: motivation The Lévy measure provides the expected number of jumps by time unit, i.e. in a time interval of the form: [t, t + 1], and of a certain size Example: ν([1, )) is the expected number of jumps of size greater than 1 occurring by time unit. Now, if we want to describe the jump behaviour, we need not only to know the expected number by time unit: we need info about the number of jumps occurring during any given time period, and have information about the jump size, and this for any trajectory. So to study the jump behaviour, we need info about WHEN do jump occur, and WHICH SIZE they have. 1/49
: motivation So we need to be able to associate to any state of the world ω, and any pair (]t 1, t 2], A) where A R Borel set: (ω, ]t 1, t 2] A) # jumps on ]t 1, t 2] with size in A for trajectory associated to ω This number of jumps is clearly stochastic, as it depends on the trajectory. This leads to the introduction of the jump measure, a particular case of Poisson random measure 2/49
: Definition Let (Ω, F, P) be a probability space, E R 2 and µ a positive Radon measure on (E, B). A on E with intensity measure µ is a mapping: M : Ω B N : (ω, B) M(ω, B) such that: 1 For almost all ω Ω, M(ω,.) is a Radon measure with integer values on (E, B) 2 For all B E measurable, M(., B) = M(B) is a random variable Poisson distributed with intensity parameter λ = µ(b): k N : P[M(B) = k] = e µ(b) µ(b) k k! 3 If B 1,..., B n B are disjoint, then M(B 1),..., M(B n) are independent random variables In the framework of Lévy processes, M(B) = M(]t 1, t 2] A) will be the number of jumps on ]t 1, t 2] and of size in A 3/49
of a compound Poisson process For any compound Poisson process (X t), we associate a random measure on [0, ) R that will capture the jump behaviour of (X t): J X (B) = #{(t, X t X t ) B s.t. X t X t 0} B [0, ) R J X is called the jump measure of process X. Interpretation: J X ([t 1, t 1] A) counts the number of jumps of (X t) between instants t 1 and t 2 and whose jump size is in A (i.e. such that X t A). The jump measure hence gives information on the frequency and size of jumps. We have the following result: Proposition: Let (X t) be a compound Poisson process of intensity λ and distribution of jumps F. Then its jump measure J X is a Poisson random measure on [0, ) R, of intensity µ(dt dx) = ν(dx)dt = λf (dx)dt. 4/49
of a compound Poisson process We have already interpreted the Lévy measure of a compound Poisson process as the expected number of jumps of a given size by time unit. Now, J X gives additional information: it tells us when they arrive, and what size they have, for each trajectory (not only in average). Moreover, ν appears in the intensity measure of J X. This leads a more general definition of the Lévy measure, also valid for any Lévy process (not only for compound Poisson processes): Definition: Let (X t ) be a Lévy process on R. The measure on R defined by: ν(a) = E[#{t [0, 1] : X t 0, X t A}] A B is called the Lévy measure of X. 5/49
of a compound Poisson process: computation Case of a Poisson process (N t ): Intensity: λ Jump instants :T n J N ([0, t] A) = { #{n 1 : Tn [0, t]} if 1 A 0 otherwise = n 1 I (Tn,1) [0,t] A J N = n 1 δ (Tn,1) 6/49
of a compound Poisson process: computation Case of a compound Poisson process : X t = N t n=1 Y n, jump instants denoted by T i, i N 0 J X ([0, t] A) = #{n 1 : T n [0, t] and Y n A} J X = n 1 δ (Tn,Y n) = n 1 δ (Tn, X Tn ) 7/49
of a compound Poisson process: computation In particular, as X t is equal to the sum of its jumps (pure jump process), we can write: X t = X s On the other hand, xj X (ds dx) = [0,t] A s [0,t] [0,t] A x n 1 = n 1 Y n I Tn ti Yn A = s [0,t] δ (Tn,Y n)(dtdx) X s I Xs 0, X s A 8/49
