The Wiener–Hopf Factorisation

Abstract

For a general Lévy process, it is possible to decompose its path into “excursions from the running maximum”. Conceptually, this decomposition is a priori somewhat tricky as, in principle, a general Lévy process may exhibit an infinite number of excursions from its maximum over any finite period of time. Nonetheless, when considered in the right mathematical framework, excursions from the maximum can be given a sensible definition in terms of a Poisson random measure. The theory of excursions presents one of the more mathematically challenging aspects of the theory of Lévy processes. This means that in order to keep to the level outlined in the preface of this text, there will be a number of proofs in the forthcoming sections which are excluded or discussed only at an intuitive level.

Within a very broad spectrum of probabilistic literature, the Wiener–Hopf factorisation may be found as a common reference to a multitude of statements concerning the distributional decomposition of the path of any Lévy process, when sampled at an independent and exponentially distributed time, in terms of its excursions from the maximum. The collection of conclusions which fall under the umbrella of the Wiener–Hopf factorisation turns out to provide a robust tool with which one may analyse a number of problems concerning the fluctuations of Lévy processes, in particular, problems which have relevance to the applications we shall consider in later chapters. This chapter concludes with some special classes of Lévy processes for which the Wiener–Hopf factorisation may be exemplified in more detail.

Exercises

6.1

Give an example of a Lévy process which has bounded variation with zero drift for which 0 is regular for both (0,∞) and (−∞,0). Give an example of a Lévy process of bounded variation and zero drift for which 0 is only regular for (0,∞).

6.2

Suppose that X is a spectrally negative Lévy process of unbounded variation with Laplace exponent ψ and recall the definition $\tau^{+}_{x} = \inf\{t>0 : X_{t} >x\}$. Recall also that the process $\tau^{+}:=\{\tau^{+}_{x}: x\geq0\}$ is a (possibly-killed) subordinator (see Corollary 3.14) with Laplace exponent Φ, the right inverse of ψ.

• (i) Suppose that δ is the drift of the process τ ⁺. Show that δ=0.

• (ii) Deduce that

  $$\lim_{x\downarrow0}\frac{\tau^+_x}{x}=0 $$

  almost surely, and hence that

  $$\limsup_{t\downarrow0} \frac{X_t}{t}=\infty $$

  almost surely. Conclude that 0 is regular for (0,∞) and hence that the jump measure of τ ⁺ cannot be finite.

• (iii) From the Wiener–Hopf factorisation of X show that

  $$\lim_{\theta\uparrow\infty}\mathbb{E}\bigl(\mathrm{e}^{\theta \underline {X}_{\mathbf{e}_q}}\bigr) = 0, $$

  and hence use this to give an alternative proof that 0 is regular for (0,∞).

6.3

Fix ρ∈(0,1]. Show that a compound Poisson subordinator with jump rate λρ, killed at an independent and exponentially distributed time with parameter λ(1−ρ), is equal in law to a compound Poisson subordinator killed after an independent number of jumps, which is distributed geometrically with parameter 1−ρ.

6.4

Show that the only processes for which

$$\int_0^\infty\mathbf{1}_{(\overline{X}_t = X_t)}{\mathrm{d}}t>0 \quad \text{and}\quad \int_0^\infty\mathbf{1}_{(X_t = \underline{X}_t)}{ \mathrm{d}}t >0 $$

almost surely are compound Poisson processes.

6.5

Suppose that X is spectrally negative with characteristic triple (a,σ,Π) and that $\mathbb{E}(X_{t})>0$. (Recall that, in general, $\mathbb{E}(X_{t})\in[-\infty,\infty)$.)

• (i) Show that

  $$\int_{-\infty}^{-1} \varPi(-\infty,x){\mathrm{d}}x <\infty. $$

• (ii) Using Theorem 6.15 (iv), deduce that, up to a constant,

  $$\begin{aligned} \widehat{\kappa}(0,\mathrm{i}\theta) =& \biggl(-a+\int_{(-\infty,-1)}x \varPi({\mathrm{d}}x) \biggr) \\ &{}- \frac{1}{2}\mathrm{i}\theta\sigma^2 + \int _{(-\infty,0)}\bigl(1-\mathrm{e}^{\mathrm{i}\theta x}\bigr) \varPi (-\infty,x){ \mathrm{d}}x. \end{aligned}$$

  Hence deduce that there exists a choice of local time at the maximum for which the descending ladder height process has jump measure given by Π(−∞,−x)dx on (0,∞), drift σ ²/2 and is killed at rate $\mathbb{E}(X_{1})$.

6.6

Suppose that X is a spectrally negative stable process of index 1<α<2.

