Properties of Lévy Processes

Lévy processes, which are defined as having stationary and independent increments, were introduced in the previous post. It was seen that the distribution of a d-dimensional Lévy process X is determined by the characteristics (Σ, b, ν) via the Lévy-Khintchine formula,

(1)

The positive semidefinite matrix Σ describes the Brownian motion component of X, b is a drift term, and ν is a measure on ℝ^d such that ν(A) is the rate at which jumps ΔX ∈ A of X occur. Then, equation (1) gives us the characteristic function of the increments of the process.

In the current post, I will investigate some of the properties of such processes, and how they are related to the characteristics. In particular, we will be concerned with pathwise properties of X. It is known that Brownian motion and Cauchy processes have infinite variation in every nonempty time interval, whereas other Lévy processes — such as the Poisson process — are piecewise constant, only jumping at a discrete set of times. There are also purely discontinuous Lévy processes which have infinitely many discontinuities, yet are of finite variation, on every interval (e.g., the gamma process).

Let’s start with the simple case of processes with finitely many jumps in each bounded time interval. The proof is given below in the more general case of non-stationary independent increments (see Lemma 6).

Theorem 1 Let X be a cadlag d-dimensional Lévy process with characteristics (Σ, b, ν). Then the following are equivalent.

1. With positive probability, X has finitely many jumps in some time interval (s,t] with s < t.
2. Almost surely, X has finitely many jumps in every bounded interval.
3. ν(ℝ^d) < ∞.

Furthermore, if these conditions hold then the number of jumps in the time interval (s,t] has the Poisson distribution with parameter (t - s)ν(ℝ^d).

For example, this includes homogeneous Poisson processes of rate λ > 0, where X_t – X_s has the Poisson distribution of rate λ(t - s). In that case, the Lévy measure is just ν = λδ₁, where δ₁ represents the Dirac measure at 1. More generally, we can construct pure-jump Lévy processes as follows. Fix a constant λ > 0 and probability measure μ on ℝ^d with μ({0}) = 0. Then, let S₁, S₂, … be a sequence of exponentially distributed random variables with parameter λ and Z₁, Z₂, … be ℝ^d-valued variables with measure μ, all of which are independent. Setting T_k = Σ^k_j=1S_j, then N_t ≡ max{k: S_k ≤ t} is a homogeneous Poisson process of rate λ. We can define the piecewise-constant process

which can be seen has stationary independent increments, so it a Lévy process. Then, X is known as a compound Poisson process of rate λ and jump distribution μ. Its characteristic function can be computed in terms of the characteristic function φ_μ(a) ≡ 𝔼[e^ia·Z₁] of μ.

Substituting in the expression for ψ_μ as an integral with respect to μ gives,

Comparing this with the Lévy-Khintchine formula shows that X has Lévy characteristics (0, b̃, ν), where ν = λμ and

(2)

Conversely, consider any Lévy process X with characteristics (Σ, b, ν), where ν is finite. By the decomposition into continuous and purely discontinuous parts, we can write this as

where W is a d-dimensional Brownian motion with covariance matrix Σ, b̃ is given by (2) and Y is a Lévy process with characteristics (0, b̃, ν). Letting λ = ν(ℝ^d) and μ be the probability measure λ^-1ν, we see that Y is a compound Poisson process of rate λ and jump distribution μ. This, then, describes all Lévy processes whose jumps occur at a finite rate.

We can also give conditions for a Lévy process to have finite variation on bounded intervals. The proof is left until Lemma 7 below.

Theorem 2 Let X be a cadlag d-dimensional Lévy process with characteristics (Σ, b, ν). Then the following are equivalent.

1. With positive probability, X has finite variation over some time interval (s,t] with s < t.
2. Almost surely, X has finite variation on every bounded interval
3. Σ = 0 and
   

   (3)

Furthermore, in this case, we have

(4)

where b̃ is given by (2).