of a compound Poisson process: computation In particular, if we set A = R in this formula, we get: xj X (ds dx) = X s = X t [0,t] R s [0,t] X s 0 This is a particular case of Lévy-Ito decomposition theorem (in the case of a compound Poisson process). More generally, any (measurable) function of the jumps of the process can be expressed by means of an integral with respect to the jump measure: f (x)j X (ds dx) = [0,t] R s [0,t] X s 0 f ( X s) 9/49
Lévy-Ito decomposition: Introduction Let us consider a Lévy process (Xt 0 ) with piecewise constant trajectories. This is necessarily a compound Poisson process, and we have seen that it can be represented as the sum of its jumps, as well as by means of an integral w.r.t. the jump measure: X 0 t = s [0,t] X s 0 X 0 s = xj X (ds dx) [0,t] R where J X is a with intensity measure ν(dx)dt. The Lévy measure ν is a finite measure (ν(r) = λ <,) defined by: ν(a) = E[#{t [0, 1] : X 0 t 0, X 0 t A}] 10/49
Lévy-Ito decomposition: Introduction Now, given a Brownian motion with drift, γt + σw t, independent from (X 0 t ), the sum: X t = X 0 t + γt + σw t defines another Lévy process, which can hence be decomposed in: X t = γt + σw t + xj X (ds dx) = γt + σw t + [0,t] R Is this true for any Lévy process? s [0,t] X s 0 X s 11/49
Lévy-Ito decomposition: Introduction Given a Lévy process (X t), we can still define its Lévy measure ν as above (same definition). If A is a compact set that does not contain 0, one can see that ν(a) is still finite (any cadlag process has on each bounded time interval a finite number of jumps of size ɛ for a given ɛ > 0) ν is a Radon measure on R\{0} Now, ν is not necessarily finite as it can blow up close to 0: this is the case if X can have an number of jumps of size arbitrarily small. In that case, the sum s [0,t], X s 0 X s becomes a series, and is not necessarily converging. So the result cannot be extended as such. 12/49
Theorem: Lévy-Ito decomposition Let X t be a Lévy process on R and ν its Lévy measure. Then: ν is a Radon measure on R\{0} and verifies: x 2 ν dx <, x <1 ν dx = ν x 1 < x 1 The jump measure of X, J X, is a on [0, ) R, with intensity measure ν dx dt There exists γ R, a Brownian motion B t with volatility σ 0, such that: X t = γt + B t + X t l + lim ε 0 X t ε (1) X t l = xj X ds dx = ΔX s x 1 s [0,t] ΔX s 1 s [0,t] X t ε = x ε x <1 s [0,t] J X ds dx ν dx ds The different terms in (1) are independent, and the limit exists in the sense of a.s. convergence on Ω and uniformly in t [0, T] The triplet σ, γ, ν is called the characteristic triplet or Lévy triplet of process (X t ) 13/49
Lévy-Ito decomposition: Comments The result says that X t = γt + σw t + J t where γt is a (deterministic) drift, σw t is a Brownian motion, and J t is a pure jump process, governing the jump part of X The first part γt + σw t is a continuous process (diffusion). This is the continuous part of the Lévy process, and is captured by γ and σ. The profile of jumps is described thanks to the jump measure J X, which is itself a of intensity measure ν(dx)dt. This Lévy measure ν drives hence the jump part. The triplet (σ, γ, ν) hence entirely characterizes the process (...) 14/49
Lévy-Ito decomposition: Comments The condition ν(dy) < means that X has in average a finite y 1 number of jumps of size greater than 1 by time unit. So, the second part in the decomposition : Xt l = X s s [0,t], X s 1 has actually a finite number of terms, and is a compound Poisson process (since J X is a Poisson measure), that we can denote as J (1) t. Remark that there is nothing special w.r.t. the threshold chosen for the size of jumps (1 here). Another threshold could have been chosen for expressing this decomposition result (see also later). 15/49
