Notes on Large Deviations in Economics and Financenwilliam/ldpnotes.pdfNotes on Large Deviations 1 Introduction † What is large deviation theory? † Loosely: a theory of rare events

Notes on Large Deviations

in Economics and Finance

Noah WilliamsPrinceton University

and NBER

http://www.princeton.edu/∼noahw

Notes on Large Deviations 1

Introduction

• What is large deviation theory?

• Loosely: a theory of rare events. Typically provide exponentialbound on probability of such events and characterize them.

• Large growth area in math/applied probability in last ≈ 20years. Seeing some applications in economics.

• Why is it useful?

• Often interested in characterizing extreme events in themselves(crashes, failures, busts).

• Bounding probability of extreme events can characterizelikelihood of “typical” events.


Formal Definition of LDP

Consider Zε on (Ω,F , P ), taking values in X .

A rate function S : X → [0,∞] has the property that for anyM < ∞ the level set x ∈ X : S(x) ≤ M is compact.

Definition: A sequence Zε satisfies a large deviation principle(LDP) on X with rate function S and speed ε if:

1. For each closed subset F of X ,:

lim supε→0

ε log P Zε ∈ F ≤ − infx∈F

S(x).

2. For each open subset G of X ,:

lim infε→0

ε log P Zε ∈ G ≥ − infx∈G

S(x).


Interpretation

• Under regularity, upper & lower hold with equality:

limε→0

ε log P Zε ∈ F = − infx∈F

S(x) = −S.

i.e. log PZε ∈ F ≈ C exp(−S/ε).

• Note F is independent of ε. If Zε → Z /∈ F then P → 0.

• Compare w/CLT. Let ε = 1/n, Zn =∑

i Xi/n, Xi, i.i.d., a > 0.

CLT : P

(|Zn − Z| ≥ a√

n

)→ 2Φ(−a)

LDP : P(|Zn − Z| ≥ a

) → C exp(−San)

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Cross-Out


Contexts & Applications

• Performance of decision rules (and estimators, robustness)

• Performance of portfolios (outperforming benchmarks, largelosses)

• Proofs of convergence (implies weak LLN)

• Rare events: extreme or non-equilibrium outcomes (largebusiness cycles, escape dynamics)

• Transitions between equilibria and equilibrium selection(evolutionary games)


Chebyshev

P (Xi ∈ [a,∞)) ≤ E [exp (λ (Xi − a))]

a0

1

exp(λ(X −a))

Pr(X >= a)


Basic idea: Cramer’s theorem

Take Zn =∑

Xi/n, Xi i.i.d. Fix a > E(X). Define:

H(λ) = log E exp(〈λ,X〉)S(x) = sup

λ〈λ, x〉 −H(λ)

Then Zn satisfies an LDP with rate function I and speed 1/n.Sketch of proof for upper bound:Consider scalar case, and fix a > EX.

P (Zn ∈ [a,∞)) = E [1Zn − a ≥ 0](Chebyshev) ≤ E [exp (nλ (Xi − a))] for all λ ≥ 0

= exp(−nλa)n∏

i=1

E exp(λXi)

log P (Zn ∈ [a,∞)) ≤ −n supλ>0

[λa− log E exp(λXi)] = −nS(a)

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Cross-Out

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Inserted Text

S


Illustration: Bivariate N(0, Σ)

S = minx

12x′Σ−1x s.t. ||x|| ≥ a

−1.5 −1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

y1

y2

Escape Set Iso−Cost Lines


Some notes

• Lower bound is also key: uses (exponential) change of measure.

• Low probability event → deterministic optimization problem.

• Let x∗ = arg minx∈F S(x). Then for all δ > 0:

limn→∞

P (|Zn − x∗| < δ ||Zn ∈ F ) = 1.

• Tom’s favorite quotes:1. Rare events are exponentially rare.2. “If an unlikely event occurs, it is very likely to occur in themost likely way.”


Some Extensions

• Gartner-Ellis: varying statistics. Let Zε ∼ µε. Define:

H(λ) = limε→0

ε log E exp (〈λ/ε, Zε〉)

Provided limit exists, rate function S is Legendgre of H.

• Sanov’s Theorem: Gives LDP for empirical measures of i.i.d.sample. Rate function is relative entropy w.r.t. original meas.

