Lecture 5 the Ornstein-Uhlenbeck Process

Historical Origins

Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, This process was ﬁrst introduced by Ornstein and Uhlenbeck in the Lecture 5: Lecture 5 The Ornstein-Uhlenbeck 1930s as a more accurate model of the physical phenomenon of Process Brownian motion than the Einstein-Smoluchowski-Wiener process. They argued that

David Applebaum Brownian motion = viscous drag of ﬂuid + random molecular bombardment. Probability and Statistics Department, University of Shefﬁeld, UK

July 22nd - 24th 2010

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dv(t) = −kv(t)dt + σdB(t) (0.1)

Let v(t) be the velocity at time t of a particle of mass m executing kt Brownian motion. By Newton’s second law of motion, the total force Using the integrating factor e we can then easily check that the dv(t) unique solution to this equation is the Ornstein-Uhlenbeck process acting on the particle at time t is F(t) = m .We then have dt (v(t), t ≥ 0) where dv(t) dB(t) Z t m = − mkv(t) + mσ , v(t) = e−kt v(0) + e−k(t−s)dB(s). dt dt 0 | {z } | {z } viscous drag molecular bombardment We are interested in Lévy processes so replace B by a d-dimensional where k, σ > 0. Lévy process X and k by a d × d matrix K . Our Langevin equation is dB(t) Of course, doesn’t exist, but this is a “physicist’s argument”. If dt we cancel the ms and multiply both sides by dt then we get a dY (t) = −KY (t)dt + dX(t) (0.2) legitimate SDE - the Langevin equation and its unique solution is

Z t −tK −(t−s)K Y (t) = e Y0 + e dX(s), (0.3) 0

Dave Applebaum (Sheffield UK) Lecture 5 July 2010 3 / 43 Dave Applebaum (Sheffield UK) Lecture 5 July 2010 4 / 43 d We get a Markov semigroup on Bb(R ) called a Mehler semigroup: where Y := Y (0) is a fixed F measurable random variables. We still 0 0 ( ) = ( ( ( ))| = ) call the process Y an Ornstein-Uhlenbeck or OU process. Tt f x E f Y t Y0 x Z −tK urthermore = f (e x + y)ρt (dy) (0.4) d Y has càdlàg paths. R Y is a Markov process. where ρt is the law of the stochastic integral R t e−sK dX(s) =d R t e−(t−s)K dX(s). The process X is sometimes called the background driving Lévy 0 0 This generalises the classical Mehler formula (X(t) = B(t), K = kI) process or BDLP. r ! 1 Z 1 − e−2kt y2 −kt − 2 Tt f (x) = d f e x + y e dy. d 2k (2π) 2 R

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We get nicer probabilistic properties of our solution if we make the following d d In fact (Tt , t ≥ 0) satisfies the Feller property: Tt (C0(R )) ⊆ C0(R ). We also have the skew-convolution semigroup property: Assumption K is strictly positive definite. OU processes solve simple linear SDEs. They are important in K ρs+t = ρs ∗ ρt , applications such as volatility modelling, Lévy driven CARMA processes, branching processes with immigration. where ρK (B) = ρ (etK B). Another terminology for this is s s In infinite dimensions they solve the simplest linear SPDE with additive measure-valued cocycle. noise. To develop this theme, let H and K be separable Hilbert spaces and (S(t), t ≥ 0) be a C0-semigroup on H with infinitesimal generator J. Let X be a Lévy process on K and C ∈ L(K , H).

Dave Applebaum (Sheffield UK) Lecture 5 July 2010 7 / 43 Dave Applebaum (Sheffield UK) Lecture 5 July 2010 8 / 43 Additive Processes and Wiener-Lévy Integrals We have the SPDE The study of O-U processes focusses attention on Wiener-Lévy dY (t) = JY (t) + CdX(t), R t integrals If (t) := 0 f (s)dX(s). For simplicity we assume that whose unique solution is f : Rd → Rd is continuous. Recall that Z = (Z(t), t ≥ 0) is an additive process if Z(0) = 0 (a.s.), Z Z t has independent increments and is stochastically continuous. It follows Y (t) = S(t)Y0 + S(t − s)CdX(s) , 0 that each Z (t) is infinitely divisible. | {z } stochastic convolution Theorem and the generalised Mehler semigroup is (If (t), t ≥ 0) is an additive process.