Lévy-Ito decomposition: Comments The last term is more complicated, and includes the Lévy measure ν. ν has no weight at 0 (a jump of size 0 is not a jump...) but might have a singularity in a neighborhood of 0: This corresponds to the case where there is an infinite number of jumps of size arbitrarily small, which is possible for a cadlag trajectory This is why in this result we only consider the limit for ɛ 0 and that we compensate the jump measure in this term. J X (ds dx) ν(dx)ds is by definition the compensated Poisson random measure associated to J X 16/49
Lévy-Ito decomposition: Comments Indeed, if we consider Xt ɛ defined by: Xt ɛ = X s = xj X (ds dx) ɛ x <1,0 s t s [0,t],ɛ X s <1 (so, X ɛ t, but without the compensation) this is also a compound Poisson process, having jump size between ɛ and 1. The problem is that we cannot just pass to the limit of Xt ɛ for ɛ 0 due to the potential singularities of ν close to 0 (making this sum become a series, non necessarily convergent). To get convergence, one has to consider the compensated version of this Poisson process (in order to get a 0 mean and being able to use a central limit type theorem): X ɛ t = x(j X (ds dx) ν(dx)ds) ɛ x <1,0 s t 17/49
Lévy-Ito decomposition: Comments The last term lim X t ɛ can be seen as a (possibly infinite) superposition of independent compound Poisson processes : consider for this Y n = X ɛ n+1 t for a sequence ɛ n 0 (so that lim X t ɛ following result: X ɛn t n = lim n Yi), and use the i=1 Proposition: Let (X t, Y t) be a Lévy process. If (Y t) is a compound Poisson process and (X t) and (Y t) never jump together, then they are independent. Each term Y n is hence a compound Poisson process (with support of the distribution of jumps, F n, contained in [ɛ n+1, ɛ n) ( ɛ n, ɛ n+1]), and hence never jumps simultaneously with another Y m (for n m) 18/49
Lévy-Ito decomposition: Comments This theorem in summary means that a Lévy process can be seen as a combination of a Brownian motion with drift, and of a sum (or superposition), possibly infinite, of independent compound Poisson processes, but with compensation Actually, the proof uses a central limit theorem to get convergence, applied on this sequence of centered (compensated) random variables (one can see that Yn converges a.s. uniformly for t in a compact set, n and in distribution, thanks to the compensation present in the X t ɛn ) This also means that any Lévy process can be approximated, with an arbitrarily good precision, by the sum of a Brownian motion with drift and a compound Poisson process, i.e. a jump-diffusion process (cf. Monte Carlo simulations, see later) It suffices for that to consider the decomposition for some ɛ > 0 fixed, and not passing to the limit for ɛ 0 19/49
Lévy-Ito decomposition: Comments In summary, the result says that we can write X t as: X t = γt + σw t + J t = γt + σw t + J (1) t + J (0) t where J t is a pure jump process corresponding to the jump part of X, and: J (1) t = J t Λ, Λ = {x : x 1} J (0) t = J t Λ 0, Λ 0 = {x : x < 1}\{0} is a simple compound Poisson process J (0) t is the superposition of compensated Poisson processes, that might be infinite (case of an infinite activity process) or not (case of a finite activity process), with or without finite variation (see later), in function of the behaviour of νclose to 0 In fact we can also write: J (1) t X t = J (0) t + σw t }{{} Martingale + γt + J (1) t }{{} Finite variation process which implies that X t is a semi-martingale (see later) 20/49
