The Wiener-Hopf factorisation of Lévy processes

L. Döring

M. Savov

L. Trottner

Alex Watson

22 April 2024

1 The Wiener-Hopf factorisation

An ‘elementary’ question

  • • Let \(Y>0\), \(Z<0\) be independent with full support, and \(X = Y+Z\)

  • • Is such a decomposition unique?

  • • In general, the answer is no: the Laplace distribution can be factored in many different ways

  • • \(X\) has an infinitely divisible distribution if for any \(n\ge 1\) there exist iid \(X^{(k)}\) such that \(X \overset {d}{=} X^{(1)}+\dotsb +X^{(n)}\)

  • • When restricting to infinitely divisible distributions, the answer is yes

Lévy processes

  • • \(X\) is a Lévy process if it has stationary, independent increments

  • • Its one-dimensional distributions \(X_t\) are infinitely divisible

  • • \(\psi \) given by \(\EE e^{\iu z X_t} = e^{t\psi (z)}\) is the characteristic exponent (CE) of \(X\)

  • • Lévy–Khintchine formula: \(\psi (z) = \iu a z - \frac {1}{2}\sigma ^2 z^2 + \int _{\RR } \bigl ( e^{\iu z x} - 1 - \iu z x\Ind _{[-1,1]}(x)\bigr ) \Pi (\dd x)\)

  • • \(a\) incorporates drift, \(\sigma \) the Gaussian coefficient, jumps of size \(\dd x\) occur at rate \(\Pi (\dd x)\)

Wiener-Hopf factorisation (version 1)

  • • Let \(\ee _q \sim \Exp (q)\) and \(\bar {X}_t = \sup _{s\le t} X_s\)

  • • \(X_{\ee _q} = \bar {X}_{\ee _q} + (X-\bar {X})_{\ee _q}\)

  • • \(X_{\ee _q}\) is infinitely divisible

  • • Summands are independent, infinitely divisible and have disjoint support

  • • Therefore, such a factorisation is unique

Wiener-Hopf factorisation (version 2, path picture)

(-tikz- Path of a Lévy process with the following notation labelled. X, a Lévy process. X-bar t = sup(xi s, s <= t), the running maximum. H + t, the ascending ladder height process: suprema 'stitched together'. H - t, the descending ladder height process. )

\(H^\pm \) are subordinators (increasing Lévy processes), possibly killed

Wiener-Hopf factorisation (version 2, analytic picture)

  • • Let \(\psi \) be the characteristic exponent of \(X\), and \(\kappa _\pm \) the characteristic exponents of \(H^\pm \)

  • • Then

    \[ - \psi (z) = \kappa _+(z) \kappa _-(-z), \quad z\in \symbb {R}\]

  • • Warning! This does not mean \(X_t = H^+_t - H^-_t\)!

The result: the Wiener-Hopf factorisation is unique

  • Theorem 1 (DSTW, 2023+). Let \(\kappa _\pm '\) be the characteristic exponents of two subordinators, such that

    \[ -\psi (z) = \kappa _+(z) \kappa _-(-z) = \kappa _+'(z)\kappa _-'(-z), \quad z\in \symbb {R}. \]

    Then \(\kappa _+(z) = c\kappa _+'(z)\) and \(\kappa _-'(z) = c\kappa _-(z)\) for some \(c>0\).

We also proved an analogous result for random walks.

