35
Large Chapter Deviations for
Stochastic Dierential
Equations
This last chapter revisits large deviations for stochastic dierential equations in the small-noise limit, rst raised in Chapter 22.
Section 35.1 establishes the LDP for the Wiener process (Schilders
Theorem).
Section 35.2 proves the LDP for stochastic dierential equations
where the driving noise is independent of the state of the process.
Section 35.3 states the corresponding result for SDEs when the
noise is state-dependent, and gestures in the direction of the proof.
In Chapter 22, we looked at how the diusions X which solve the SDE
dX = a(X )dt + dW, X (0) = x0
(35.1)
converge on the tra jectory x0 (t) solving the ODE
dx
= a(x(t)), x(0) = x0
dt
(35.2)
in the small noise limit, 0. Specically, Theorem 256 gave a (fairly crude)
upper bound on the probability of deviations:
lim
0
2
log P
sup (t) >
0tT
2 e2Ka T
(35.3)
where Ka depends on the Lipschitz coecient of the drift function a. The theory of large deviations for stochastic dierential equations, known as FreidlinWentzell theory for its original developers, shows that, using the metric implicit
238
239
CHAPTER 35. FREIDLIN-WENTZELL THEORY
in the left-hand side of Eq. 35.3, the family of processes X obey a large deviations principle with rate 2 , and a good rate function.
(The full Freidlin-Wentzell theory actually goes somewhat further than just
SDEs, to consider small-noise perturbations of dynamical systems of many sorts,
perturbations by Markov processes (rather than just white noise), etc. Time
does not allow us to consider the full theory (Freidlin and Wentzell, 1998), or
its many applications to nonparametric estimation (Ibragimov and Hasminskii,
1979/1981), systems analysis and signal processing (Kushner, 1984), statistical
mechanics (Olivieri and Vares, 2005), etc.)
As in Chapter 31, the strategy is to rst prove a large deviations principle
for a comparatively simple case, and then transfer it to more subtle processes
which can be represented as appropriate functionals of the basic case. Here,
the basic case is the Wiener process W (t), with t restricted to the unit interval
[0, 1].
35.1
Large Deviations of the Wiener Pro cess
We start with a standard d-dimensional Wiener process W , and consider its dilation by a factor , X (t) = W (t). There are a number of ways of establishing
that X obeys a large deviation principle as 0. One approach (see Dembo
and Zeitouni (1998, ch. 5) starts with establishing an LDP for continuous-time
random walks, ultimately based on the Grtner-Ellis Theorem, and then showa
ing that the convergence of such processes to the Wiener process (the Functional
Central Limit Theorem, Theorem 174 of Chapter 16) is suciently fast that the
LDP carries over. However, this approach involves a number of surprisingly
tricky topological issues, so I will avoid it, in favor of a more probabilistic path,
marked out by Freidlin and Wentzell (Freidlin and Wentzell, 1998, sec. 3.2).
Until further notice, w will denote the supremum norm in the space of
continuous curves over the unit interval, C([0, 1], Rd ).
Denition 459 (Cameron-Martin Spaces) The Cameron-Martin space HT
consists of al l continuous sample paths x C([0, T ], Rd ) where x(0) = 0, x is
absolutely continuous, and its Radon-Nikodym derivative x is square-integrable.
Lemma 460 Cameron-Martin spaces are Hilbert spaces, with norm x
T
2
|x(t)| dt.
0
CM
=
Proof: An exercise (35.1) in verifying that the axioms of a Hilbert space are
satised.
Denition 461 The eective Wiener action of an continuous function x
C([0, t], Rd ) is
1
2
x CM
(35.4)
JT (x)
2
240
CHAPTER 35. FREIDLIN-WENTZELL THEORY
if x HT , and otherwise. In particular,
J1 (x)
1
2
1
0
2
|x(t)| dt
(35.5)
For every j > 0, let LT (j ) = {x : JT (x) j }.