Z Proof. (sketch) Independent increments follows from the fact that for Tt f (x) = f (S(t)x + y)ρt (dy). d r ≤ s ≤ t R R s If (s) − If (r) = r f (u)dX(u) is σ{X(b) − X(s); r ≤ a < b ≤ s} - From now on we will work in ﬁnite dimensions and assume the strict measurable, R t positive-deﬁniteness of K . If (t) − If (s) = s f (u)dX(u) is σ{X(d) − X(c); s ≤ c < d ≤ t} - measurable, 2

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Theorem If X has characteristics (b, A, ν), it follows that If (t) has characteristics d f f f If X has Lévy symbol η then for each t ≥ 0, u ∈ R , (bt , At , νt ) where Z t (ei(u,If (t))) = exp η(f (s)T u) . Z t Z Z E f 0 bt = f (s)bds + f (s)x(1Bb(x) − 1Bb(f (s)x))ν(dx)ds, 0 0 d −{0} n R t o R Proof. (sketch) Define Mf (t) = exp i u, 0 f (s)dX(s) and use Itô’s formula to show that Z t Af = f (s)T Af (s)ds, t t Z 0 Mf (t) = 1 + i u, Mf (s−)f (s)dB(s) 0 Z t Z Z t i(u,f (s)x) ˜ T Z t + Mf (s−)(e − 1)N(ds, dx) + Mf (s−)η(f (s) u)ds. νf (B) = ν(f (s)−1(B)). d t 0 R −{0} 0 0 Now take expectations of both sides to get It follows that every OU process Y conditioned on Y0 = y is an additive Z t −sK f T process. It will have characteristics as above with f (s) = e and bt E(Mf (t)) = 1 + E(Mf (s))η(f (s) u)ds, −tK 0 translated by e y. and the result follows. 2 Dave Applebaum (Sheffield UK) Lecture 5 July 2010 11 / 43 Dave Applebaum (Sheffield UK) Lecture 5 July 2010 12 / 43 Invariant Measures, Stationary Processes, Ergodicity: General Theory Equivalently for all Borel sets B Z pt (x, B)µ(dx) = µ(B). (0.6) We want to investigate invariant measures and stationary solutions for d OU processes. First a little general theory. R First let (Tt , t ≥ 0) be a general Markov semigroup with transition To see that (0.5) ⇒ (0.6) rewrite as R probabilities pt (x, B) = Tt 1B(x) so that Tt f (x) = d f (y)pt (x, dy) for d R Z Z Z f ∈ Bb( ). We say that a probability measure µ is an invariant R f (y)pt (x, dy)µ(dx) = f (x)µ(dx), d d d d measure for the semigroup if for all t ≥ 0, f ∈ Bb(R ), R R R Z Z and put f = 1B. For the converse - approximate f by simple functions Tt f (x)µ(dx) = f (x)µ(dx) (0.5) and take limits. d d R R

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e.g. A Lévy process doesn’t have an invariant probability measure but Proof. If the process is stationary then µ is invariant since Lebesgue measure is invariant in the sense that for f ∈ L1(Rd ) Z µ(B) = P(Z (0) ∈ B) = P(Z(t) ∈ B) = pt (x, B)µ(dx). Z Z Z Z d R Tt f (x)dx = f (x + y)pt (dy)dx = f (x)dx. d d d d R R R R For the converse, its sufﬁcient to prove that E(f1(Z(t1 + h)) ··· fn(Z(tn + h)))) is independent of h for all A process Z = (Z (t), t ≥ 0) is (strictly) stationary if for all d f1,... fn ∈ Bb(R ). Proof is by induction. Case n = 1. Its enough to + n ∈ N, t1,..., tn, h ∈ R , show

d (Z (t1),..., Z(tn)) = (Z(t1 + h),..., Z (tn + h)) E(f (Z (t)) = E(E(f (Z (t)|F0))) = E(Tt f (Z(0))) Z = (Tt f (x))µ(dx) Theorem d R A Markov process Z wherein µ is the law of Z(0) is stationary if and Z = f (x)dx = E(f (Z (0))). only if µ is an invariant measure. d R

Dave Applebaum (Shefﬁeld UK) Lecture 5 July 2010 15 / 43 Dave Applebaum (Shefﬁeld UK) Lecture 5 July 2010 16 / 43 Let µ be an invariant probability measure for a Markov semigroup (Tt , t ≥ 0). µ is ergodic if In general use Tt 1B = 1B(µ a.s.) ⇒ µ(B) = 0 or µ(B) = 1. If µ is ergodic then “time averages” = “space averages” for the E(f1(Z (t1 + h)) ··· fn(Z (tn + h))) corresponding stationary Markov process, i.e.