Corollary: Lévy-Khinchin representation Corollary Let (X t ) be a Lévy process on R with characteristic triplet (σ, γ, ν). Then Φ t (z) = E[e izxt ] = e tψ(z) z R where the characteristic exponent ψ(z) has the form: ψ(z) = 1 2 σ2 z 2 + iγz + (e izx 1 izxi x <1 )ν(dx) R 21/49
Corollary: Lévy-Khinchin representation Proof: We have seen that for a compound Poisson process, the characteristic exponent ψ was equal to: ψ(z) = (e izx 1)ν(dx), ν(dx) = λdf (x) R and for a compensated compound Poisson process ( X t = X t E[X t]): ψ(z) = (e izx 1 izx)ν(dx). R By Lévy - Ito decomposition theorem, for each t, γt + σw t + X l t + X ɛ t ɛ 0 X t a.s. which implies the convergence in distribution. Now convergence in distribution is equivalent to convergence of the characteristic functions 22/49
Corollary: Lévy-Khinchin representation The characteristic function of γt + σw t + X l t + X ɛ t is: E[e iz(γt+σwt +X l t + X ɛ t ) ] = E[e iz(γt+σwt ) ]E[e izx l t ]E[e iz X ɛ t ] as the different terms are independent (cf. proof of L-I). Now, E[e iz(γt+σwt ) ] = e σ2 z 2 t+itγz 2 (the characteristic function of X N(µ, σ 2 ) is Φ X (z) = exp( σ2 z 2 + iµz)), 2 { } and E[e izx l t ] = exp E[e iz X ɛ t ] = exp { t t (e izx 1)ν(dx) x 1 (e izx 1 izx)ν(dx) ɛ x <1 Clearly, the product of (1), (2), (3) converges to the formula of the Lévy-Khinchin theorem. } (1) (2) (3) 23/49
Infinite activity vs Finite activity ν(a) measures the expected number of jumps by time unit with size in A ν(r) is the expected number of jumps by time unit For instance, for a Poisson process of intensity parameter λ, ν(r) = λ <. A process for which ν(r) = is said to be of infinite activity In this case, any (even bounded) time interval, even very small, contains an infinite number of jumps in average One can see that it is not only in average: the number of jumps in any time interval will be infinite a.s. Moreover, one can show that the set of jump instants is countable, and dense in [0, ) 24/49
Remark: truncation convention The truncation of jumps at 1 in Lévy-Ito decomposition is purely conventional. We could express the result with any other truncation threshold ɛ > 0. Lévy-Khinchin formula becomes then: ψ(z) = 1 2 σ2 z 2 + iγ ɛ z + (e izx 1 izxi x <ɛ )ν(dx) with R γ ɛ = γ xi ɛ x <1 ν(dx) R (by using the fact that I x <1 = I ɛ x <1 + I x <ɛ ) We truncate here with the function I x <1. 25/49
Determination of the characteristic triplet (σ, γ, ν): examples Standard Brownian motion If X N(µ, σ 2 ), then Φ X (z) = exp( σ2 z 2 2 + iµz). So if W t N(0, t) is a standard B.M., its characteristic function is equal to: Φ Wt (u) = e u2 2 t = e tψ(z) ψ(u) = u2, and hence the characteristic triplet of a standard Brownian 2 motion is: (σ, γ, ν) = (1, 0, 0) In particular, the Lévy measure is 0 (no jump part). Similarly, the triplet of X t = µt + σw t is (σ, µ, 0). 26/49
Determination of the characteristic triplet: examples Poisson process N t of intensity λ We have seen that E[e iznt ] = exp{λt(e iz 1)}. This implies ψ(z) = λ(e iz 1) Now, we need to re-write this by making appear as in Lévy-Khinchin formula the expression: (e izx 1 izxi x <1 )ν(dx) R From the definition of ν, as jumps have only a size 1, we see that ν(dx) = λδ 1 (Dirac measure at 1). So : (e izx 1 izxi x <1 )ν(dx) = λ(e iz 1) = ψ(z) R So the triplet is simply: (σ, γ, ν) = (0, 0, λδ 1). 27/49