Property (3) is particularly interesting in the case where ν(ℝ^d) is infinite. In that case, X jumps infinitely often but has finite variation in every nonzero bounded time interval. When X is an FV process, it is often convenient to use the decomposition (4), so that we break it up into a constant drift term and a pure jump process Y satisfying Y_t = Σ_s≤tΔY_s. Such pure jump processes have a particularly simple expression for the characteristic function 𝔼[e^ia·Y_t] = exp(tψ_Y(a)),

This follows from Theorem 2 by noting that, in this case, b̃ = b so that the Lévy-Khintchine formula simplifies. Comparing with the form of the Lévy-Khintchine formula given in (1), we see that this avoids the rather messy ia·x/(1 + ‖x‖) term in the integral. However, some purely discontinuous Lévy processes, such as the Cauchy process, do have infinite variation on bounded time intervals, so the more general form in (1) is used.

Real valued and nondecreasing Lévy processes starting from 0 are known as subordinators. In particular, these are finite variation processes, so (3) is satisfied. Subordinators are easily described in terms of their characteristics.

Corollary 3 Let X be a cadlag real valued Lévy process with characteristics (0, b, ν) satisfying (3), and let b̃ be defined by (2).

Then, X is nondecreasing if and only if ν is supported by the positive reals and b – b̃ is nonnegative.

Proof: If X is nondecreasing then its jumps must be nonnegative, so ν(ℝ_–) = 0. Also, (4) shows that X has continuous part X_t – X₀ – Σ_s≤tΔX_s = (b - b̃)t which must be nondecreasing, so b – b̃ ≥ 0. Conversely, if b – b̃ ≥ 0 and ν(ℝ_–) = 0 then X has nonnegative jumps and (4) shows that X is nondecreasing. ⬜

An example of a subordinator can be constructed from hitting times of a standard Brownian motion B started from 0. For each a ≥ 0, let τ_a be the stopping time

Considering a as a time index, the process a ↦ τ_a is right-continuous and increasing. By the strong-Markov property, B̃_t ≡ B_τa+t – a is also a standard Brownian motion independently of τ_a which passes through level b at time τ_a+b – τ_a, so τ_a+b – τ_a ∼ τ_b. From this, we see that a ↦ τ_a is a subordinator.

At each time t ≥ 0, the Brownian motion B is almost surely strictly less than its maximum B^∗_t = sup_s≤tB_s, which implies that t is in the union of the intervals (τ_a–, τ_a). So, the union ⋃_b≤a(τ_b–, τ_b) almost surely covers almost all of the interval [0, τ_a]. This means that τ_a = Σ_b≤aΔτ_a is a pure jump process. We can compute its Lévy measure. We have {τ_a < u}= {B^∗_u > a} and, by the reflection principle, this has probability exactly twice that of B_u being greater than a. So, in the limit as a goes to zero,

The Lévy measure of τ satisfies ν([u, ∞)) = (2/πu)^1/2, so dν(x) = x^-3/2dx/√2π. Also, using the fact that B̃_t ≡ u^-1/2B_ut is a Brownian motion hitting a at time u^-1τ_√ua, we see that τ_a ∼ u^-1τ_√ua. In particular, the sum of two independent copies of τ_a has the same distribution as τ_2a ∼ 4τ_a. This is equivalent to saying that τ_a is a stable random variable with stability parameter 1/2. Lévy processes whose increments have stable distributions are known as stable processes and, in particular, τ is a stable subordinator. Writing the moment generating function as 𝔼[e^–uτ_a] = exp(-aφ(u)), the Lévy-Khintchine formula can be used to calculate φ(u) = ∫(1 - e^–ux)dx/(√2πx^3/2), from which we obtain φ(u) = √2u. This verifies the well-known formula for Brownian motion hitting times,

Other examples of subordinators include the gamma process. A gamma process X with mean μ and variance σ² per unit time is a Lévy process with X₀ = 0 such that X_t has the gamma distribution with mean μt and variance σ²t. Setting λ = μ/σ² and γ = μ²/σ², this has probability density function

From this, we see that it has Lévy measure dν(x) = 1_{x>0}γx^-1e^–λx dx. Furthermore, computing

we see that the drift term in (4) is zero, so that this is a pure jump process with X_t = Σ_s≤tΔX_s. As it has the gamma distribution, the characteristic function of X is

