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Regular Variation and Extreme Events forStochastic Processes

FILIP LINDSKOG

Royal Institute of Technology, Stockholm

2005

based on joint work with Henrik Hult

www.math.kth.se/∼lindskog

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Extremes for stochastic processes

We study a multivariate stochastic process (Xt)t∈[0,1] representing someinteresting quantities. We want to understand:

• What is the behavior of the process when it is “extreme”?What does the sample path look like given that supt∈[0,1] |Xt| is large?

• How can we approximate probabilities of the type P(Xt ∈ A for some t), whereA is a set far away from the origin?

• How can we approximate the probability P(h(X) ∈ A) where h is a functionaland {h(X) ∈ A} an “extreme” event?

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Extremal behavior

Start with “simple” processes, e.g. random walks or processes with stationary,independent increments (Levy processes). Typically we have two different cases:

• “Light tails” – Many increments contribute to make the process large(Ex: Brownian motion).

• “Heavy tails” – Large values are due to one single large increment(Ex: α-stable Levy motion).

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Simulation example

Left: The most “extreme” out of 1000 simulations of Brownian motion.Right: The most “extreme” out of 100 simulations of α-stable Levy motion.

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Regular variation

A function f : (0,∞) → (0,∞) is regularly varying at infinity with index −α iffor every x > 0,

limu→∞

f(xu)f(u)

= x−α.

A nonnegative random variable X is said to be regularly varying (at infinity) withindex α if for every x > 0

limu→∞

P(X > xu)P(X > u)

= x−α,

i.e. if 1− F (F is the distribution function of X) is regularly varying with index−α. (Bingham, Goldie, Teugels 1987)

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Multivariate regular variation

A Rd-valued random vector X is regularly varying if there exists a measure µ(finite on sets bounded away from 0) such that

limu→∞

P(X ∈ u A)P(|X| > u)

= µ(A), A bounded away from 0, µ(∂A) = 0.

Equivalently: X is regularly varying if there exist a sequence (an), 0 < an ↑ ∞,such that

limn→∞

n P(a−1n X ∈ A) = µ(A), A bounded away from 0, µ(∂A) = 0.

µ has scaling property: µ(uA) = u−αµ(A).We write X ∈ RV(α, µ). (Resnick 1987, 2004)

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• Multivariate regular variation serves as domain of attraction condition forpartial sums of iid random vectors (Rvaceva 1962) and as maximum domain ofattraction condition for component-wise maxima (Resnick 1987).

• Under general conditions, the solution Y∞ to a stochastic recurrenceequation Yt = AtYt−1 + Bt is regularly varying (Kesten 1973).

One example is the GARCH-process (Basrak, Davis, Mikosch 2002):

Xt = σtZt, σ2t = α0 + α1X

2t−1 + β1σ

2t−1 = (α1Z

2t−1 + β1)σ2

t−1 + α0,

Yt =(

X2t

σ2t

), At =

(α1Z

2t β1Z

2t

α1 β1

), Bt =

(α0Z

2t

α0

).

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Sums and products

Let (Xk) be an iid sequence with X1 ∈ RV(α, µ), let M ∈ N be a constant.

limn→∞

n P( M∑

k=1

Xk ∈ anA)

= Mµ(A),

limn→∞

n P(MXk ∈ anA

)= µ(A/M) = Mαµ(A).

Let M ∈ N be stochastic with light tails.

limn→∞

n P( M∑

k=1

Xk ∈ anA)

= E(M)µ(A)

limn→∞

n P(MXk ∈ anA

)= E(Mα)µ(A).

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Regular variation in finance

• Financial log-returns: Let St be the price of a stock. The distribution of

Xt = log St+∆t − log St

is often assumed to be regularly varying (supported by empirical studies).

• Independent logreturns: (log St) is a Levy process (independent, stationaryincrements) with log S1 regularly varying.

• Stochastic volatility: log St =∫ t

0σu dLu

where (σu) is a volatility process and (Lu) a Levy process.

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Extreme events

• How do the regular variation of the input noise affect the extremal behavior ofthe associated stochastic processes?

• How do we compute the probability of certain extreme events in these models,e.g. the probability that a functional of the sample path of the process is large(supremum, average, etc.)?

• We will look at a general framework for studying these problems.

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Regularly varying stochastic processes

D = D([0, 1], Rd) is the space of cadlag functions with the J1-metric.

A stochastic process X is regularly varying if there exist a sequence (an),0 < an ↑ ∞, and a measure m (finite on sets bounded away from 0) such that

limn→∞

n P(a−1n X ∈ A) = m(A),

A ∈ B(D) bounded away from 0,m(∂A) = 0.

m has a scaling property: m(uA) = u−αm(A).

