Regular Variation and Extreme Events for Stochastic Processes

Transcription

1 1 Regular Variation and Extreme Events for Stochastic Processes FILIP LINDSKOG Royal Institute of Technology, Stockholm 2005 based on joint work with Henrik Hult lindskog

2 2 Extremes for stochastic processes We study a multivariate stochastic process (X t ) t [0,1] representing some interesting 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 sup t [0,1] X t is large? How can we approximate probabilities of the type P(X t A for some t), where A is a set far away from the origin? How can we approximate the probability P(h(X) A) where h is a functional and {h(x) A} an extreme event?

3 3 Extremal behavior Start with simple processes, e.g. random walks or processes with stationary, independent increments (Lévy 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 Lévy motion).

4 4 Simulation example Left: The most extreme out of 1000 simulations of Brownian motion. Right: The most extreme out of 100 simulations of α-stable Lévy motion.

5 5 Regular variation A function f : (0, ) (0, ) is regularly varying at infinity with index α if for every x > 0, lim u f(xu) f(u) = x α. A nonnegative random variable X is said to be regularly varying (at infinity) with index α if for every x > 0 lim u 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)

6 6 Multivariate regular variation A R d -valued random vector X is regularly varying if there exists a measure µ (finite on sets bounded away from 0) such that lim u 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 (a n ), 0 < a n, such that lim n n P(a 1 n X A) = µ(a), A bounded away from 0, µ( A) = 0. µ has scaling property: µ(ua) = u α µ(a). We write X RV(α, µ). (Resnick 1987, 2004)

7 7 Multivariate regular variation serves as domain of attraction condition for partial sums of iid random vectors (Rvačeva 1962) and as maximum domain of attraction condition for component-wise maxima (Resnick 1987). Under general conditions, the solution Y to a stochastic recurrence equation Y t = A t Y t 1 + B t is regularly varying (Kesten 1973). One example is the GARCH-process (Basrak, Davis, Mikosch 2002): X t = σ t Z t, σ 2 t = α 0 + α 1 X 2 t 1 + β 1 σ 2 t 1 = (α 1 Z 2 t 1 + β 1 )σ 2 t 1 + α 0, Y t = ( X 2 t σ 2 t ), A t = ( α1 Z 2 t β 1 Z 2 t α 1 β 1 ), B t = ( α0 Z 2 t α 0 ).

8 8 Sums and products Let (X k ) be an iid sequence with X 1 RV(α, µ), let M N be a constant. ( lim n P M ) X k a n A = Mµ(A), n k=1 ( ) lim n P MX k a n A = µ(a/m) = M α µ(a). n Let M N be stochastic with light tails. ( lim n P M ) X k a n A = E(M)µ(A) n k=1 ( ) lim n P MX k a n A = E(M α )µ(a). n

9 9 Regular variation in finance Financial log-returns: Let S t be the price of a stock. The distribution of X t = log S t+ t log S t is often assumed to be regularly varying (supported by empirical studies). Independent logreturns: (log S t ) is a Lévy process (independent, stationary increments) with log S 1 regularly varying. Stochastic volatility: log S t = t 0 σ u dl u where (σ u ) is a volatility process and (L u ) a Lévy process.

10 10 Extreme events How do the regular variation of the input noise affect the extremal behavior of the 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.

11 11 Regularly varying stochastic processes D = D([0, 1], R d ) is the space of càdlàg functions with the J 1 -metric. A stochastic process X is regularly varying if there exist a sequence (a n ), 0 < a n, and a measure m (finite on sets bounded away from 0) such that lim n n P(a 1 n X A) = m(a), A B(D) bounded away from 0, m( A) = 0. m has a scaling property: m(ua) = u α m(a). The convergence may be formulated in terms of convergence of boundedly finite measures on D 0 = (0, ] {x D : sup t [0,1] x t = 1}. (de Haan and Lin 2001, Hult and Lindskog 2005a,b)

12 12 Regularly varying stochastic processes cont. Suppose that X is regularly varying: n P(a 1 n 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 sup t [0,1] X t is large. Mapping theorem: for a mapping h : D E with m(disc h ) = 0 and A such that h 1 (A) is bounded away from 0, n P(h(a 1 n 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 = R k ) with limit measure m h 1.

13 13 Sufficient conditions for regular variation A stochastic process (X t ) t [0,1] is regularly varying if It has regularly varying finite dimensional distributions: (X t1,..., X tk ) are regularly varying. the random vectors A relative compactness condition holds: one large jump in X does not trigger further jumps or oscillations of same magnitude within an arbitrarily small time interval. For Markov processes with weakly dependent increments, much simpler sufficient conditions can be formulated. If X is a Lévy process, then X is regularly varying if and only if X 1 is a regularly varying random vector.

14 14 Example: a regularly varying Lévy process Let X be a Lévy process satisfying X 1 RV(α, µ). The process X can be decomposed into a sum of two independent Lévy processes: X t = N t k=1 J k + X t the Lévy-Itô decomposition, where N t Po(ν(B c 0,1)) is the number of jumps J k with norm larger than one. The process ( X t ) is a Lévy process with bounded jumps. It has light tails: E( X t p ) < for all p <. Markov s inequality yields: lim sup n n P( sup X t > a n ε) = 0 for every ε > 0. t [0,1]

15 15 Example continued If X 1 RV(α, µ), then J 1 RV(α, µ/ν(b c 0,1)). It follows that ( Nt lim n P ) J k a n A n k=1 = E(N t) ν(b0,1 c µ(a) = t µ(a). ) if A is bounded away from 0 and µ( A) = 0. Hence, as u, P ( N t k=1 ) J k ua ν(b c 0,1) t P(J 1 ua) t P(X 1 ua) P(X t ua). which means that X t is large due to only one of the jumps J k being large.