Subordinators are often used to apply stochastic time changes to a process. If Z is a Lévy process and, independently, t ↦ τ_t is a subordinator, then X_t ≡ Z_τt is another Lévy process. For example, the variance gamma process is a Brownian motion time-changed by a gamma process. Let Z_t = B_t + θt be a standard Brownian motion with drift θ and t ↦ τ_t be a gamma process with mean 1 and variance σ² per unit time. As τ is a pure jump process, then the variance gamma process is also a pure jump process satisfying X_t = Σ_s≤tΔX_s. Its characteristic function can be calculated from the characteristic functions of the normal and gamma distributions,

where γ = σ^-2. Factoring the quadratic into linear terms gives

However, this is just the product of the characteristic function of a gamma process and minus a gamma process, from which we see that variance gamma processes are the difference of independent gamma processes. Therefore, X has Lévy measure

An example of a purely discontinuous Lévy process with infinite variation over all nonzero time intervals is given by the Cauchy process, which is a stable Lévy process with stability parameter α = 1. If X is a Cauchy process, then X_t has the symmetric Cauchy distribution at time t. The probability density function is

So, X has Lévy measure dν(x) = π^-1x^-2 dx. In particular, as ∫¹_-1|x| dν(x) = ∞, inequality (3) does not hold and, consequently, X is not a finite variation process.

It is possible to determine the integrability of a Lévy process from its Lévy measure. In fact, integrability is equivalent to the apparently much weaker property of local integrability, and is also equivalent to the seemingly much stronger condition of integrability of its maximum process. The proof of the following theorem will be given in Lemma 8 below for the more general situation of inhomogeneous independent increments processes.

Theorem 4 Let X be a cadlag d-dimensional Lévy process with X₀ = 0 and let p be a real number in [1, ∞). Then, the following are equivalent.

1. X is locally L^p-integrable.
2. X is L^p-integrable.
3. X^∗_t ≡ sup_s≤t‖X_s‖ is L^p-integrable.
4. The Lévy measure ν satisfies ∫_‖x‖≥1‖x‖^p dν(x) < ∞.

So each of the examples of Lévy processes mentioned above are integrable, with the exceptions of the 1/2-stable process of Brownian hitting times and the Cauchy process.

Note that if X is an integrable Lévy process then X – 𝔼[X] has independent increments of mean zero, so is a martingale. It is therefore possible, and often convenient, to decompose such processes as the sum of a constant drift term and a martingale Lévy process.

Lemma 5 Let X be an L^p-integrable Lévy process with characteristics (Σ, b, ν). Then it uniquely decomposes as

(5)

where b′ ∈ ℝ^d, W is a Brownian motion with covariance matrix Σ and M is an L^p-integrable martingale with characteristic function 𝔼[e^ia·M_t] = exp(tψ_M(a)) where,

(6)

As |e^ia·x – 1 – ia·x| ≤ |a·x|+2, the final statement of Theorem 4 implies that the integral in (6) is well-defined. The proof of this Lemma follows quickly from previous results of these notes. First, for any b′ ∈ ℝ^d, decomposition (5) follows from the decomposition of X into its purely continuous and discontinuous parts, W and M respectively. Then, M is an L^p-integrable Lévy process with characteristics (0, b - b′, ν). Setting

then (6) follows from the Lévy-Khintchine formula. It only remains to show that M is a martingale and, by the independent increments property, it is enough to show that it has zero mean. Using dominated convergence to commute the differentiation with the integral in (6), we can calculate

So, differentiating (6) at a = 0 gives 𝔼[M_t] = t∇ψ_M(0) = 0.

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Inhomogeneous independent increments processes

All of the results given above apply equally to time-inhomogeneous independent increments processes, and I will now go through their proofs at this level of generality. Throughout this section, it is assumed that X is a cadlag d-dimensional process with independent increments, and which is continuous in probability. As previously shown, the increments of such a process have characteristic function of the form

(7)

Here, Σ̃: ℝ₊ → ℝ^d×d is a map from time t to a symmetric d × d matrix Σ̃_t with Σ̃₀ = 0 and is increasing in the sense that Σ̃_t – Σ̃_s is positive semidefinite for all t ≥ s. Also, b̃: ℝ₊ → ℝ^d is a continuous function starting from zero, and μ is a Borel measure on ℝ₊ × ℝ^d such that μ(ℝ₊ × {0}) = μ({t} × ℝ^d) = 0 for all t ≥ 0, and

In the case where X also has stationary increments, so that it is a Lévy process, then (Σ̃, b̃, μ) are related to the Lévy characteristics (Σ, b, ν) by Σ̃_t = tΣ, b̃ = tb and dμ(t, x) = dt dν(x). In that case, equation (7) reduces to the standard Lévy-Khintchine formula.