Determination of the characteristic triplet: examples Compound Poisson process X t of intensity λ and jump distribution F We have seen that Φ(z) = E[e izxt ] = exp{tλ R (eizx 1)dF (x)} and ν(dx) = λdf (x). So: ψ(z) = λ (e izx 1)dF (x) R = λ (e izx 1 izxi x <1 )df (x) + izλ R = σ2 z 2 + iγz + λ 2 xi x <1 df (x) R (e izx 1 izxi x <1 )df (x) R required that: γ = λ σ = 0 xi x <1 df (x) = λ R x df (x) x <1 28/49
Determination of the characteristic triplet: examples Gamma process: X t Γ(αt, λ), where the density of a Gamma distribution Γ(a, b) is: { b a f Γ (x; a, b) = x a 1 e bx x > 0 Γ(a) 0 else where the Gamma function is defined for z C with R(z) > 0 by: Γ(z) = 0 t z 1 e t dt (function generalizing the factorial : if z = n N, Γ(n) = (n 1)!, and satisfying the recurrence: Γ(z + 1) = zγ(z)). 29/49
Determination of the characteristic triplet: examples Gamma process (X t Γ(αt, λ)): The Gamma law can also be defined by its characteristic function: ( Φ Γ (z; a, b) = E[e izx ] = 1 iz ) a. b From this expression, we easily see that the Gamma law is infinitely divisible, as its characteristic function can be written as the n th power of another characteristic function of an identical law: E[e iux ] = where Y i i.i.d. Γ( a n, b). ( 1 iu b ) a = ( ( 1 iu b = Π n k=1 E[eiuY k ] = E[e iu(y 1+...+Y n) ] In consequence, it defines a Lévy process. ) ) a n n ( = Π n k=1 1 iu ) a n b 30/49
Determination of the characteristic triplet: examples Gamma process (X t Γ(αt, λ)): By using an integral representation of ψ(z) = ln ( 1 iz λ ) α: ( α ln 1 iz ) = α λ 0 (e izx 1)x 1 e λx dx (Frullani integral 1 ), and using Lévy-Khinchin representation theorem, we get: and γ = ν(dx) = α x e λx I x>0dx 1 and finally that σ = 0 (no diffusion part!). 0 x α x e λx dx = α 1 e λ λ 1 If f is C 1 and if the integral equal to (f (0) f ( )) ln b a 0 f (ax) f (bx) dx converges, then this integral is x 31/49
Determination of the characteristic triplet: examples Gamma process (X t Γ(αt, λ)): We see that for any ɛ > 0: ν( x ɛ) = x ɛ α x e λx I x>0dx < which means that the number of jumps of size ɛ is finite. However, α ν(r) = ν(dx) = x e λx dx =. R R + The Gamma process is hence an infinite activity process: the average number of jumps by time unit is infinite. 32/49
Infinite activity processes: remark One can see generally that for any infinite activity process (i.e. a process s.t. ν(r) = ), any time interval contains an infinity of jumps. Moreover, one can see that the set of jumps instants is countable and dense in [0, ). The countable character is due to the cadlag property of trajectories. The density (as well as the infinity of the jumps number) can be proven by the following: Consider a time interval [a, b] and { ɛ(n) = sup r : x r ν(dx) = ν( x r) n } ν( x r) is clearly a decreasing function of r ɛ(n) represents a jump size: it is the greatest admissible jump size for which we will find at least n jumps of that size or greater by time unit in average ɛ(n) is a decreasing function of n 33/49
Infinite activity processes: remark One then defines Y n = J X (dx dt) ɛ(n) x <ɛ(n 1),t [a,b] = J X (([ɛ(n), ɛ(n 1)) ( ɛ(n 1), ɛ(n)]) [a, b]) which corresponds to the number of jumps on [a, b] with absolute size between ɛ(n) and ɛ(n 1). By Lévy Ito decomposition theorem, the random variables Y n are independent Poisson distributed random variables. These variables are also identically distributed: they are Poisson distributed, with a mean equal to the expected number of jumps of size within [ɛ(n), ɛ(n 1)), which is 1 by construction of the (ɛ(n)). Yn hence provides the total number of jumps on [a, b], which n=1 appears as an infinite sum of i.i.d variables with mean 1. By the law of large numbers, this series converges to. This is true for any interval [a, b], which implies the density property. 34/49
