, we find
1
1
Z
Z
Ex φ(X1h ) exp βf (Xsh )ds − Ex φ(X1 ) exp βf (Xs )ds
0
0
= Ex u(1, X1h ) exp
Z1
0
= Ex u(1, X1h ) exp
βf (Xsh )ds − u(0, x)
Z1
0
+ Ex u(1, X1h ) exp
βf (Xsh )ds − u(0, x) 1(γn < 1)
Z1
0
βf (Xsh )ds − u(0, x) 1(γn ≥ 1).
Here the first expectation is estimated due to the bound (4),
¯
¯
1
¯
¯
Z
¯
¯
h
h
¯Ex u(1, X1 ) exp βf (Xs )ds − u(0, x) 1(γn < 1)¯
¯
¯
¯
¯
0
≤ CkφkC(R1 ) P (γn < 1) ≤ CkφkC(R1 ) exp(−nc ).
Hence it remains to estimate the second term with 1(γn > 1) (recall that P (γn =
1) = 0). Denote Λn = 1(γn > 1), and let us use the identity,
1
Z
Ex u(1, X1h ) exp βf (Xsh )ds − u(0, x) Λn
0
=
n−1
X
k=0
¢
¡
h
exp
Ex u (k + 1/n), X(k+1)/n
Denote
¢
¡
h
exp
−u k/n, Xk/n
Zk/n
0
(k+1)/n
Z
0
βf (Xsh )ds
βf (Xsh ) ds Λn .
Lz = a(z)∂x2 .
is the generator of the process Xsh on k/n ≤ s ≤ (k + 1)/n.
The operator LXk/n
h
Due to the Itˆo formula on the set (k + 1)/n ≤ γn which holds true for any k
including k = n − 1 (see below – at the end of the paper – about the latter case)
(k+1)/n
Z
¢
¡
h
exp
βf (Xsh )ds
Ex u (k + 1)/n, X(k+1)/n
0
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
³
−u k/n, X hk
= Ex Λn
(k+1)/n
Z
(k+1)/n
Z
k/n
= E x Λn
k/n
=
(k+1)/n
Z
k/n
=−
(k+1)/n
Z
k/n
−
(k+1)/n
Z
k/n
n
Zt
exp
0
exp
Ex Λn exp
Ex Λn exp
Ex Λn exp
Zt
0
Zt
Zt
0
Zt
0
0
´
Zk/n
βf (Xsh )ds Λn
exp
387
0
βf (Xsh )ds
βf (Xsh )ds
βf (Xsh )ds
·
∂t + LX hk +
·
¸
f (Xth )
n
−Lu(t, Xth )
+ LX hk
u(t, Xth ) dt
¸
u(t, Xth )
n
dt
i
h
h
h
−a(Xt ) + a(X k ) uxx (t, Xth ) dt
n
t
Z
βf (Xsh ) ds
a(X hk )axx (Xsh ) ds uxx (t, Xth ) dt
n
k/n
t
Z
βf (Xsh )ds
ax (Xsh )σ(X hk ) dBs uxx (t, Xth ) dt
n
k/n
≡ −I1 − I2 .
We have used the identity due to Itˆo’s formula on the set t ≤ γn (in the
sequel, k/n ≤ t ≤ (k + 1)/n),
h
a(Xth ) − a(Xk/n
)=
Zt
h
LXk/n
h a(Xs ) ds +
k/n
Zt
h
ax (Xsh )σ(Xk/n
) dBs .
k/n
3. Consider first the case k/n ≤ 1/2. We have
¯
¯
t
t
¯
¯
Z
Z
¯
¯
¯
h ¯
h
LXk/n
h a(Xs ) ds uxx (t, Xt )¯ ≤ Ch.
¯exp βf (Xsh )ds
¯
¯
¯
¯
0
k/n
Hence, the integral I1 possesses the bound
¯
¯
t
t
¯ (k+1)/n
¯
Z
Z
¯ Z
¯
¯
¯
h
h
h
|I1 | = ¯
Ex Λn exp
LXk/n
βf (Xs )ds
h a(Xs ) ds uxx (t, Xt ) dt¯
¯
¯
¯
¯ k/n
0
k/n
≤ C h2 .