We start by giving a proof of Theorem 1 which, just being a re-statement of results previously covered in these note, is particularly simple.

Lemma 6 The following are equivalent.

1. With positive probability, X has finitely many jumps.
2. With probability one, X has finitely many jumps.
3. μ(ℝ₊ × ℝ^d) is finite.

In this case, the number of jumps of X is Poisson distributed with rate μ(ℝ₊ × ℝ^d).

Proof: Letting η be the total number of jumps of X, then this is just a restatement of the fact that η is Poisson distributed with rate λ = μ(ℝ₊ × ℝ^d) whenever λ is finite, and η is almost surely infinite whenever λ is infinite. ⬜

Theorem 1 is an immediate consequence of this. If X is a Lévy process with characteristics (Σ, b, ν), then the first statement of Theorem 1 implies that there is a non-trivial time interval [s,t] on which, with positive probability, X has finitely many jumps. Applying Lemma 6 to X restricted to this interval implies that μ([s, t] × ℝ^d) = (t - s)ν(ℝ^d) is finite. So, ν(ℝ^d) < ∞ and it follows that μ([s, t] × ℝ^d) is finite for all s < t and, again applying Lemma 6, the number of jumps in any finite time interval [s,t] is almost surely finite with the Poisson distribution of rate (t - s)ν(ℝ^d).

I now give a proof of the following generalization of Theorem 2.

Lemma 7 Let f: ℝ₊ × ℝ^d → ℝ be a measurable function satisfying f(t, 0) = 0, and set V = Σ_t>0|f(t, ΔX_t)|. The following are equivalent,

1. With positive probability, V is finite.
2. With probability one, V is finite.
3. μ(|f| ∧ 1) < ∞.

In this case, setting U = Σ_t>0f(t, ΔX_t), then

(8)

for all real a.

Proof: This result actually holds in the generality of Poisson point processes although, here, we are only concerned with the application to independent increments processes. Let us start by considering the case where μ(|f| ∧ 1) is finite. In that case, the identity

shows that V is almost-surely finite. So, Y_t ≡ Σ_s≤tf(s, ΔX_s) is a well-defined independent increments process. Therefore, its characteristic function is of the form 𝔼[e^iaY_t] = exp(ψ̃_t(a)) where

is a square integrable martingale. Taking expectations and letting t increase to infinity, so that Y_t → U, gives

as required. Note that the inequality |e^iaf – 1| ≤ |af| ∧ 2 implies that e^iaf – 1 is μ-integrable, so this expression is always well defined.

It only remains to show that all this holds whenever V is finite with nonzero probability. Consider, for the moment, the case where f is nonnegative and μ(f ∧ 1) < ∞. Then, e^iaf is uniformly bounded for all complex a with nonnegative imaginary part and, by analytic continuation (8) extends to all such a. So, we can replace a by ia to get

(9)

for all a > 0. However, this identity extends to all nonnegative f. Simply apply it to a sequence of nonnegative functions f_n ↑ f with μ(f_n ∧ 1) < ∞ and use dominated convergence on the left and monotone convergence on the right to take the limit as n goes to infinity. Suppose that the third property of the statement of the Lemma did not hold, so that μ(f ∧ 1) = ∞. Then, the inequality 1 – e^–af ≥ 12 ((af) ∧ 1) gives μ(1 - e^–af) = ∞ for all positive a. So, the right hand side of (9) is zero. Taking the limit as a decreases to 0,

Conversely, if U ≡ Σ_t>0f(t, ΔX_t) has nonzero probability of being finite, then we see that μ(f ∧ 1) must be finite. Applying this to |f| for arbitrary (not necessarily positive) f shows that the first condition of the Lemma implies the third. ⬜