Path properties The following proposition is a consequence of the results seen up to now Proposition A Lévy process has piecewise constant trajectories it is a compound Poisson process its characteristic triplet satisfies: σ = 0 ν(dx) < R γ = its characteristic exponent is of the form: with ν(r) <. ψ(z) = xν(dx) x <1 (e izx 1)ν(dx) R 35/49
Path properties: finite variation One defines the total variation of a function f : [a, b] R by TV (f ) = sup n f (t i) f (t i 1) where the sup is taken on all finite subdivisions / partitions of [a, b]: i=1 a = t 0 < t 1 <... < t n = b In particular, any increasing or decreasing function has a finite variation, and every finite variation function can be written as the difference of two increasing functions. A Lévy process is said of finite variation if its trajectories are of finite variation with probability one 36/49
Path properties: finite variation Proposition: A Lévy process is of finite variation its triplet satisfies σ = 0 and x ν(dx) < x 1 Idea of the proof:( ): Use Lévy Ito decomposition: X t = γt + X ɛ t = x 1,0 s t ɛ x <1,0 s t which can be rewritten here without compensation: X ɛ t = X t = bt + ɛ x <1,0 s t x 1,0 s t xj X (ds dx) + lim ɛ 0 X ɛ t x(j X (ds dx) ν(dx)ds) xj X (ds dx) + lim Xt ɛ ɛ 0 xj X (ds dx) = where b is the absolute drift: b = γ 0< x <1 xν(dx). ɛ X s <1,s t X s 37/49
Path properties: finite variation as by hypothesis, xν(dx) exists, so that this part can be separated from x <1 the limit (and grouped with γ to provide a new drift b). The first two terms have a finite variation. We then consider the variation of the third one on [0, t]: TV (Xt ɛ ) = X s = x J X (ds dx) ɛ x <1,0 s t ɛ X s <1,s t We take the expectation of both members and use Fubini theorem: E[TV (X ɛ t )] = t x ν(dx) t ɛ x <1 so that E[TV (X ɛ t )] converges towards a finite limit. x ν(dx) < if ɛ 0 x <1 38/49
Path properties: finite variation On the other hand, TV (Xt ɛ ) X s a.s. 0< X s <1,s t if ɛ 0. One easily sees that this last series is equal to TV ( ) X s = TV xj X (ds dx) x <1,0 s t 0< X s <1,s t By monotone convergence ( thm, E[TV (Xt ɛ )] E[limTV (Xt ɛ )] = E[TV (lim Xt ɛ )] = E[TV xj x <1,0 s t X (ds dx)) ]. Hence TV (lim Xt ɛ ) is a random variable with a finite expectation, and is hence finite a.s. 39/49
Path properties: finite variation ( ): TV (X t) TV(pure jump part) (i.e. the part moving only by jumps), and is supposed to be finite. The TV of the pure jump part restricted to jumps larger than ɛ is: = x J X (ds dx) ɛ x <1,s [0,t] x ν(dx) + ɛ x <1 x (J X (ds dx) ν(dx)ds) ɛ x <1,s [0,t] We then show that the second term converges a.s. to something finite (similar arguments as in the proof of Lévy-Ito decomposition). This implies that the first term converges to a finite limit. Indeed, if this is not the case, then the TV of the jump part would not be finite, a contradiction. 40/49
Path properties: finite variation So the limit of this first term, x ν(dx), is <, and this implies that 0 x <1 xj [0,t] R X (ds dx) is well defined and has a finite variation. So we can decompose X t in : X t = Xt c + xj [0,t] R X (ds dx), where Xt c is the continuous part of process X, and this implies that Xt c must also be of finite TV. This implies that σ = 0 (as a Brownian motion has an infinite variation). 41/49
Path properties: finite variation Corollary (Lévy-Ito and Lévy-Khinchin theorems in the finite variation case): Let (X t ) be a Lévy process of finite variation with Lévy triplet (0, ν, γ). Then X can be expressed as the sum of its jumps between 0 and t and a linear drift: X t = bt + x J X (ds dx) = bt + X s [0,t] R where b = γ x <1 x ν(dx). Its characteristic function can be expressed as: Φ t (z) = e t{ibz+ R (eizx 1)ν(dx)}. s [0,t], X s 0 42/49