The convergence may be formulated in terms of convergence of boundedly finitemeasures on D0 = (0,∞]× {x ∈ D : supt∈[0,1] |xt| = 1}.

(de Haan and Lin 2001, Hult and Lindskog 2005a,b)

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Regularly varying stochastic processes cont.

Suppose that X is regularly varying: n P(a−1n X ∈ A) → m(A), A ∈ B(D).

• The measure m describes the extremal behavior of the process X.

• The support of m tells us which sample paths to expect given that supt∈[0,1] |Xt|is large.

• Mapping theorem: for a mapping h : D → E with m(Disch) = 0 and A suchthat h−1(A) is bounded away from 0,

n P(h(a−1n X) ∈ A) → m ◦ h−1(A) as n →∞

In particular (for nice mappings h) we have: h(X) is a regularly varying process(if E = D) or a regularly varying vector (if E = Rk) with limit measure m ◦ h−1.

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Sufficient conditions for regular variation

A stochastic process (Xt)t∈[0,1] is regularly varying if

• It has regularly varying finite dimensional distributions: the random vectors(Xt1, . . . ,Xtk

) are regularly varying.

• A relative compactness condition holds: one large jump in X does not triggerfurther jumps or oscillations of same magnitude within an arbitrarily small timeinterval.

For Markov processes with weakly dependent increments, much simpler sufficientconditions can be formulated.• If X is a Levy process, then X is regularly varying if and only if X1 is a regularlyvarying random vector.

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Example: a regularly varying Levy process

Let X be a Levy process satisfying X1 ∈ RV(α, µ). The process X can bedecomposed into a sum of two independent Levy processes:

Xt =Nt∑

k=1

Jk + Xt the Levy-Ito decomposition,

where Nt ∼ Po(ν(Bc0,1)) is the number of jumps Jk with norm larger than one.

• The process (Xt) is a Levy process with bounded jumps. It has light tails:

E(|Xt|p) < ∞ for all p < ∞. Markov’s inequality yields:

lim supn→∞

n P( supt∈[0,1]

|Xt| > anε) = 0 for every ε > 0.

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Example continued

• If X1 ∈ RV(α, µ), then J1 ∈ RV(α, µ/ν(Bc0,1)). It follows that

limn→∞

n P( Nt∑

k=1

Jk ∈ anA)

=E(Nt)ν(Bc

0,1)µ(A) = t µ(A).

if A is bounded away from 0 and µ(∂A) = 0. Hence, as u →∞,

P( Nt∑

k=1

Jk ∈ uA)∼ ν(Bc

0,1) t P(J1 ∈ uA)

∼ t P(X1 ∈ uA) ∼ P(Xt ∈ uA).

which means that Xt is large due to only one of the jumps Jk being large.

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Example continued

• In particular n P(a−1n X ∈ A) → m(A) with

m({x ∈ D : x = y1[v,1]}c) = 0, mt := m ◦ π−1t = tm1, t ∈ [0, 1].

• This means that the measure m is concentrated on step functions with one step.• Combining this with the mapping theorem we can easily show that (here d = 1,A bounded away from 0) as u →∞,

P( ∫ 1

0

Xtdt ∈ uA)∼ P(X1 ∈ uA)

α + 1,

P( supt∈[0,1]

Xt ∈ uA) ∼ P(X1 ∈ uA), A ⊂ (0,∞).

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Simulation

Compound Poisson process with t3-jumps and intensity 100. Out of 1000simulations, the sample path with largest supremum is plotted.

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More complicated processes

Many interesting processes are driven by Levy processes.

• Filtered processes: Yt =∫ t

0f(t, s)dXs, f deterministic.

• Stochastic integrals: Yt =∫ t

0σsdXs, σ stochastic.

• Stochastic differential equations:

dYt = atdt + btdXt, a, b stochastic.

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Approximating extreme sample paths

Let X be a regularly varying Levy process and let d denote the complete J1-metric(i.e. the metric on D). Then for every ε > 0,

P(d(u−1X, u−1Y) > ε | sup

t∈[0,1]

|Xt| > u)→ 0 as u →∞,

where Y = ∆Xτ1[τ,1] with τ being the time of the jump with largest norm.

Hence, the following approximation is justified (for u large)

u−1X ≈ u−1∆Xτ1[τ,1] if supt∈[0,1]

|Xt| > u.

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Filtered processes

Let X be a regularly varying Levy process and consider the process Y given by

Yt =∫ t

0

f(t, s)dXs, f deterministic.

Example: Ornstein-Uhlenbeck process f(t, s) = e−θ(t−s), θ > 0.