388
A. YU. VERETENNIKOV
Let us estimate I2 :
¯
¯
t
t
¯
¯ (k+1)/n
Z
Z
Z
¯
¯
¯
¯
ax (Xsh )σ(X hk ) dBs uxx (t, Xth ) dt¯
Ex Λn exp βf (Xsh )ds
¯
n
¯
¯
¯
¯ k/n
0
k/n
¯
¯
¯
¯ Zt
(k+1)/n
Z
¯
¯
¯
¯
h
h
≤C
Ex ¯
ax (Xs )σ(X k ) dBs ¯ dt ≤ Ch3/2 .
n
¯
¯
¯
¯ k/n
k/n
Overall, one half of the whole sum is estimated by
¯
(k+1)/n
Z
n/2 ¯
X
¯
¢
¡
h
¯Ex u (k + 1/n), X(k+1)/n
exp
βf (Xsh )ds
¯
k=0 ¯
0
k/n
¯
¯
Z
¯
¢
¡
h
h
−u k/n, Xk/n exp
βf (Xs ) ds Λn ¯¯ ≤ Ch1/2 .
¯
0
4. Consider the case k/n > 1/2, the term I1 . Note that generally speaking
uxx is not bounded as t → 1, therefore, we must get rid of terms like this.
We will replace it by ux 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 Xth,ǫ the solution of the SDE
Zt
h,ǫ
h,ǫ
Xt = x + σ(X[s/h]h
) (dBs + ǫ ds),
(14)
0
∂Xth,ǫ
∂Yth,ǫ
, Zth,ǫ =
∂ǫ
∂ǫ
(both derivative processes are understood in the classical sense, see [2]), and
Yth,ǫ =
Yth = Yth,0 , Zth = Zth,0 .
In this stage we use essentially the factor Λn in all expectations, and hence can
h
, etc. We are going to
restrict ourselves to considering stopped processes, Xt∧γ
n
show the following properties and bounds: for some h0 > 0,
•
h
> 0 for any t > 0, h < h0 ;
(15)
Yt∧γ
n
•
¡
¢
(16)
lim sup sup sup EΛn (Yth )p < ∞, ∀p > 0;
n→∞
•
h 0;
n→∞
h 0.
lim sup sup sup EΛn |Zt∧γ
n
n→∞
=
Zt
(18)
h 0. From (19) we conclude,
h
h
Y(k+1)h
= Ykh
(1 + dk ∆B(k+1)h ) + bk h,
∆B(k+1)h := B(k+1)h − Bkh .
Hence, by induction,
h
Ykh
=
k−1
X
hbj
j=0
k−1
Y
(1 + di ∆B(i+1)h ).
i=j+1
A similar representation holds true for kh ≤ t < (k + 1)h:
Yth
= (1 + dk (Bt − Bkh ))
k−1
X
j=0
hbj
k−1
Y
i=j+1
(1 + di ∆B(i+1)h ) + bk (t − kh).
Now choose h > 0 so small that ρ supk |dk | < 1; it suffices that ρ kσx kC < 1.
Then all values Yth (as t ≤ γn ) are positive.
5. Let us show the bound
sup EΛn |Yth |−p < ∞.
h 1),
n
m>0
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 ≤ γn }, ln(1 + di ∆B(i+1)h ) =
390
A. YU. VERETENNIKOV
di ∆B(i+1)h − d2i (∆B(i+1)h )2 /2 + oi (h), where supi |oi (h)|/h → 0. We can choose
h > 0 so small that inf i (n oi (h)) ≥ − ln 2. Let κ−1 ∈ (0, inf i bi ). Then we get
h
P (|Ykh
| ≤ 1/m; γn > 1) = P
=P
à k−1
X
hbj exp
≤P
k−1
X
j=0
hbj exp
j=0
Ã
hbj
k−1
Y
i=j+1
!