• Donsker-Varadhan: extend to empirical measures of Markov


Some Relationships

• Entropy: information theory, statistical mechanics

• Perturbation methods and sharp LD:

log P ≈ C + εS1 + ε2S2 + . . .

• Risk sensitivity and (asymptotic) robustness via Varadhan.Let L be a (continuous) loss function L : X → R.

limε→0

ε log E[exp(L(Zε)/ε)] = supx∈X

L(x)− S(x)

• Idempotent probability/Possibility theory/fuzzy measures.Puhalskii: “Deviabilities” Π(F ) = supx∈F exp(−S(x)).A nonadditive measure: Π(A ∪B) = Π(A) ∨Π(B).

LDP : limε→0

(∫

Xh(x)

1ε dµε(x)

)ε

= supx∈X

h(x)Π(x)


Applications: i.i.d. or cross-section

• Hypothesis testing: Say accept hypothesis when (sample) loglikelihood ratio above a cutoff. Asymptotics of type I and IIerrors for fixed cutoffs determined by LDP (Chernoff).

• Krasa-Villamil (1992) - Suppose a bank pools (i.i.d.) risks.Probability of default decreases exponentially with size. If costincreases slower w/size, interior optimal size.

• Dembo-Deuschel-Duffie (2003) - Portfolio consists of positionssubject to losses Zi at exposures Ui. Evaluate portfolio lossesLn =

∑i ZiUi, characterize P (Ln ≥ nx).

• Stutzer (2003): Minimize probability that portfolio will grow atrate lower than some benchmark. Use LDP to derive long rundecay rate. Show that yields result similar to max utility ofwealth with endogenous preference power.


Convergence of Exchange Economies

• Random excess demands zi(ω, p) : Ω× Rd 7→ Rd. Aggregate:Zn(ω, p) =

∑ni=1 zi(ω, p). Prices: πn(ω) = p : Zn(ω, p) = 0.

• “Expectation Economy”: z(p) = limn→∞ Zn(ω, p)/n. Prices π.

• Define H(λ, p), S(x, p) as in Gartner-Ellis. Then under somecontinuity conditions, Nummelin (2000) shows that πnsatisfies an LDP with speed 1/n and rate function S(0, p).

• Some simple conditions insure π is nonempty, gives asymptoticexistence (a.s. equilibrium exists eventually) and convergenceto expectation economy.


Sample Path Large Deviations

• So far: LDP for single realization or sample average.Now: Realization of event in a (dynamic) sample path.

• Most complete theory: Freidlin-Wentzell for diffusions

• Simplest to present: discrete time analoguesxσ

t+1 = gσ(xt) + σWt+1. Wt ∼ N(0, 1), gσ → g0, x∗ = g0(x∗).

• Gartner-Ellis: given xt = x, 1-Step LDP at rate σ2 for xt+1

with rate function: S1(x, y) = 12 (y − g0(x))2

• For multi-step rate function, sum up the one-step transitions:

S(x, y, T ) = infxtT

t=0

12

T−1∑t=0

(xt+1 − g0(xt))2 s.t. x0 = x, xT = y


Multi-Step Sample Path LDP

• LDP conditional on xσ0 = x is then:

limσ→0

σ2 log P

(sup

1≤t≤T

∣∣xσt − x0

t

∣∣ > a

)= − inf

y:|y−x0t |>a

S(x, y, T )

• Define escape time from stable point x∗:

τσ = min t > 0 : |xσt − x∗| ≥ a, |xσ

0 − x∗| < a

• Let S = infy∈x∗±a,T<∞ S(x, y, T ), y∗ the minimizer.

• Then for η > 0 we have:

limσ→0

P

(exp

(S − η

σ2

)≤ τσ ≤ exp

(S + η

σ2

))= 1.

limσ→0

P (|xστσ − y∗| < η) = 1.

The Exit/Escape Problem I

*

Average Motion/Mean Dynamics

F

The Exit/Escape Problem IWhen pushed away, tend to return to equilibrium

*


F

Sample Path

The Exit/Escape Problem I

*


Rare Motion/Escape Dynamics

FOccasionally, escape to boundary of set

The Exit/Escape Problem II: Multiple Equilibria

2 stable equilibria with basins of attraction

*

Average Motion/Mean DynamicsF1

*

F2

Separatrix


*



F1

*

F2

SeparatrixOccasionally escape from one eq to the other


*



F1

*

F2

Separatrix

Generates a Markov chain over equilibria


Application: Large Business Cycles

• “Small Noise Asymptotics . . . ” studies stochastic growthmodel. Characterize prob. and freq. of large movements awayfrom deterministic steady state.