= E(f1(Z (t1 + h)) ··· E(fn(Z (tn + h))|Ftn−1+h)) 1 Z T Z = E(f1(Z (t1 + h) ··· Ttn−tn−1 fn(Z (tn−1 + h)))). lim f (Z(s))ds = f (x)µ(dx) a.s. T →∞ T d 0 R 2 Fact: The invariant measures form a convex set and the ergodic measures are the extreme points of this set. It follows that if an invariant measure is unique then it is ergodic.

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The Self-Decomposable Connection

Recall that a random variable Z is self-decomposable if for each 0 < a < 1 there exists a random variable Wa that is independent of Z such that d Z t Z t Z = aZ + Wa d −k(t−s) d −ks Y (t) = Y0 and e dX(s) = e dX(s) 0 0 a a −1 or equivalently ρZ = ρ ∗ ρW , where ρ (B) = ρ(a B). Z a Z ⇒ Y =d e−kt Y + W . Now suppose that Y is a stationary Ornstein-Uhlenbeck process on R. 0 0 a(t) −kt R t −ks Then Y0 is self decomposable with a = e and Wa(t) = 0 e dX(s) since

Z t −kt −(t−s)K Y (t) = e Y0 + e dX(s) 0 and by stationary increments of the process X

Dave Applebaum (Shefﬁeld UK) Lecture 5 July 2010 19 / 43 Dave Applebaum (Shefﬁeld UK) Lecture 5 July 2010 20 / 43 Now suppose that µ is self-decomposable - more precisely that

ekt µ = µ ∗ ρt , So we have shown that: where ρ is the law of W . Then t a(t) Theorem The following are equivalent for the O-U process Y . Z Z Z −kt Tt f (x)µ(dx) = f (e x + y)ρt (dy)µ(dx) Y is stationary. R ZR ZR The law of Y (0) is an invariant measure. ekt = f (x + y)ρt (dy)µ (dx) R t −ks The law of Y (0) is self-decomposable (with Wa(t) = 0 e dX(s)). ZR R ekt = f (x)(µ ∗ ρt )(dx) ZR = f (x)µ(dx). R So µ is an invariant measure.

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We seek some condition on the Lévy process X which ensures that Y is stationary. When does lim R t e−ksdX(s) exist in distribution? Use the Lévy-Itô Fact: If Y := R ∞ e−ksdX(s) exists in distribution then it is t→∞ 0 ∞ 0 decomposition. self-decomposable. Z To see this observe that (using stationary increments of X) X(t) = bt + M(t) + xN(t, dx). |x|≥1

Z ∞ Z ∞ Z t R t −ks 2 −ks −ks −ks ( ) e dX(s) = e dX(s) + e dX(s) It is not difﬁcult to see that limt→∞ 0 e dM s exists in L -sense. Z t Z 0 t 0 −ks Z ∞ Z t Fact: lim e xN(ds, dx) exists in distribution if and only if d −k(t+s) −ks t→∞ 0 |x|≥1 = e dX(s) + e dX(s) R 0 0 |x|≥1 log(1 + |x|)ν(dx) < ∞. Z ∞ Z t = e−kt e−ksdX(s) + e−ksdX(s) 0 0

Dave Applebaum (Shefﬁeld UK) Lecture 5 July 2010 23 / 43 Dave Applebaum (Shefﬁeld UK) Lecture 5 July 2010 24 / 43 In fact, if an invariant measure µ exists then it is unique. For suppose that Y is stationary, then To prove this you need Z ∞ d −kt −ks 1 P n ( ) = ( ) + ( ). If (ξn, n ∈ N) are i.i.d. then n=1 c ξn converges a.s. (0 < c < 1) Y 0 e Y 0 te dX s 0 if and only if E(log(1 + |ξ1|)) < ∞. 2 R iuy Now let ρ be the law of Y (0) and Φρ(u) := e ρ(dy). Then for all Z n Z n−1 R −ks d X −kj u ∈ R, by independence e xN(ds, dx) = e Mj 0 |x|≥1 j=0 Z t −kt −ks Φρ(u) = Φρ(e u) exp − η(e u)ds . j+1 R R −k(s−j) 0 where Mj := j |x|≥1 e xN(ds, dx). Note that (Mj , j ∈ N) are i.i.d. Take limits as t → ∞ to get f f f In this case, Y has characteristics (b∞, A∞, ν∞). 1 Z ∞ e.g. Brownian motion case. X(t) = B(t). µ ∼ N 0, . −ks 2k Φρ(u) = exp − η(e u)ds . 0

So ρ is the law of Y∞.