Classification This leads to the following classification (for the jump part): Finite activity process : ν(r) <. This is the jump diffusion case. It has a finite variation if it has moreover no Brownian component (its jump part is anyway of finite variation), and can be expressed as the sum of its jumps between 0 and t (+ linear drift) Infinite activity but of finite variation process: ν(r) = but x ν(dx) < 0< x <1 infinite number of jumps on any time interval BUT the process (without drift) can be expressed as the sum of its jumps between 0 and t (no need to compensate to make the series converge). Infinite activity and infinite variation process: only condition: x 2 ν(dx) < 0< x <1 43/49
Classification The infinite activity and/or finite variation property depends on the behavior of ν in a neighborhood of 0. 44/49
Path properties: Examples Compound Poisson process: ν(dx) = λδ 1 (dx) in the non compound case, and ν(dx) = λdf (x) in the general case. It is clearly of finite activity, and hence also of finite variation ( x ν(dx) < clearly) x <1 Gamma process: we have seen that it is of infinite activity (ν(r) = ) BUT it is of finite variation! Indeed: ν(dx) = ax 1 e bx I x>0 dx, hence x ν(dx) = ae bx I x>0 and hence x ν(dx) = ae bx dx < x <1 0<x<1 The same will hold for the Inverse Gaussian process (see later). Before seeing other examples, the introduce the notion of subordinator: 45/49
Path properties: Subordinators Definition: A subordinator is an increasing Lévy process. Subordinators are important in financial modelling as they can be used as time change within other Lévy processes, and in particular Brownian motion: W (t) W (Z t ), Z t subordinator This is like observing a Brownian motion on a new time scale, a stochastic one, that can be interpreted as a kind of business time i.e. the integrated rate of information arrival 46/49
Path properties: Subordinators Proposition Let(X t ) be a Lévy process on R. Then the following points are equivalent: 1 X t 0 a.s. for some t > 0 2 X t 0 a.s. for all t > 0 3 (X t ) is a subordinator: sample paths of (X t ) are a.s. non decreasing 4 The characteristic triplet satisfies: 0 σ = 0 ν((, 0]) = 0 min(1, x)ν(dx) < b 0 where b = γ x <1 xν(dx). 47/49
Path properties: Subordinators Examples (1) Proposition: If X is a Lévy process on R, f : R [0, ) such that f (x) = O( x 2 ) for x 0. Then S t = f ( X s) is a Lévy process, and a subordinator. s t, X s 0 Proof: exercise (hint: show that E[S t] < to show that S t is well defined) In particular, by choosing f (x) = x 2, we get that the sum of the squared jumps of a Lévy process leads to a subordinator: S t = X s 2 s t, X s 0 This new process is called the discontinuous quadratic variation of X (see later). 48/49
Path properties: Subordinators Examples (2) What we have seen on the Gamma process implies that it is a subordinator (the Gamma distribution is positive). We have see that: ν(dx) = αx 1 e x I x>0dx ν has hence a positive support, as announced by proposition of slide 47. Moreover, we have seen that there is no Brownian component (σ = 0), and that its (relative) drift γ is equal to: Hence its absolute drift b is equal to: γ = α 1 e λ λ b = γ 1 0 αe λx dx = γ + α λ [e λx ] 1 0 = γ + α λ (e λ 1) = 0 By subordinating a Brownian motion with drift with a Gamma process, we obtain a new process, the variance gamma process. This is still a Lévy process (see later). 49/49