Prop osition 462 Fix a function f H1 , and let Y = X f . Then L (Y ) =
is absolutely continuous with respect to L (X ) = , and the Radon-Nikodym
derivative is
1
d
( w) = exp
d
1
0
w(t) dW
1
22
1
0
2
|w(t)| dt
(35.6)
Proof: This is a special case of Girsanovs Theorem. See Corollary 18.25 on p.
365 of Kallenberg, or, more transparently perhaps, the relevant parts of Liptser
and Shiryaev (2001, vol. I).
Lemma 463 For any , , K > 0, there exists an
P( X x
) e
0
> 0 such that, if
J1 (x)+
2
<
0,
(35.7)
provided x(0) = 0 and J1 (x) < K .
Proof: Using Proposition 462,
P( X x
)
= P( Y 0
=
w
= e
<
)
d
( w)d ( w)
d
J1 (x)
2
e
w
<
1
R1
0
xdW
(35.8)
(35.9)
d (w)
(35.10)
From Lemma 254 in Chapter 22, we can see that P ( W < ) 1 as 0.
So, if is suciently small, P ( W < ) 3/4. Now, applying Chebyshevs
inequality to the integrand,
11
22
P
x dW
J1 (x)
(35.11)
0
1
P
2
0
1
0
E
=
1
x dW
x dW
8 2 J1 (x)
1
0
22
J1 (x)
(35.12)
2
(35.13)
2
|x| dt
1
=
8J1 (x)
4
(35.14)
241
CHAPTER 35. FREIDLIN-WENTZELL THEORY
using the It isometry (Corollary 196). Thus
o
P e
e
w
<
1
1
R1
0
R1
xdW
0
xdW
P( X x
e
22
3
4
(35.15)
1 22 J1 (x)
e
2
1 J1 (x) 22 J1 (x)
2
e
2
d (w) >
J1 (x)
) >
(35.16)
(35.17)
where the second term in the exponent can be made less than any desired by
taking small enough.
Lemma 464 For every j > 0, > 0, let U (j, ) be the open neighborhood of
L1 (j ), i.e., al l the trajectories coming within of a trajectory whose action is
less than or equal to j . Then for any > 0, there is an 0 > 0 such that, if
< 0 and
s
P (X U (j, )) e 2
(35.18)
Proof: Basically, approximating the Wiener process by a continuous piecewiselinear function, and showing that the approximation is suciently ne-grained.
Chose a natural number n, and let Yn, (t) be the piecewise linear random function which coincides with X at times 0, 1/n, 2/n, . . . 1, i.e.,
Yn, (t) = X ([tn]/n) + t
[tn]
n
X ([tn + 1]/n)
(35.19)
We will see that, for large enough n, this is exponentially close to X . First,
though, lets bound the probability in Eq. 35.18.
P (X U (j, ))
= P X U (j, ), X Yn,
+P X U (j, ), X Yn,
P X U (j, ), X Yn,
P (J1 (Yn, ) > j ) + P
<
< +P
X Yn,
(35.20)
X Yn,
(35.21)
(35.22)
J1 (Yn, ) can be gotten at from the increments of the Wiener process:
J1 (Yn, )
=n
=
2
2
n
2
i=1
|W (i/n) W ((i 1)/n)|
2 dn
2
i
i=1
(35.23)
(35.24)
CHAPTER 35. FREIDLIN-WENTZELL THEORY
242
where the i have the 2 distribution with one degree of freedom. Using our
results on such distributions and their sums in Ch. 22, it is not hard to show
that, for suciently small ,
P (J1 (Yn, ) > j )
1 j
e2
2
(35.25)
To estimate the probability that the distance between X and Yn, reaches or
exceeds , start with the independent-increments property of X , and the fact
that the two processes coincide when t = i/n.
P
X Yn,
n
max
P
(i1)/nti/n
i=1
= nP
= nP
nP
|X (t) Yn, (t)|
max |X (t) Yn, (t)|
(35.27)
max | W (t) n W (1/n)|
(35.28)
0t1/n
0t1/n
max | W (t)|
0t1/n
4dnP W1 (1/n)
2
(35.29)
2d
(35.30)
n 2
2d
4dn
e 8d2 2
n
(35.31)
again 2 freely using our calculations from Ch. 22. If n > 4d2 j / 2 , then P
j
12
,
2e
(35.26)
and we have overall
P (X U (j, )) e
j
2
X Yn,
(35.32)
as required.