(1 + di ∆B(i+1)h ) ≤ 1/m; γn > 1
!
ln(1 + di ∆B(i+1)h )
i=j+1
j=0
à k−1
X
Ã
à k−1
X
!
≤ 1/m; γn > 1
!
k−1
X
£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2 + oi (h)
i=j+1
!
≤ 1/m; γn > 1
≤P
Ã
=P
Ã
inf
exp
inf
Ã
0≤j≤k−1
0≤j≤k−1
Ã
!
!
k−1
X
£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2 ≤ 2/(mt); γn > 1
i=j+1
k−1
X
i=j+1
!
£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2
!
≤ − log(mt/2); γn > 1)
≤P
Ã
sup
0≤j≤k−1
Ã
!
j
X
£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2
i=0
!
≥ log(mt/2)/(2κ); γn > 1
+P
Ã
!
k−1
X
£
¤
−
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2 ≥ log(mt/2)/(2κ); γn > 1
i=0
:= P1 + P2 .
6. Let us estimate P2 (we use a constant λ > 0 to be fixed later):
P2 ≤ P
Ã
Ã
!
k−1
X
£
¤
exp −λ
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2
i=0
≥ exp ((λ/(2κ)) log(mt/2))
≤ exp (−(λ/(2κ)) log(mt/2))
!
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
Ã
!
k−1
X
£
¤
× E exp −
λdi ∆B(i+1)h ∓ λ2 d2i h − λ(di ∆B(i+1)h )2 /2
i=0
Ã
Ã
≤ exp (−(λ/(2κ)) log(mt/2)) E exp −2λ
Ã
Ã
× exp(Cλ2 ) E exp C
³
Since E exp −2λ
Ã
E exp Cλ
we get
k−1
X
391
Pk−1
i=0
k−1
X
λ(∆B(i+1)h )2
i=0
k−1
X
i=0
di ∆B(i+1)h − 2λ2 d2i h
!!1/2
!!1/2
.
´
di ∆B(i+1)h − 2λ2 d2i h = 1, and for h > 0 small enough
2
(∆B(i+1)h )
i=0
!
=
µ
1
√
1 − λCh
P2 ≤ Cλ,t m−λ/2 ,
¶k
≤
µ
λC
1+
2n
¶n
≤ eλC/2 ,
Cλ,t < ∞.
7. Now let us estimate P1 . Using the Kolmogorov–Doob inequality, we get a
similar bound:
!
Ã
à j
X£
¤
di ∆B(i+1)h − (di ∆B(i+1)h )2 /2
P1 ≤ P
sup exp
0≤j≤k−1
i=0
≥ exp ((λ/(2κ)) log(mt/2))
!
≤ exp(−(λ/(2κ)) log(mt/2))
à k−1
!
X£
¤
λdi ∆B(i+1)h ∓ λ2 d2i h − λ(di ∆B(i+1)h )2 /2
× E exp
i=0
¡
¢
≤ exp −(λ/(2κ)) log(mt/2) + Cλ2
Ã
à k−1
!!1/2
k−1
X
X
× E exp 2λ
di ∆B(i+1)h − (4λ2 /2)
d2i h
i=0
≤ Cλ,t m
−λ/2
,
i=0
Cλ,t < ∞.
Hence, choosing λ > 2p + 2, we estimate,
X
h −p
h
EΛn |Ykh
| ≤
(m + 1)p P (|Ykh
| ≤ 1/m; γn > 1)
m>0
≤
X
(m + 1)p Cλ,t m−λ/2 < ∞.
m>0
Thus (17) is established; we remind that the general case t 6= kh is considered
similarly.
392
A. YU. VERETENNIKOV
8. Now (18) follows from (17) and a representation
Zth,ǫ
=2
Zt
h,ǫ
h,ǫ
σx (X[s/h]h
)Y[s/h]h
0
+
ds +
Zt
0
Zt
´2
³
h,ǫ
h,ǫ
(dBs + ǫ ds)
σxx (X[s/h]h
) Y[s/h]h
h,ǫ
h,ǫ
σx (X[s/h]h
)Z[s/h]h
(dBs + ǫ ds).