• Nonlinearity in policy function determines expected directionof large movements. Convex: positive movements more likely.Concave: negative more likely.

• The way state, policy defined in model implies sharp recessions(slightly) more likely than large booms.


“Intuition” for Asymmetry

Consider 2 step for simplicity. Say x∗ = 0, x1 = 0.5, a = x2 = 1.Start of date 1: at x∗ = 0. End of date 1: at 0.5.Start of date 2: at g0(0.5). End of date 2: at 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance From Stable Point

State Evolution

LinearConvexConcave45 Degree


Application: Escape Dynamics

• In some learning models, recurrent escapes from a stable limitpoint are a key feature of time series.

• Example: Single agent learning a vector γ. Truth depends onagent’s action, and beliefs.

Belief : yin = γixin + ηin, i = 1, 2,

Action : xn = bn + Wn, Wn ∼ N(0, σ2I)

Decision : bn = b(γ)

Truth : yin = fi(bn) + γixin

Updating : γn+1 = γn + ε(yn − γxn)xn

• Specify so for σ > 0 unique stable equilibrium γ.


An Example

Convergence: as ε → 0, γn ⇒ γ = [.75, .25]. Recurrent escapes.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.2

0.4

0.6

0.8

1

Simulated Time Path of Beliefs, ε=.07

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.2

0.4

0.6

0.8

1

Simulated Time Path of Beliefs, ε=.05


Overview: Escape Dynamics

• Similar type of LDP applies, gives predictions of probabilityand frequency of escape, location of dominant escape path.

• With multiple equilibria, can calculate limit transition ratesbetween equilibria. Limit distribution over equilibria collapseson “stochastically stable” or long-run equilibrium.

• With unique equilibrium, can still have interesting escapedynamics (as in the example). Often have escapes toward a“near equilibrium”.


Analysis: “Near Equilibrium”

As noise → 0, there are 2 (stable) equilibria, 1 of which gets“discovered” by escape dynamics.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

γ1

γ 2

Level Sets of Rate Function

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Rate Function Along Diagonal

γ1

γ1−barγ

1*


Comparing the Predictions to Simulations

0.05 0.1 0.15 0.20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Gain (ε)

Loca

tion

Exit Distribution

γ2

γ1

Mean Predicted1 SD Band

5 10 15 20 25

5

6

7

8

9

10

11

Log Mean Escape Time

1/Gain (1/ε)

Mean Predicted1 SD Band


Some References

• Books: Dembo-Zeitouni (1998, Springer, 2nd ed.), Freidlin-Wentzell (1998,

Springer, 2nd ed.), Puhalskii (2001, Chapman), Dupuis-Ellis (1997, Wiley)

• Econ - decisions etc.: Krasa-Villamil (1992 JET, 1992 Oxford Ec. Papers),

Nummelin (2000, Ann. App. Prob.), Williams (2003, “Small Noise”)

• Econ - multiple eq.: Foster-Young (1990, J. Pop Bio),

Kandori-Mailath-Rob (1993, Ec.), Young (1993, Ec.), Ellison (1993, Ec.,

2000 ReStud), Kasa (2004, IER), Williams (2002, unpub “Stochastic

Ficititious Play”), Benaım-Weibull (2003, Ec.)

• Econ - escape dynamics: Sargent (1999, Conquest book),

Cho-Williams-Sargent (2002, ReStud), Williams (2004, unpub, “Escape

Dynamics”), Bullard-Cho (2002, unpub), Cho-Kasa (2003, unpub)

• Finance: Stutzer (2003, J. Econometrics), Dembo-Deuschel-Duffie (2004,

Fin. & Stoch.), Pham (2003, Fin. & Stoch.), Callen-Govindaraj-Xu (2000,

Ec. Theory)

Documents

Notes on Large Deviations in Economics and Financenwilliam/ldpnotes.pdfNotes on Large Deviations 1 Introduction † What is large deviation theory? † Loosely: a theory of rare events