16 16 Example continued In particular n P(a 1 n X A) m(a) with m({x D : x = y1 [v,1] } c ) = 0, m t := m π 1 t = tm 1, 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, ( 1 P 0 ) X t dt ua P(X 1 ua), α + 1 P( sup X t ua) P(X 1 ua), t [0,1] A (0, ).

17 17 Simulation Compound Poisson process with t 3 -jumps and intensity 100. Out of 1000 simulations, the sample path with largest supremum is plotted.

18 18 More complicated processes Many interesting processes are driven by Lévy processes. Filtered processes: Y t = t 0 f(t, s)dx s, f deterministic. Stochastic integrals: Y t = t 0 σ sdx s, σ stochastic. Stochastic differential equations: dy t = a t dt + b t dx t, a, b stochastic.

19 19 Approximating extreme sample paths Let X be a regularly varying Lévy process and let d denote the complete J 1 -metric (i.e. the metric on D). Then for every ε > 0, P ( ) d(u 1 X, u 1 Y) > ε sup X t > u t [0,1] 0 as u, where Y = X τ 1 [τ,1] with τ being the time of the jump with largest norm. Hence, the following approximation is justified (for u large) u 1 X u 1 X τ 1 [τ,1] if sup X t > u. t [0,1]

20 20 Filtered processes Let X be a regularly varying Lévy process and consider the process Y given by Y t = t 0 f(t, s)dx s, 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 is regularly varying and (for u large) u 1 Y u 1( ) f(t, τ) X τ 1 [τ,1] (t) t [0,1] if sup Y t > u. t [0,1] We expect an extreme sample path to look like t e θ(t τ) X τ 1 [τ,1] (t).

21 21 Simulated Ornstein-Uhlenbeck process Simulated Ornstein-Uhlenbeck process driven by a Compound Poisson process with t 2 -distributed jumps and intensity λ = 100. Out of 1000 simulations the sample path with largest supremum is plotted.

22 22 Stochastic integration w.r.t. a Lévy process We now consider a stochastic integral (Y t ) t [0,1] given by Y t = t 0 σ u dx u, where X is a Lévy process satisfying X 1 RV(α, µ), and σ is a predictable càglàd process satisfying E(sup t [0,1] σ t γ ) < for some γ > α. First we consider the simpler case: univariate: d = 1, X t = N t k=1 J k, i.e. compound poisson.

23 23 Stochastic integral - continued Then Y t = t 0 σ s dx s = N t k=1 σ τk J k, where σ τk and J k are independent for each k. Hence, Y t is a sum of dependent terms σ τk J k with each term being a product of independent factors. If X and σ are independent, then ( lim n P N1 ) σ τk J k a n A n k=1 where T is uniformly distributed on (0, 1). ( ) = E(N 1 )E(σT α ) lim n P J 1 a n A n = E(σ α T )µ(a),

24 24 Stochastic integral - continued We now return to the original more general case... The stochastic integral Y = (Y t ) t [0,1] is given by Y t = t 0 σ u dx u, where X is a Lévy process satisfying X 1 RV(α, µ), and σ is a predictable càglàd process satisfying E(sup t [0,1] σ t γ ) < for some γ > α. Then Y is regularly varying: lim n n P(a 1 n Y A) = E(µ{x R d : xσ T 1 [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.

25 25 Stochastic integral - continued Moreover, P ( ) d(u 1 Y, u 1 σ τ X τ 1 [τ,1] ) > ε sup Y t > u t [0,1] 0 as u, where τ is the time of the jump of X with largest norm, which justifies the approximation (for u large) u 1 Y u 1 σ τ X τ 1 [τ,1] if sup Y t > u. t [0,1]

26 26 Stochastic integral - continued We can choose σ = X β, β (0, 1). Hence, σ and X can be dependent. However, since σ is predictable, for every fixed t, σ t and X t are independent. For a sufficiently light-tailed process (Q t ) we have t 0 σ s dx s = N t k=1 σ τk J k + Q t, σ τk and J k independent for every k.

27 27 Simulated stochastic integral Simulated stochastic integral t 0 X s 1/2 dx s driven by a Compound Poisson process X, with Cauchy-distributed jumps and intensity λ = 100. Out of 1000 simulations the sample path with largest supremum is plotted.

28 28 References Basrak, Davis, Mikosch (2002). Regular variation of GARCH processes, Stochastic Process. Appl Bingham, Goldie, Teugels (1987). Regular Variation, No 27 in Encyclopedia of Mathematics and its Applications, Cambridge University Press. de Haan, Lin (2001). On convergence towards and extreme value limit in C[0, 1], Ann. Probab Hult, Lindskog (2005a). Extremal behavior of regularly varying stochastic processes, Stochastic Process. Appl. 115(2)

29 29 Hult, Lindskog (2005b). Extremal behavior of stochastic integrals driven by regularly varying Lévy processes, Technical Report No. 1423, School of ORIE, Cornell University, Kesten (1973). Random difference equations and renewal theory for products of random matrices, Acta Math 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 Rvačeva (1962). On domains of attraction of multi-dimensional distributions, Select. Transl. Math. Statist. and Probability American Mathematical Society, Providence, R.I