Using the mapping theorem we can show that if f is continuous then Y isregularly varying and (for u large)

u−1Y ≈ u−1(f(t, τ)∆Xτ1[τ,1](t)

)t∈[0,1]

if supt∈[0,1]

|Yt| > u.

We expect an extreme sample path to look like t 7→ e−θ(t−τ)∆Xτ1[τ,1](t).

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Simulated Ornstein-Uhlenbeck process

Simulated Ornstein-Uhlenbeck process driven by a Compound Poisson processwith t2-distributed jumps and intensity λ = 100. Out of 1000 simulations thesample path with largest supremum is plotted.

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Stochastic integration w.r.t. a Levy process

We now consider a stochastic integral (Yt)t∈[0,1] given by

Yt =∫ t

0

σudXu,

where X is a Levy process satisfying X1 ∈ RV(α, µ), and σ is a predictable cagladprocess satisfying E(supt∈[0,1] |σt|γ) < ∞ for some γ > α.

First we consider the simpler case:

• univariate: d = 1,

• Xt =∑Nt

k=1 Jk, i.e. compound poisson.

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Stochastic integral - continued

Then

Yt =∫ t

0

σsdXs =Nt∑

k=1

στkJk,

where στkand Jk are independent for each k.

Hence, Yt is a sum of dependent terms στkJk with each term being a product of

independent factors. If X and σ are independent, then

limn→∞

n P( N1∑

k=1

στkJk ∈ anA

)= E(N1)E(σα

T ) limn→∞

n P(J1 ∈ anA

)= E(σα

T )µ(A),

where T is uniformly distributed on (0, 1).

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Stochastic integral - continued

We now return to the original more general case...

The stochastic integral Y = (Yt)t∈[0,1] is given by

Yt =∫ t

0

σudXu,

where X is a Levy process satisfying X1 ∈ RV(α, µ), and σ is a predictable cagladprocess satisfying E(supt∈[0,1] |σt|γ) < ∞ for some γ > α. Then Y is regularlyvarying:

limn→∞

n P(a−1n Y ∈ A) = E(µ{x ∈ Rd : xσT1[T,1] ∈ A}) =: m(A),

A ∈ B(D) bounded away from 0,m(∂A) = 0,

where T is uniformly distributed on (0, 1) and independent of σ.

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Stochastic integral - continued

Moreover,

P(d(u−1Y, u−1στ∆Xτ1[τ,1]) > ε | sup

t∈[0,1]

|Yt| > u)→ 0 as u →∞,

where τ is the time of the jump of X with largest norm, which justifies theapproximation (for u large)

u−1Y ≈ u−1στ∆Xτ1[τ,1] if supt∈[0,1]

|Yt| > u.

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Stochastic integral - continued

• We can choose σ = Xβ−, β ∈ (0, 1). Hence, σ and X can be dependent.

• However, since σ is predictable, for every fixed t, σt and ∆Xt are independent.

• For a sufficiently light-tailed process (Qt) we have

∫ t

0

σsdXs =Nt∑

k=1

στkJk + Qt, στk

and Jk independent for every k.

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Simulated stochastic integral

Simulated stochastic integral∫ t

0|Xs−|1/2dXs driven by a Compound Poisson

process X, with Cauchy-distributed jumps and intensity λ = 100. Out of 1000simulations the sample path with largest supremum is plotted.

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References

• Basrak, Davis, Mikosch (2002). Regular variation of GARCH processes,Stochastic Process. Appl. 99 95–116 .

• Bingham, Goldie, Teugels (1987). Regular Variation, No 27 in Encyclopedia ofMathematics and its Applications, Cambridge University Press.

• de Haan, Lin (2001). On convergence towards and extreme value limit in C[0, 1],Ann. Probab. 29 467–483.

• Hult, Lindskog (2005a). Extremal behavior of regularly varying stochasticprocesses, Stochastic Process. Appl. 115(2) 249–274.

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• Hult, Lindskog (2005b). Extremal behavior of stochastic integrals driven byregularly varying Levy processes, Technical Report No. 1423, School of ORIE,Cornell University, 2005.

• Kesten (1973). Random difference equations and renewal theory for products ofrandom matrices, Acta Math. 131 207–248.

• Resnick (1987). Extreme Values, Regular Variation, and Point Processes,Springer-Verlag, New York.

• Resnick (2004). On the foundations of multivariate heavy-tail analysis, J. Appl.Probab. 41A 191–212.

• Rvaceva (1962). On domains of attraction of multi-dimensional distributions,Select. Transl. Math. Statist. and Probability American Mathematical Society,Providence, R.I. 2 183–205.