To show that this result implies Theorem 2, consider a Lévy process X with characteristics (Σ, b, μ). As the continuous part of its quadratic variation, [X]^c_t = tΣ is constant over any interval on which X has finite variation, the first condition of Theorem 2 implies that Σ is zero. If X has finite variation with positive probability over an interval [s,t], Lemma 7 applied to V ≡ Σ_s<u≤t‖ΔX_u‖ shows that

is finite. So, ν(‖x‖ ∧ 1) < ∞. Then, applying the lemma to any interval [0,t] shows that Σ_s≤t‖ΔX_s‖ is almost surely finite, so Z_t ≡ X_t – Σ_s≤tΔX_s is well defined. As this is a continuous Lévy process with zero quadratic variation, it is simply of the form Z_t = X₀ + ct for a constant c ∈ ℝ^d. So, X almost surely has finite variation over all bounded time intervals. To calculate c, we can apply (8) with f(t, x) = ia·x to obtain

Comparing this with the Lévy-Khintchine formula (1) gives c = b – b̃ with b̃ as in equation (2). So (4) holds, and Theorem 2 follows.

Finally, we give a proof of Theorem 4 for independent increments processes.

Lemma 8 Suppose that X₀ = 0 and choose any 1 ≤ p < ∞. Then, the following are equivalent.

1. X is locally L^p-integrable.
2. X is L^p-integrable.
3. X^∗_t ≡ sup_s≤t‖X_s‖ is L^p-integrable.
4. For all times t ≥ 0
   

   (10)

Proof: First, consider any nonnegative measurable function f: ℝ^d → ℝ such that

is finite for all t ≥ 0. Then, taking Y_t = Σ_s≤t1_{ΔXs≠0}f(ΔX_s), we know that Y – c is a martingale so, by optional stopping,

(11)

for all stopping times τ. Note that this extends to all nonnegative measurable f. Just apply it to a sequence 0 ≤ f_n ≤ f increasing to f such that ∫_[0,t]×ℝ^df_n(x) dμ(s, x) is finite, and use monotone convergence as n goes to infinity. In particular, consider f(x) = 1_{{‖x‖≥1}}‖x‖^p. In that case ΔY ≤ ‖ΔX‖^p. If property 1 holds so that X is locally L^p-integrable, then Y is locally integrable. Therefore, there is a stopping time τ with ℙ(τ ≥ t) > 0 and 𝔼[Y_τ] < ∞. Applying (11) to this gives

So c_t is finite and (10) holds.

Conversely, supposing that (10) holds, it only remains to be shown that X^∗_t is L^p-integrable for each fixed time t. To do this, fix a constant K > 0 and define stopping times τ₀ = 0 and

for n ≥ 0. For each time s ≤ t, there almost surely exists an n with τ_n ≤ s < τ_n+1, so ‖X_s – X_τn‖< K. Therefore, we can bound X^∗_t in the L^p norm by

(12)

Noting that X_τn+1∧t – X_τn∧t is bounded by K + ‖ΔX_τn+1∧t‖ whenever τ_n < t and zero otherwise, its L^p norm satisfies the bound

(13)

The terms on the right hand side can be bounded as follows. By the independent increments property, for all times s < t,

Using α to denote the term on the right hand side, we can assume that K has been chosen large enough that this is strictly less than 1 (actually, this is true for all positive K). Then, as the space-time process (s, X_s) is a Feller process and, hence, satisfies the strong Markov property, this inequality holds when s is replaced by a stopping time. Replacing s by τ_n ∧ t gives ℙ(τ_n+1 ≤ t |ℱ_τn) ≤ α. So, ℙ(τ_n+1 < t) ≤ αℙ(τ_n < t) and, by induction, we see that ℙ(τ_n < t) is bounded by αⁿ.

Again using the independent increments property for s < t,

Denote the right hand side by c which, by (10), is finite. Applying the strong Markov property again to replace s by τ_n ∧ t, this gives

Putting these bounds back into (13),

Finally putting this back into (12) gives the required bound

⬜