0
At ǫ = 0 this reads as
Zth = 2
+
Zt
h
h
σx (X[s/h]h
)Y[s/h]h
ds +
0
t
Z
Zt
0
¡ h ¢2
h
σxx (X[s/h]h
) Y[s/h]h
dBs
h
h
σx (X[s/h]h
)Z[s/h]h
dBs .
0
By virtue of (16) and (17) this implies (18).
9. Let Λǫn = 1(γnǫ > 1), where γnǫ is defined as γn for the process Bt +ǫt, t ≥ 0.
To get rid of uxx in our integrals, consider Girsanov’s transformation,
t
t
Z
Z
h,ǫ
a(Xk/n
)axx (Xsh,ǫ ) ds ux (t, Xth,ǫ )
Ex Λǫn exp βf (Xsh,ǫ ) ds
0
k/n
¶
µ
´−1
³
ǫ2
h,ǫ
= const.
exp −ǫB1 −
× Yt
2
We differentiate this identity with respect to ǫ at ǫ = 0 to get the following:
t
t
Z
Z
Λǫ − Λn
h
exp βf (Xsh ) ds
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
0 = limEx n
ǫ→0
ǫ
0
k/n
t
Z
Zt
h
a(Xk/n
)axx (Xsh ) ds uxx (t, Xth )
+ Ex Λn exp βf (Xsh ) ds
0
k/n
t
Zt
Z
h
+ Ex Λn exp βf (Xsh ) ds
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
0
×
¡
¢−1
Yth
Zt
0
βfx (Xsh )Ysh ds
k/n
ON APPROXIMATE LARGE DEVIATIONS FOR 1D DIFFUSION
+ Ex Λn exp
¡ ¢−1
× Yth
Zt
Zt
0
βf (Xsh ) ds
Zt
k/n
393
h
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
h
h
ax (Xk/n
)Yk/n
axx (Xsh ) ds
k/n
+ Ex Λn exp
×
¡
¢−1
Yth
Zt
Zt
0
t
Z
h
βf (Xsh ) ds
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
k/n
h
a(Xk/n
)axxx (Xsh )Ysh ds
k/n
+ Ex Λn exp
¡
¢ ³
h −1
¡
¢
h −1
Zt
0
t
Z
h
βf (Xsh ) ds
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
k/n
¡
h
¢ ´
h −2
−Zt Yt
t
t
Z
Z
h
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )
+ Ex Λn exp βf (Xsh,ǫ ) ds
× Yt
0
× Yt
k/n
(−B1 ).
Here all the terms with ux but the first one possess the bound by absolute value,
≤ C(1 − t)−1/2 h.
10. Let us to show the bound
¯
¯
t
t
¯
¯
Z
Z
¯ Λǫ − Λ
¯
¯
¯
n
h
exp βf (Xsh ) ds
lim sup ¯Ex n
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )¯
¯
ǫ
ǫ→0 ¯
¯
¯
0
k/n
√
c
≤ Che−n / 1 − t.
We have,
¯
¯
t
t
¯
¯
Z
Z
¯ Λǫ − Λ
¯
¯
¯
n
h
exp βf (Xsh ) ds
lim sup ¯Ex n
a(Xk/n
)axx (Xsh ) ds ux (t, Xth )¯
¯
ǫ
ǫ→0 ¯
¯
¯
0
k/n
¯ ǫ
¯
¯ Λ − Λn ¯
¯.
≤ Ch(1 − t)−1/2 lim sup Ex ¯¯ n
¯
ǫ
ǫ→0
394
A. YU. VERETENNIKOV
The random variables Λn and Λǫn may be represented as follows:
Ã
!
n
Y
Λn =
1
sup |Bt − Bih | ≤ ρ ,
Λǫn =
i=1
n
Y
ih