• (i) Deduce, with the help of Theorem 3.12, that up to a multiplicative constant

  $$\kappa(\theta,0) = \theta^{1/\alpha}, \quad\theta\geq0, $$

  and hence that $\mathbb{P}(X_{t} \geq0)=1/\alpha$ for all t≥0.

• (ii) By reconsidering the Wiener–Hopf factorisation, show that, for each t≥0 and θ≥0,

  $$\mathbb{E}\bigl(\mathrm{e}^{-\theta\overline{X}_t } \bigr)=\sum _{n=0}^{\infty} \frac{(-\theta t^{1/\alpha})^n}{\varGamma(1+n/\alpha)}. $$

  This identity is taken from Bingham (1971, 1972).

6.7

(The second factorisation identity)

In this exercise, we derive what is commonly called the second factorisation identity, which is due to Percheskii and Rogozin (1969). It uses the Laplace exponents κ and $\widehat{\kappa}$ to give an identity concerning the problem of first passage above a fixed level $x\in\mathbb{R}$. The derivation we use here makes use of calculations in Darling et al. (1972) and Alili and Kyprianou (2005). We shall use the derivation of this identity later to solve some optimal stopping problems.

Define as usual

$$\tau^+_x = \inf\{t>0: X_t >x\}, \quad x\geq0, $$

where X is any Lévy process.

• (i) Using the same technique as in Exercise 5.7, prove that, for all α>0, β≥0 and $x\in \mathbb{R}$, we have

  $$ \mathbb{E} \bigl( \mathrm{e}^{-\alpha\tau_{x}^{+}-\beta X_{\tau _{x}^{+}}}\mathbf{1}_{(\tau_{x}^{+}<\infty)} \bigr) = \frac{\mathbb{E} ( \mathrm{e}^{-\beta \overline{X}_{\mathbf{e}_{\alpha}}}\mathbf{1}_{(\overline {X}_{\mathbf {e}_{\alpha}}>x)} ) }{\mathbb{E} ( \mathrm{e}^{-\beta\overline {X}_{\mathbf{e}_{\alpha}}} ) }. $$

  (6.46)

  Note that the identity is still true when α=0 if $\mathbb {P}(\overline{X}_{\infty}<\infty) = 1$.

• (ii) Establish the second factorisation identity as follows: If X is not a subordinator then, for α,β≥0,

  $$ \int_{0}^{\infty}\mathrm{e}^{-qx}\mathbb{E} \bigl( \mathrm{e}^{-\alpha\tau_{x}^{+}-\beta(X_{\tau _{x}^{+}}-x)}\mathbf{1}_{(\tau^+_x <\infty)} \bigr) {\mathrm{d}}x = \frac{\kappa( \alpha,q ) -\kappa( \alpha,\beta ) }{(q-\beta)\kappa( \alpha,q ) }. $$

6.8

Suppose that X is any Lévy process which is not a subordinator and e _p is an independent random variable which is exponentially distributed with parameter p>0. Note that 0 is regular for (0,∞) if and only if $\mathbb{P}(\overline{X}_{\mathbf{e}_{p}}=0)=0$.

• (i) Use the Wiener–Hopf factorisation to show that 0 is regular for (0,∞) if and only if

  $$ \int_{0}^{1}\frac{1}{t} \mathbb{P}(X_{t}>0){\mathrm{d}}t=\infty. $$

• (ii) Now noting that 0 is irregular for [0,∞) if and only if $\mathbb{P}(\overline{G}_{\mathbf{e}_{p}}=0)>0$, show that 0 is regular for [0,∞) if and only if

  $$ \int_{0}^{1}\frac{1}{t} \mathbb{P}(X_{t}\geq0){\mathrm{d}}t=\infty. $$

6.9

This exercise gives the random walk analogue of the Wiener–Hopf factorisation. In fact, this is the original setting of the Wiener–Hopf factorisation. We give the formulation in Greenwood and Pitman (1980a). However, one may also consult Feller (1971) and Borovkov (1976) for other accounts.

Suppose that, under P, S={S _n :n≥0} is a random walk with S ₀=0 and increment distribution F. We assume that S can jump both upwards and downwards, in other words min{F(−∞,0),F(0,∞)}>0 and that F has no atoms. Denote by Γ _p an independent random variable which has a geometric distribution with parameter p∈(0,1) and let

$$G = \min\Bigl\{ k = 0,1,\ldots, \boldsymbol{\Gamma}_p : S_k = \max_{j =1,\ldots,\boldsymbol{\Gamma}_p} S_j\Bigr\} . $$

Note that S _G is the last maximum over times {0,1,…,Γ _p }. Define N=inf{n>0:S _n >0} the first-passage time into (0,∞), or equivalently the first strict ladder time. Our aim is to characterise the joint laws (G,S _G ) and (N,S _N ) in terms of F, the basic data of the random walk.