Prior art: killed Lévy process, I

  • • Fix \(q>0\)

  • • \(z\mapsto \psi (z)-q\) is the characteristic exponent of \(X\) killed at rate \(q\)

  • • Path and analytic pictures still valid

  • • \(\kappa _\pm (q,\cdot )\) are characteristic exponents of some ladder height processes, and

    \[ q-\psi (z) = \kappa _+(q,z)\kappa _-(q,-z), \quad z \in \symbb {R}. \]

  • Theorem 2 (Rogozin (1966) or earlier). Fix \(q>0\) and let \(\kappa _\pm (q,\cdot )\) and \(\kappa '_\pm (q,\cdot )\) be characteristic exponents of subordinators, such that

    \[ q -\psi (z) = \kappa _+(q,z) \kappa _-(q,-z) = \kappa _+'(q,z)\kappa _-'(q,-z), \quad z\in \symbb {R}. \]

    Then \(\kappa _+(q,z) = c\kappa _+'(q,z)\) and \(\kappa _-'(q,z) = c\kappa _-(q,z)\) for some \(c>0\) and all \(z \in \RR \).

Prior art: killed Lévy process, II

Sketch proof

  • • \(\kappa (q,z)\) can be extended to holomorphic functions on \(\Im z \ge 0\) (where \(\kappa \in \{\kappa _+,\kappa _-,\kappa _+',\kappa _-'\}\))

  • • \(\Re \kappa (q,z) \le \kappa (q,0) < 0\)

  • • \(\kappa (q,z) = O(z)\) as \(\lvert z\rvert \to \infty \)

  • • \(F(z) = \begin {cases} \kappa _+(q,z)/\kappa _+'(q,z), & \Im z \ge 0, \\ \kappa _-'(q,-z)/\kappa _-(q,-z), & \Im z \le 0 \end {cases}\)

  • • \(F\) is entire and non-zero

  • • \(\log F(z) = \log \kappa _+(q,z) - \log \kappa _+'(q,z)\) for \(\Im z \ge 0\)

  • • \(\log F(z) = o(z)\) as \(\lvert z\rvert \to \infty \)

  • • Liouville’s theorem: \(F(z) = c\)

(-tikz- diagram)

Prior art: killed Lévy process, III

  • • When \(q = 0\), may have \(\liminf _{\lvert z\rvert \to \infty , z\in \RR }\lvert \kappa (z)\rvert = 0\)

  • • Then \(\log \kappa (z)\) is unbounded and Liouville argument fails

2 Why is it important?

The inverse problem

  • • Let \(H^\pm \) be a pair of subordinators with CEs \(\kappa _\pm \)

  • • When is there a Lévy process \(X\) with CE \(\psi \) such that \(-\psi (z) = \kappa _+(z)\kappa _-(-z)\)?

  • • When such \(X\) exists, we call \(H^\pm \) friends

Philanthropy

A subordinator \(H\) is called a philanthropist if its Lévy measure admits a decreasing density.

  • Theorem 3 (Vigon, 2002). Any two philanthropists can be friends.

Construction of Lévy processes

If we have philanthropists with CEs \(\kappa _\pm \), then

\[ \psi (z) = -\kappa _+(z)\kappa _-(-z) \]

is the CE of a Lévy process.

  • Example 4 (Kuznetsov and Pardo, 2013). Let \(\beta _\pm \ge 0, \gamma _\pm \in (0,1)\). Then

    \[ \kappa _\pm (z) = \protect \frac {\Gamma (\beta _\pm +\gamma _\pm -\mathsf {i}z)}{\Gamma (\beta _\pm -\mathsf {i}z)} \]

    gives rise to a hypergeometric Lévy process.

Fluctuations of constructed Lévy processes

  • • First passage times: \(\tau _Z(x) = \inf \{ t\ge 0: Z_t > x \}\), where \(Z \in \{X, H^+\}\)

  • • First passage distributions: \(\PP (X_{\tau _X(x)} \in \cdot ) = \PP (H^+_{\tau _{H^+}(x)} \in \cdot )\)

  • • If \(X\) is constructed via friendship, does \(H^+\) have CE \(\kappa _+\)?