Prop osition 465 The Cameron-Martin norm has compact level sets.
Proof: See Kallenberg, Lemma 27.7, p. 543.
Theorem 466 (Schilders Theorem) If W is a d-dimensional Wiener process on the unit interval, then X = W obeys an LDP on C([0, 1], Rd ), with
rate 2 and good rate function J1 (x), the eective Wiener action over [0, 1].
Proof: It is easy to show that Lemma 463 implies the large deviation lower
bound for open sets. (Exercise 35.2.) The tricky part is the upper bound. Pick
any closed set C and any > 0. Let s = J1 (C ) . By Lemma 465, the set
K = L1 (s) = {x : J1 (x) s} is compact. By construction, C K = . So
243
CHAPTER 35. FREIDLIN-WENTZELL THEORY
= inf xC,yK x y
use Lemma 464
> 0. Let U be the closed -neighborhood of K . Then
P (X C ) P (X U )
s
2
e
e
lim sup
2
0
(35.34)
(35.35)
J1 (C ) 2
(35.36)
log P (X C ) J1 (C ) 2
log P (X C ) J1 (C ) 2
(35.37)
(35.38)
log P (X C )
2
J1 (C )2
2
(35.33)
2
Since was arbitrary, this completes the proof.
Remark: The trick used here, about establishing results like Lemmas 464
and 463, and then using compact level sets to prove large deviations, works
more generally. See Theorem 3.3 in Freidlin and Wentzell (1998, sec. 3.3).
Corollary 467 Schilders theorem remains true for Wiener processes on [0, T ],
for al l T > 0, with rate function JT , the eective Wiener action on [0, T ].
Proof: If W is a Wiener process on [0, 1], then, for every T , S (W ) = T W (t/T )
is a Wiener process on [0, T ]. (Show this!) Since the mapping S is continuous from C([0, 1], Rd ) to C([0, T ], Rd ), by the Contraction Principle (Theorem
410) the family S (W ) obey an LDP with rate 2 and good rate function
JT (x) = J1 (S 1 (x)). (Notice that S is invertible, so S 1 (x) is a function, not
a set of functions.) Since x H1 i y = S (x) HT , its easy to check that for
such, y (t) = T 1/2 x(t/T ), meaning that
y
CM
T
=
0
T
2
|y (t)| dt =
0
2 dt
|x(t/T )|
T
=x
CM
(35.39)
which completes the proof.
Corollary 468 Schilders theorem remains true for Wiener processes on R+ ,
with good rate function J given by the eective Wiener action on R+ ,
J (x)
1
2
0
2
|x(t)| dt
(35.40)
if x H , J (x) = otherwise.
Proof: For each natural number n, let n x be the restriction of x to the
interval [0, n]. By Corollary 467, each of them obeys an LDP with rate function
2
1n
2 0 |x(t)| dt. Now apply the pro jective limit theorem (420) to get that J (x) =
supn Jn (x), which is clearly Eq. 35.40, as the integrand is non-negative.
CHAPTER 35. FREIDLIN-WENTZELL THEORY
35.2
244
Large Deviations for SDEs with State-Indep endent
Noise
Having established an LDP for the Wiener process, it is fairly straightforward
to get an LDP for stochastic dierential equations where the driving noise is
independent of the state of the diusion process.
Denition 469 (SDE with Small State-Indep endent Noise) An SDE with
small state-independent noise is a stochastic dierential equation of the form
dX = a(X )dt + dW
X (0) = 0
(35.41)
(35.42)
where a : Rd Rd is uniformly Lipschitz continuous.
Notice that any non-random initial condition x0 can be handled by a simple
change of coordinates.
Denition 470 (Eective Action: State-Indep endent Noise) The eective action of a trajectory x H is
J (x)
1
2
t
0
2
|x(t) a(x(t))| dt
(35.43)
and = if x C \ H .
