22
Large Chapter Deviations for
Small-Noise Sto chastic
Dierential Equations
This lecture is at once the end of our main consideration of diffusions and stochastic calculus, and a rst taste of large deviations
theory. Here we study the divergence between the tra jectories produced by an ordinary dierential equation, and the tra jectories of
the same system perturbed by a small amount of white noise.
Section 22.1 establishes that, in the small noise limit, the SDEs
tra jectories converge in probability on the ODEs tra jectory. This
uses Feller-process convergence.
Section 22.2 upper bounds the rate at which the probability of
large deviations goes to zero as the noise vanishes. The methods are
elementary, but illustrate deeper themes to which we will recur once
we have the tools of ergodic and information theory.
In this chapter, we will use the results we have already obtained about
SDEs to give a rough estimate of a basic problem, frequently arising in practice1
namely taking a system governed by an ordinary dierential equation and seeing
how much eect injecting a small amount of white noise has. More exactly,
we will put an upper bound on the probability that the perturbed tra jectory
goes very far from the unperturbed tra jectory, and see the rate at which this
probability goes to zero as the amplitude of the noise shrinks; this will be
1 For applications in statistical physics and chemistry, see Keizer (1987). For applications
in signal processing and systems theory, see Kushner (1984). For applications in nonparametric regression and estimation, and also radio engineering (!) see Ibragimov and Hasminskii
(1979/1981). The last b o ok is especially recommended for those who care about the connections b etween sto chastic pro cess theory and statistical inference, but unfortunately exp ounding the results, or even just the problems, would require a too-long detour through asymptotic
statistical theory.
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CHAPTER 22. SMALL-NOISE SDES
2
O(eC ). This will be our rst illustration of a large deviations calculation. It
will be crude, but it will also introduce some themes to which we will return
(inshallah!) at greater length towards the end of the course. Then we will see
that the ma jor improvement of the more rened tools is to give a lower bound
to match the upper bound we will calculate now, and see that we at least got
the logarithmic rate right.
I should say before going any further that this example is shamelessly ripped
o from Freidlin and Wentzell (1998, ch. 3, sec. 1, pp. 7071), which is the
book on the sub ject of large deviations for continuous-time processes.
22.1
Convergence in Probability of SDEs to ODEs
To begin with, consider an unperturbed ordinary dierential equation:
d
x(t) = a(x(t))
dt
x(0) = x0 Rd
(22.1)
(22.2)
Assume that a is uniformly Lipschitz-continuous (as in the existence and uniqueness theorem for ODEs, and more to the point for SDEs). Then, for the given,
non-random initial condition, there exists a unique continuous function x which
solves the ODE.
Now, for > 0, consider the SDE
dX = a(X )dt + dW
(22.3)
where W is a standard d-dimensional Wiener process, with non-random initial condition X (0) = x0 . Theorem 216 clearly applies, and consequently so
does Theorem 220, meaning X is a Feller diusion with generator G f (x) =
2
ai (x)i f (x) + 2 2 f (x).
Write X0 for the deterministic solution of the ODE.
d
Our rst assertion is that X X0 as 0. Notice that X0 is a Feller
process2 , whose generator is G0 = ai (x)i . We can apply Theorem 170 on
convergence of Feller processes. Take the class of functions with bounded second
derivatives. This is clearly a core for G0 , and for every G . For every function
f in this class,
G f G0 f
=
=
ai i f (x) +
2
2
2
f (x)
2
2
2
f (x) ai i f (x)
(22.4)
(22.5)
which goes to zero as 0. But this is condition (i) of the convergence theorem,
which is equivalent to condition (iv), that convergence in distribution of the
d
2 You can amuse yourself by showing this. Rememb er that X (t) X (t) is equivalent to
y
x
E [f (Xt )|X0 = y ] E [f (Xt )|X0 = x] for all b ounded continuous f , and the solution of an
ODE dep ends continuously on its initial condition.
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CHAPTER 22. SMALL-NOISE SDES
initial condition implies convergence in distribution of the whole tra jectory.
Since the initial condition is the same non-random point x0 for all , we have
d
P
X X0 as 0. In fact, since X0 is non-random, we have that X X0 .
That last assertion really needs some consideration of metrics on the space of
continuous random functions to make sense (see Appendix A2 of Kallenberg),
but once thats done, the upshot is
Theorem 253 Let (t) = |X (t) X0 (t)|. For every T > 0, > 0,
lim P
0
sup (t) >
=0
(22.6)
0tT
Or, using the maximum-process notation, for every T > 0,
P
(T ) 0
(22.7)
Proof: See above.
This is a version of the weak law of large numbers, and nice enough in its
own way. One crucial limitation, however, is that it tells us nothing about the
rate of convergence. That is, it leaves us clueless about how big the noise can
be, while still leaving us in the small-noise limit. If the rate of convergence were,
say, O( 1/100 ), then this would not be very useful. (In fact, if the convergence
were that slow, we should be really suspicious of numerical solutions of the
unperturbed ODE.)
22.2
Rate of Convergence; Probability of Large
Deviations
Large deviations theory is essentially a study of rates of convergence in probabilistic limit theorems. Here, we will estimate the rate of convergence: our
methods will be crude, but it will turn out that even more rened estimates
wont change the rate, at least not by more than log factors.
Lets go back to the dierence between the perturbed and unperturbed trajectories, through going our now-familiar procedure.