3 Proving uniqueness

Tempered distributions

  • • Rapidly decaying functions: \(\mathcal {S} = \{ \phi \colon \RR \to \CC : \forall \alpha ,\beta \in \NN \cup \{0\} \ \lim _{\lvert x\rvert \to \infty } \lvert x^\alpha \phi ^{(\beta )}(x) \rvert < \infty \}\)

  • • Tempered distributions: \(\mathcal {S}' = \{ \text {continuous linear functionals } h \colon \mathcal {S}\to \CC \}\)

  • • Notation: \(\langle h,\phi \rangle = h(\phi ) = \langle h(x), \phi (x) \rangle \)

Examples

  • • Radon measures with slow growth near \(\pm \infty \): \(\langle \mu ,\phi \rangle = \int \phi (x) \, \mu (\dd x)\)

  • • \(\langle \delta , \phi \rangle = \phi (0)\)

  • • \(\langle D\delta , \phi \rangle = -\phi '(0)\)

Tempered distributions: operations

  • • Reflection: \(\langle \hat {h},\phi \rangle = \langle h(x), \phi (-x)\rangle \)

  • • Differentiation: \(\langle Dh, \phi \rangle = -\langle h, \phi '\rangle \)

  • • Fourier transform:

    • – \(\mathcal {F}\phi (z) = \int e^{\iu x z} \phi (x) \, \dd x\); then \(\mathcal {F} \colon \mathcal {S} \to \mathcal {S}\)

    • – \(\langle \mathcal {F}h, \phi \rangle = \langle h, \mathcal {F}\phi \rangle \)

Some useful representations

Lévy–Khintchine formula

For \(H^+\) we have

\[ \kappa _+(z) = -q_+ + \mathsf {i}d_+ z + \int _{(0,\infty )} (e^{\mathsf {i}x z} - 1) \, \mu _+(\mathrm {d}x). \]

Let

\[ \langle G_+, \phi \rangle = -q_+\langle \delta , \phi \rangle - d_+\langle D\delta , \phi \rangle + \int _{(0,\infty )} \protect \bigl ( \phi (x)-\phi (0)\protect \bigr ) \mu _+(\mathrm {d}x). \]

Then \(G_+ \in \mathcal {S}'\) and \(\mathcal {F} G_+(z) = \kappa _+(z)\).

Potentials

The potential of \(H^+\): \(U_+(\dd x) = \int _0^\infty \PP (H^+\in \dd x) \, \dd t\).

Then \(U_+\in \mathcal {S}'\) and \(\mathcal {F}U_+ = 1/\kappa _+\) (away from zero)

Sketch of the proof

  • • Focus on non-lattice case (only zero of \(\psi \) is \(0\))

  • • \(\CC \setminus \{0\} \ni z \mapsto F(z) = \begin {cases} \kappa _+(z)/ \kappa _+'(z), & \Im z \ge 0, \\ \kappa _-'(-z)/ \kappa _-(-z), & \Im z \le 0 \end {cases}\)
    (to show: \(F\) is constant)

  • • \(W_+ = G_+*U_+'\) and \(W_- = \widehat {G_-'*U_-}\)

  • • \(\mathcal {F}W_+(z) = F(z) = \mathcal {F}W_-(z)\) for \(z\in \RR \) ‘away from zero’

(-tikz- diagram)

Sketch of the proof, II

  • • \(\mathcal {F}(W_+-W_-)\) has support \(\{0\}\) and hence \(\mathcal {F}(W_+-W_-) = \sum _{n=0}^N a_n D^n\delta \)

  • • \((W_+-W_-)(x) = \sum _{n=0}^N \frac {a_n}{2\pi } (-\iu x)^n\), and in fact \(a_1=\dotsb = a_N = 0\). (The probability happens here!)