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In fact - it is possible to go further. Given any self-decomposable distribution µ there exists a stationary Ornstein-Uhlenbeck process Y Example: Let (X(t), t ≥ 0) be a compound Poisson process such that the law of Y (0) is µ. Let’s sketch the proof of this - due to PN(t) Jurek and Vervaat (1983). Let X be a self-decomposable random X(t) = i=1 Wi where the Wi s are i.i.d. exponential with common −ax density fW (x) = ae 1x>0. Then variable with distribution µ. Then for each t ≥ 0

Z ∞ λa d −t η(u) = λa (eiux − 1)e−ax dx = . X = e X + Xt , 0 a − iu where X and Xt are independent. −1 −λ You can check that Φρ(u) = (1 − ia u) as so ρ has a gamma(c, λ) The key step is the observation that we can construct an additive distribution. process (Z(t), t ≥ 0) such that

d d −t Z (t) = Xt and Z (t + h) − Z(t) = e Xh.

Dave Applebaum (Sheffield UK) Lecture 5 July 2010 27 / 43 Dave Applebaum (Sheffield UK) Lecture 5 July 2010 28 / 43 This follows by Kolmogorov’s theorem since We then find that Z(t) = R t e−sdY (s) and so d −(t+h) 0 X = e X + Xt+h Z t Z t d −t −h d −t −s d −t −(t−s = e (e X + Xh) + Xt X = e X + e dY (s) = e X + e )dY (s), 0 0 ⇒ X =d e−t X + X . t+h h t using stationary increments of Y which is extended to an Lévy process on the whole of R. R t s In the 1990s, Sato showed that µ is self-decomposable if and only if it It follows that Y (t) = 0 e dZ (s) also has independent increments. But Y is a Lévy process since is the law of W (1) where (W (t), t ≥ 0) is a self-similar additive process. Recall W self-similar (index H) means for all c ≥ 0 Z t+h s Y (t + h) − Y (t) = e dZ (s) W (ct) =d cH W (t). t Z h = eset dZ (s + t) 0 So we can embed selfdecomposable distributions into stationary OU Z h d processes and self-similar additive processes. Is there a connection? = eset e−t dZ (s) = Y (h) 0

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The next step is due to Jeanblanc, Pitman, Yor (SPA 100, 223(2002)) To understand the connection between the two “embeddings” of µ we Start with a self-similar additive process (W (t), t ≥ 0). Then we know need the that W (1) is self-decomposable. There exist two independent, identically distributed Lévy processes (X −, t ≥ 0) and (X +, t ≥ 0) such Lamperti Transform. There is a one-to-one correspondence between t t that self-similar processes (W (t), t ≥ 0) and stationary processes t Z 1 dW (r) Z e dW (r) − = , + = . (Z(t), t ≥ 0) given by Xt H Xt H e−t r 1 r W (t) = tH Z (log(t)) or equivalently Z (t) = e−tH W (et ). Let (Z(t), t ≥ 0) be the stationary Lamperti transform of W . Then it is an Ornstein-Uhlenbeck process and Indeed if W self-similar Z t −tH −(t+s)H + −(t+h)H t+h Z(t) = e W (1) + e dXs , Z(t + h) = e W (e ) 0 d − − = e tH e hH ehH W (et ) Z t −tH −(t+s)H − = ( ). Z (−t) = e W (1) − e dXs . Z t 0

In the last part of the lecture we’ll brieﬂy look at some recent developments.

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Assume

1 Rank[B, AB,..., An−1B] = n, We’ve seen that each Y (t) is inﬁnitely divisible so if the Lévy process where [B, AB,..., An−1B] is the matrix of the linear mapping from X(t) has a Gaussian component then so does Y (t) in which case it Rnd to Rn given by has a density by Fourier inversion. n−1 More generally, Priola and Zabczyk (BLMS, 41, 41,(2009)) study (u0, u1,..., un−1) → Bu0 + ABu1 + ... + A Bun−1.

dY (t) = AY (t)dt + BdX(t), 2 The restriction of the Lévy measure ν to Br (0) has a density for d n where each Y (t) is R -valued but X(t) is R -valued (n ≥ d). So A is some r > 0. an n × n matrix and B is an n × d matrix. Then Y (t) has a density for t > 0.