X (t) X0 (t) =
(t)
t
0
t
0
[a(X (s)) a(X0 (s))] ds + W (t)
(22.8)
|a(X (s)) a(X0 (s))| ds + |W (t)|
(22.9)
t
Ka
(T )
0
(s)ds + |W (t)|
t
W (T ) + Ka
(s)ds
(22.10)
(22.11)
0
Applying Gronwalls Inequality (Lemma 214),
(T )
W (T )eKa T
(22.12)
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CHAPTER 22. SMALL-NOISE SDES
The only random component on the RHS is the supremum of the Wiener process,
so were in business, at least once we take on two standard results, one about
the Wiener process itself, the other just about multivariate Gaussians.
Lemma 254 For a standard Wiener process, P (W (t) > a) = 2P (|W (t)| > a).
Proof: Proposition 13.13 (pp. 256257) in Kallenberg.
Lemma 255 If Z is a d-dimensional standard Gaussian (i.e., mean 0 and covariance matrix I ), then
2
2z d2 ez /2
P (|Z | > z ) d/2
2 (d/2)
(22.13)
for suciently large z .
Proof: Each component of Z , Zi N (0, 1). So |Z | =
density function (see, e.g., (Cramr, 1945, sec. 18.1, p. 236))
e
f (z ) =
d
i=1
2
Zi has the
z2
2
z d1 e 22
2d/2 d (d/2)
This is the d-dimensional Maxwell-Boltzmann distribution, sometimes called the
-distribution, because |Z |2 is 2 -distributed with d degrees of freedom. Notice
2
that P (|Z | z ) = P |Z | z 2 , so we will be able to solve this problem in
2
terms of the 2 distribution. Specically, P |Z | z 2 = (d/2, z 2 /2)/(d/2),
where (r, a) is the upper incomplete gamma function. For said function, for
every r, (r, a) ar1 ea for suciently large a (Abramowitz and Stegun,
1964, Eq. 6.5.32, p. 263). Hence (for suciently large z )
= P |Z | z 2
2
(22.14)
=
P (|Z | z )
(d/2, z 2 /2)
(d/2)
(22.15)
z2
d/21 1d/2 z 2 /2
2
e
(d/2)
(22.16)
2
=
Theorem 256 In the limit as
2z d2 ez /2
2d/2 (d/2)
(22.17)
0, for every > 0, T > 0,
log P ( (T ) > ) O(
2
)
(22.18)
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CHAPTER 22. SMALL-NOISE SDES
Proof: Start by directly estimating the probability of the deviation, using
preceding lemmas.
P ( (T ) > ) P |W | (T ) >
eKa T
(22.19)
eKa T
=
2P |W (T )| >
4
2 e2Ka T
2
2d/2 (d/2)
(22.20)
d/21
e
2 e2Ka T
22
(22.21)
if is suciently small, so that 1 is suciently large to apply Lemma 255.
Now take the log and multiply through by 2 :
2
log P ( (T ) > )
4
+
2 log d/2
2 (d/2)
lim
0
2
(22.22)
2
d
1
2
log 2 e2Ka T 2 log
log P ( (T ) > ) 2 e2Ka T
2 e2Ka T
(22.23)
since 2 log 0, and the conclusion follows.
Notice several points.
1. Here gauges the size of the noise, and we take a small noise limit. In many
forms of large deviations theory, we are concerned with large-sample (N
) or long-time (T ) limits. In every case, we will identify some
asymptotic parameter, and obtain limits on the asymptotic probabilities.
There are deviations inequalities which hold non-asymptotically, but they
have a dierent avor, and require dierent machinery. (Some people are
made unc ortable by an 2 rate, and prefer to write the SDE dX =
omf
a(X )dt + dW so as to avoid it. I dont get this.)
2. The magnitude of the deviation does not change as the noise becomes
small. This is basically what makes this a large deviations result. There
is also a theory of moderate deviations, which with any luck well be able
to at least touch on.
3. We only have an upper bound. This is enough to let us know that the
probability of large deviations becomes exponentially small. But we might
be wrong about the rate it could be even faster than weve estimated.
In this case, however, itll turn out that weve got at least the order of
magnitude correct.
4. We also dont have a lower bound on the probability, which is something
that would be very useful in doing reliability analyses. It will turn out
that, under many circumstances, one can obtain a lower bound on the
probability of large deviations, which has the same asymptotic dependence
on as the upper bound.
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CHAPTER 22. SMALL-NOISE SDES
5. Suppose were right about the rate (which, it will turn out, we are), and
it holds both from above and below. It would be nice to be able to say
something like
P ( (T ) > ) C1 (, T )eC2 (,T )
rather than
2
2
log P ( (T ) > ) C2 (, T )
(22.24)
(22.25)
The diculty with making an assertion like 22.24 is that the large deviation probability actually converges on any function which goes to asymptotically to zero! So, to extract the actual rate of dependence, we need to
get a result like 22.25.
More generally, one consequence of Theorem 256 is that SDE tra jectories
which are far from the tra jectory of the ODE have exponentially small probabilities. The vast ma jority of the probability will be concentrated around the
unperturbed tra jectory. Reasonable sample-path functionals can therefore be
well-approximated by averaging their value over some small ( ) neighborhood of
the unperturbed tra jectory. This should sound very similar to Laplaces method
for the evaluate of asymptotic integrals in Euclidean space, and in fact one of
the key parts of large deviations theory is an extension of Laplaces method to
innite-dimensional function spaces.
In addition to this mathematical content, there is also a close connection
to the principle of least action in physics. In classical mechanics, the system
follows the tra jectory of least action, the action of a tra jectory being the
integral of the kinetic minus potential energy along that path. In quantum
mechanics, this is no longer an axiom but a consequence of the dynamics: the
action-minimizing tra jectory is the most probable one, and large deviations from
it have exponentially small probability. Similarly, the theory of large deviations
can be used to establish quite general stochastic principles of least action for
Markovian systems.3
3 For
a fuller discussion, see Eyink (1996),Freidlin and Wentzell (1998, ch. 3).