  • • So \((W_+-W_-)(x) = \frac {a_0}{2\pi }\)

(-tikz- diagram)

Sketch of the proof, III

(-tikz- diagram)

  • • So \(W_+ - \frac {a_0}{2\pi }\Ind _{\RR _+} = W_- + \frac {a_0}{2\pi }\Ind _{\RR _-}\)

  • • LHS has support \([0,\infty )\) and RHS has support \((-\infty ,0]\), so both have support \(\{0\}\)

  • • Repeat idea from before and more trickery yields \(W_+ = \frac {a_0}{2\pi } \Ind _{\RR _+} + b_0\delta \)

  • • \(F(z) = \mathcal {F}W_+(z) = -\frac {a_0}{2\pi \iu z} + b_0\) for \(z\in \RR \) ‘away from zero’

  • • Asymptotics of \(F\) near zero and comparison with \(W_-\) (again) yield \(F(z) = b_0\): we are done.

The probability happens where?

  • • Let \(h_+ = (\bar {\mu }_+ + d_+\delta + q_+\Ind _{\RR _+})*U_+'\), a measure with support \([0,\infty )\)

    where \(\bar {\mu }_+(x) = \mu _+(x,\infty )\)

  • • \(- Dh_+ = W_+\)

  • • The renewal theorem implies \(\int _{[0,\infty )} (1\wedge x^{-(2+\epsilon )}) h_+(\dd x) < \infty \)

  • • So \((W_+-W_-)(x) = \sum _{n=0}^N \frac {a_n}{2\pi } (-\iu x)^n = \frac {a_0}{2\pi }\)

Swept under the carpet

  • • Convolvability of distributions is tricky

  • • When \(X\) is lattice valued, \(\psi \) has zeroes on \(\eta \ZZ \) for some \(\eta >0\): support arguments are trickier

4 The outlook

Return to the probabilistic approach

  • • We saw that \(X_{\mathrm {e}_q} = \bar {X}_{\mathrm {e}_q} + (X-\bar {X})_{\mathrm {e}_q}\) is a unique decomposition

  • • But \(\not \exists \lim _{q\to 0} X_{\mathrm {e}_q} \in \RR \)

  • • \(q^{-1} \PP (X_{\mathrm {e}_q} \in A) \to U(A) \coloneqq \int _0^\infty \PP (X_t\in A)\,\dd t = U_+*\widehat {U_-}(A)\),

    where \(U_+\) and \(\widehat {U_-}\) are potentials of \(H^+\) and \(-H^-\) (but if \(X\) is recurrent, \(U\) is very bad!)

  • • In Fourier space, this decomposition reads:

    \[ - \protect \frac {1}{\psi (z)} = - \protect \frac {1}{\kappa _+(z)}\times -\protect \frac {1}{\kappa _-(-z)} \]

    and we prove that it is unique (among potentials of subordinators)

Factorising the potential

  • • Consider \(U = U_+*\widehat {U_-}\)

  • • For infinitely divisible finite measures the Lévy–Khintchine formula is invaluable

  • • \(U\), \(U_+\) and \(\widehat {U_-}\) are all ‘infinitely divisible infinite measures’

  • • Is there a representation of these, other than \(\mathcal {F}U_+ = 1/\kappa _+\) etc.?

Markov additive processes

  • • A Markov additive process (MAP) with finite phase space is a collection of Lévy processes with Markovian regime-switching.

  • • There are ‘matrix exponents’ with a WHF of the form

    \[ -\symbfup {\Psi }(z) = \symbfup {\Delta }_{\symbfup {\pi }}^{-1} \symbfup {\kappa }_-(-z)^\top \symbfup {\Delta }_{\symbfup {\pi }} \symbfup {\kappa }_+(z) \]

    where \(\mathbfup {\pi }\) is the stationary distribution of the phase, and there is a notion of friendship (DTW 2023+).

  • • Uniqueness holds when the MAP is killed and under certain absolute continuity conditions (DTW 2023+).

  • • Does it hold in general?

Further reading

  • [1]  L. Döring, M. Savov, L. Trottner and A. R. Watson
    The uniqueness of the Wiener–Hopf factorisation of Lévy processes and random walks
    arXiv:2312.13106 [math.PR]

Thank you!