• (i) Show that (even without the restriction that min{F(0,∞),F(−∞,0)}>0),

  $$E\bigl(s^{\boldsymbol{\Gamma}_p} \mathrm{e}^{\mathrm{i}\theta S_{\boldsymbol{\Gamma}_p} }\bigr) = \exp\Biggl\{ - \int _{\mathbb{R}} \sum_{n=1}^\infty \bigl(1 - s^n \mathrm{e}^{\mathrm{i}\theta x} \bigr)q^n \frac{1}{n}F^{*n}({\mathrm{d}}x) \Biggr\} $$

  where 0<s≤1, $\theta\in\mathbb{R}$, q=1−p and E is expectation under P. Deduce that the pair $(\boldsymbol{\Gamma}_{p}, S_{\boldsymbol{\Gamma}_{p}})$ is infinitely divisible.

• (ii) Let ν be an independent random variable which is geometrically distributed on {0,1,2,…} with parameter P(N>Γ _p ). Using a path decomposition in terms of excursions from the maximum, show that the pair (G,S _G ) is equal in distribution to the component-wise sum of ν independent copies of (N,S _N ) conditioned on the event {N≤Γ _p }, and hence it is an infinitely divisible two-dimensional random variable.

• (iii) Show that (G,S _G ) and $(\boldsymbol{\Gamma}_{p} - G, S_{\boldsymbol{\Gamma}_{p}} - S_{G})$ are independent. Further, show that the latter pair is equal in distribution to (D,S _D ), where

  $$D= \max\Bigl\{ k = 0,1,\ldots,\boldsymbol{\Gamma}_p : S_k = \min_{j=1,\ldots,\boldsymbol{\Gamma}_p} S_j\Bigr\} . $$

• (iv) Deduce that

  $$E\bigl(s^{G} \mathrm{e}^{\mathrm{i}\theta S_G }\bigr) = \exp\Biggl\{ - \int _{(0,\infty)} \sum_{n=1}^\infty \bigl(1 - s^n \mathrm{e}^{\mathrm{i}\theta x} \bigr)q^n \frac{1}{n}F^{*n}({\mathrm{d}}x) \Biggr\} , $$

  for 0<s≤1 and $\theta\in\mathbb{R}$. Note, when s=1, this identity was established by Spitzer (1956).

• (v) Show that

  $$E\bigl(s^{G} \mathrm{e}^{\mathrm{i}\theta S_G }\bigr) = \frac {P(\mathbf {\varGamma}_p < N)}{1 - E((qs)^N \mathrm{e}^{\mathrm{i}\theta S_N}) } $$

  and hence deduce the Spitzer–Baxter identity

  $$\frac{1}{1-E(s^N \mathrm{e}^{\mathrm{i}\theta S_N})} = \exp\Biggl\{ \int_{(0,\infty)} \sum _{n=1}^\infty s^n \mathrm{e}^{\mathrm{i}\theta x} \frac{1}{n}F^{*n}({\mathrm{d}}x) \Biggr\} . $$

  See, for example, Bingham (2001).

6.10

Suppose that X is a spectrally negative Lévy process with Laplace exponent ψ whose right inverse is denoted by Φ.

• (i) Use the Frullani integral to show that, for λ,q>0,

  $$\frac{\varPhi(q)}{\varPhi(q)+ \lambda} = \exp\biggl\{ \int_0^\infty {\mathrm{d}}x\int_{[0,\infty)}\bigl(\mathrm{e}^{-\lambda x} - 1 \bigr)\frac{\mathrm{e}^{-qt}}{x}\mathbb{P}\bigl(\tau^+_x\in \mathrm{d}t\bigr) \biggr\} , $$

  where $\tau^{+}_{x} = \inf\{t>0 : X_{t} >x\}$.

• (ii) Next use Theorem 6.15 to show that, for q,λ≥0,

  $$\frac{\varPhi(q)}{\varPhi(q)+ \lambda} = \exp\biggl\{ \int_0^\infty \mathrm{d}t\int_{[0,\infty)}\bigl(\mathrm{e}^{-\lambda x} - 1\bigr) \frac{\mathrm{e}^{-qt}}{t}\mathbb{P}(X_t \in\mathrm{d}x) \biggr\} . $$

• (iii) Hence deduce Kendall’s identity, that

  $$t\mathbb{P}\bigl(\tau^+_x\in\mathrm{d}t \bigr)\mathrm{d}x =x\mathbb{P}(X_t \in\mathrm{d}x)\mathrm{d}t $$

  on [0,∞)×[0,∞).