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Application - Volatility Modelling Barndorff-Nielsen and Shephard (JRSS B 63, 167 (2001)) proposed the OU model

dσ2(t) = −λσ2(t) + dX(λt), Consider the Black-Scholes model for a stock price where λ > 0 and X is a subordinator. Then σ2(t) > 0 (a.s.) since S(t) = S(0) exp{µt + σB(t)}, Z t 2 −λt 2 −λ(t−s) where µ ∈ R is stock drift and σ > 0 is volatility. By Itô’s formula σ (t) = e σ (0) + e dX(λt) 0   1 dS(t) = σS(t)dB(t) + S(t) µ + σ2 dt. −λt  2 X −λu  2 = e σ (0) + e ∆X(λu) .  0≤u≤t  In stochastic volatility models the parameter σ2 is replaced by a R ∞ Assume that 1 log(1 + x)ν(dx) < ∞. Then there is a unique invariant stochastic process (σ2(t), t ≥ 0). measure µ which is self-decomposable and has characteristic function Z ∞ iux k(x) µb(u) = exp (e − 1) dx , 0 x where k is decreasing. Dave Applebaum (Shefﬁeld UK) Lecture 5 July 2010 35 / 43 Dave Applebaum (Shefﬁeld UK) Lecture 5 July 2010 36 / 43 Generalised Ornstein-Uhlenbeck Processes

2 Let X = (X1, X2) be a Lévy process on R . Then each Xi is a real-valued Lévy process. Let Y0 be independent of X. The Problem. Based on discrete-time observations generalised Ornstein-Uhlenbeck process is σ2(0), σ2(∆), σ2((N − 1)∆) ﬁnd estimates of the parameter λ and k. Z t For a non-parametric approach - see Jongbloed et. al. Bernoulli, 11, −X1(t) −X1(s) Y (t) = e Y0 + e dX2(s) . 759 (2005). 0

The usual OU process is obtained by taking X1(t) = λt (λ > 0). Necessary and sufﬁcient conditions for stationarity solutions were found by Lindner and Maller (SPA 115, 1701 (2005)). Almost sure R t −X1(s) convergence of 0 e dX2(s) as t → ∞ is a sufﬁcient condition but the general story is more complicated.

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References and Further Reading

In fact a necessary and sufﬁcient condition for stationary solutions is t the almost sure convergence of R e−X1(s)dL(s) as t → ∞ where the 0 These lectures have been broadly based on my recent book: one-dimensional Lévy process (L(t), t ≥ 0) is deﬁned by D.Applebaum Lévy Processes and Stochastic Calculus, Cambridge X −∆X1(s) ˜ L(t) := X2(t) + (e − 1)∆X2(s) − tA1,2, University Press (second edition) (2009) 0≤s≤t and from an earlier course of lectures partly derived from it, which ˜ have been separately published as where A1,2 is the off-diagonal entry of the covariance matrix of the Gaussian component of the bivariate Lévy process (X1, L). For further D.Applebaum, Lévy processes in Euclidean spaces and groups in work on generalised O-U processes see Lindner and Sato (AP 37, 250 Quantum Independent Increment Processes I: From Classical (2009)). Probability to Quantum Stochastic Calculus, Springer Lecture Notes in Mathematics , Vol. 1865 M Schurmann, U. Franz (Eds.) 1-99,(2005)

Dave Applebaum (Sheffield UK) Lecture 5 July 2010 39 / 43 Dave Applebaum (Sheffield UK) Lecture 5 July 2010 40 / 43 A comprehensive account of the structure and properties of Lévy processes is: For an insight into the wide range of both theoretical and applied K-I.Sato, Lévy Processes and Infinite Divisibility, Cambridge University recent work wherein Lévy processes play a role, consult Press (1999) O.E.Barndorff-Nielsen,T.Mikosch, S.Resnick (eds), Lévy Processes: A shorter account, from the point of view of the French school, which Theory and Applications, Birkhäuser, Basel (2001) concentrates on fluctuation theory and potential theory aspects is For stochastic calculus with jumps, the authoritative treatise is J.Bertoin, Lévy Processes, Cambridge University Press (1996) P.Protter, Stochastic Integration and Differential Equations (second edition), Springer-Verlag, Berlin Heidelberg (2003) From this point of view, you should also look at A.Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer-Verlag (2006)

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For ﬁnancial modelling I recommend: R.Cont, P.Tankov, Financial Modelling with Jump Processes, Chapman and Hall/CRC (2004) which is extremely comprehensive and also contains a lot of valuable background material on Lévy processes. W.Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, Wiley (2003) is shorter and aimed at a wider audience than mathematicians and statisticians.

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