Lemma 471 The map F : C(R+ , Rd ) C(R+ , Rd ) given by
t
x(t) = w(t) +
a(w(s))ds
(35.44)
0
when x = F (w) is continuous.
Proof: This goes rather in the same manner as the proof of existence and
uniqueness for SDEs (Theorem 216). For any w1 , w2 C(R+ , Rd ), set x1 =
F (w1 ), x2 = F (w2 ). From the Lipschitz property of a,
|x1 (t) x2 (t)| w1 w2 + Ka
t
0
|x1 (s) x2 (s)| ds
(35.45)
(writing |y (t)| for the norm of Euclidean vectors y , and x for the supremum
norm of continuous curves). By Gronwalls Inequality (Lemma 214), then,
x1 x2 w1 w2 eKa T
(35.46)
on every interval [0, T ]. So we can make sure that x1 x2 is less than any
desired amount by making sure that w1 w2 is suciently small, and so F
is continuous.
245
CHAPTER 35. FREIDLIN-WENTZELL THEORY
Lemma 472 If w H , then x = F (w) is in H .
Proof: Exercise 35.3.
Theorem 473 (Freidlin-Wentzell Theorem: State-Indep endent Noise)
The It processes X of Denition 469 obey the large deviations principle with
o
rate 2 and good rate function given by the eective action J (x).
Proof: For every , X = F ( W ). Corollary 468 tells us that W obeys
the large deviation principle with rate 2 and good rate function J . Since
(Lemma 471) F is continuous, by the Contraction Principle (Theorem 410) X
also obeys the LDP, with rate given by J (F 1 (x)). If F 1 (x) H = ,
this is . On the other hand, if F 1 (x) does contain curves in H , then
J (F 1 (x)) = J (F 1 (x) H ). By Lemma 472, this implies that x H ,
too. For any curve w F 1 (x) H , x = w + a(x), or w = x a(x).
2
J (w) = 0 |x a(x)| dt is however the eective action of the tra jectory x
(Denition 470).
35.3
Large Deviations for State-Dep endent Noise
If the diusion term in the SDE does depend on the state of the process, one
obtains a very similar LDP to the results in the previous section. However, the
approach must be modied: the mapping from W to X , while still measurable,
is no longer necessarily continuous, so we cant use the contraction principle as
before.
Denition 474 (SDE with Small State-Dep endent Noise) An SDE with
small state-dependent noise is a stochastic dierential equation of the form
dX = a(X )dt + b(X )dW
X (0) = 0
(35.47)
(35.48)
where a and b are uniformly Lipschitz continuous, and b is non-singular.
Denition 475 (Eective Action: State-Dep endent Noise) The eective
action of a trajectory x H is given by
J (x)
where
L(q , p) =
L(x(t), x(t))dt
1
(pi ai (q )) Bij 1 (q ) (pj aj (q ))
2
and
with J (x) = if x C \ H .
(35.49)
0
B (q ) = b(q )bT (q )
(35.50)
(35.51)
CHAPTER 35. FREIDLIN-WENTZELL THEORY
246
Theorem 476 (Freidlin-Wentzell Theorem: State-Dep endent Noise) The
processes X obey a large deviations principle with rate 2 and good rate function equal to the eective action.
Proof: Considerably more complicated. (See, e.g., Dembo and Zeitouni (1998,
sec. 5.6, pp. 213220).) The essence, however, is to consider an approximating It process Xn , where a(Xt ) and b(Xt ) are replaced in Eq. 35.47 by
o
a(Xn ([tn]/n)) and b(Xn ([tn]/n)). Here the mapping from W to Xn is continuous, so its not too hard to show that the latter obey an LDP with a reasonable
rate function, and also that theyre exponentially equivalent (in n) to X .
35.4
Exercises
Exercise 35.1 Prove Lemma 460.
Exercise 35.2 Prove that Lemma 463 implies the large deviations lower bound
for open sets.
Exercise 35.3 Prove Lemma 472. Hint: Use Gronwal ls Inequality (Lemma
214) again to show that F maps HT into HT , and then show that H =
n=1 Hn .