Markov additive friendships

Leif Döring

Lukas Trottner

Alex Watson

18 July 2022

1 Lévy processes and the theory of friends

Wiener-Hopf factorisation (path picture)

(-tikz- Path of a Lévy process with the following notation labelled. xi, a Lévy process. xi-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).

Wiener-Hopf factorisation (analytic picture)

  • • Let \(\Psi \) be the characteristic exponent of \(\xi \) (i.e., \(\EE e^{\iu \theta \xi _t} = e^{t\Psi (\theta )}\))

  • • Let \(\Psi ^\pm \) be characteristic exponents of \(H^\pm \)

  • • Then

    \[ \Psi (\theta ) = - \Psi ^-(-\theta ) \Psi ^+(\theta ). \]

The inverse problem

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

  • • When is there a Lévy process \(\xi \) with CE \(\Psi \) such that \(\Psi (\theta ) = -\Psi ^-(-\theta )\Psi ^+(\theta )\)?

  • • When such \(\xi \) exists, we call \(H^\pm \) friends and \(\xi \) the bonding process

The theorem of friends

  • • \(d^\pm \) = drift of \(H^\pm \)

  • • \(\Pi ^\pm \) = Lévy measure of \(H^\pm \)

  • • \(H^\pm \) are compatible if \(d^\mp > 0\) implies \(\Pi ^\pm \) is absolutely continuous and its density \(\partial \Pi ^\pm \) is the tail of a signed measure

  • Theorem 1 (Vigon). Define

    \[ \Upsilon (x) = \protect \begin {cases} \int _{(0,\infty )} \protect \bigl (\Pi ^-(y,\infty ) - \Psi ^-(0)\protect \bigr )\, \Pi ^+(x+ \mathsf {d}y) + d^- \partial \Pi ^+(x), & x > 0,\\ \int _{(0,\infty )} \protect \bigl (\Pi ^+(y,\infty ) - \Psi ^+(0)\protect \bigr )\, \Pi ^-(-x + \mathsf {d}y) + d^+ \partial \Pi ^-(-x), & x < 0. \protect \end {cases} \]

    \(H^\pm \) are friends if and only if they are compatible and \(\Upsilon \) is decreasing on \((0,\infty )\) and increasing on \((-\infty ,0)\).

    Then, \(\Upsilon \) is a.e. the right/left tail of the Lévy measure of the bonding process.

Philanthropy

  • • Let \(H^+_t = d^+t\). A subordinator \(H^-\) is called a philanthropist if it is a friend of \(H^+\).

  • • Equivalently, a subordinator is called a philanthropist if its Lévy measure admits a decreasing density.

  • Theorem 2 (Vigon). Any two philanthropists can be friends.

Example (spectrally negative processes)

  • • Let \(H^+\) be a (killed) pure drift

  • • Let \(H^-\) be a philanthropist

  • • Then \(H^\pm \) are friends and the bonding process is a spectrally negative Lévy process

  • • All spectrally negative Lévy processes are of this form

2 Markov additive processes

Markov additive processes

  • • A process \((\xi ,J)\) with state space \((\RR \cup \{\partial \}) \times \{1,\dotsc , N\}\) is a Markov additive process (MAP) if

    \begin{multline*} \text {given } \{ J_t = i \}, \ (\xi _{t+s}-\xi _t, J_{t+s}) \text { is independent of the past up to }t, \\ \text {and has the same distribution as } (\xi _s,J_s) \text { under } \PP ^{0,i} \end{multline*}

  • • Equivalently, a Markov additive process is a regime-switching Lévy process:

    • – \(\xi ^{(i)}\) is a Lévy process for each \(i\)

    • – \(J\) is a Markov chain with transition matrix \(Q\)

    • – when \(J\) is in state \(i\), run \(\xi ^{(i)}\)

    • – when \(J\) moves to \(j\), make a jump from distribution \(F_{ij}\) and run \(\xi ^{(j)}\)

Markov additive processes (notation)

  • • Let \(\Pi _{ii}\) be the Lévy measure of \(\xi ^{(i)}\) and \(\Pi _{ij} = q_{ij} F_{ij}\) when \(i\ne j\)

  • • Call \(\mathbfup {\Pi } = (\Pi _{ij})_{i,j=1,\dotsc ,N}\) the matrix Lévy measure of \(\xi \)

  • • Matrix characteristic exponent \(\mathbfup {\Psi }\): \(\EE ^{0,i}[ e^{\iu \theta \xi _t}; J_t = j] = (e^{t\mathbfup {\Psi }(\theta )})_{ij}\)

  • • Structure:

    \[ \symbfup {\Psi }(\theta ) = \protect \begin {pmatrix} \Psi _1(\theta ) + q_{11} & \widehat {\Pi }_{12}(\theta ) & \protect \cdots \\ \widehat {\Pi }_{21}(\theta ) & \Psi _2(\theta ) + q_{22} & \protect \cdots \\ \vdots & \vdots & \ddots \protect \end {pmatrix} \]

    where \(\Psi _i\) is CE of \(\xi ^{(i)}\) and \(\widehat {\Pi }_{ij}\) is the characteristic function of \(\Pi _{ij}\)

  • • Let \(\pi \) be the invariant measure of \(J\)

Wiener-Hopf factorisation

  • • \((H^+,J^+)\) and \((H^-,J^-)\): ladder height processes of \((\xi ,J)\)

  • • They are MAP subordinators (increasing MAPs) with matrix exponents \(\mathbfup {\Psi }^\pm \)

  • • Path picture is the same

  • Theorem 3 (Dereich, Döring and Kyprianou).

    \[ \symbfup {\Psi }(\theta ) = -\Delta _\pi ^{-1} \symbfup {\Psi }^-(-\theta )^T \Delta _\pi \symbfup {\Psi }^+(\theta ), \]

    where \(\Delta _\pi \) is the diagonal matrix containing \(\pi \).

3 Markov additive friendship

The inverse problem

  • • Two MAP subordinators \((H^\pm ,J^\pm )\) are \(\pi \)-friends if there is a MAP for which they satisfy the above matrix equation

  • • Are there necessary and sufficient conditions for friendship?

  • • Is there a theory of philanthropy?

Compatibility

\((H^\pm ,J^\pm )\) are \(\pi \)-compatible if

  • • \(d^\mp _i > 0\) implies \(\Pi ^\pm _i\) is absolutely continuous and its density is the tail of a signed measure

  • • Elements of the vector \(-\Delta _\pi ^{-1}\mathbfup {\Psi }^-(0)^T \Delta _\pi \mathbfup {\Psi }^+(0)\mathbfup {1}\) are non-positive (row sums of \(Q\)-matrix of the putative bonding MAP)

  • • ...and an additional positivity condition involving intensity of zero-jumps

Compatibility is a necessary condition for \(\pi \)-friendship!

The theorem of friends

  • Theorem 4. Define the matrix-valued function

    \[ \symbfup {\Upsilon }(x) = \protect \begin {cases} \int _{0+}^\infty \Delta _{\pi }^{-1}\protect \Big (\symbfup {\Pi }^-(y,\infty ) - \symbfup {\Psi }^-(0)\protect \Big )^\top \Delta _{\pi } \, \symbfup {\Pi }^+(x+\mathsf {d}y) + \Delta ^-_{\symbfup {d}} \partial \symbfup {\Pi }^+(x), & x > 0,\\ \int _{0+}^\infty \Delta _{\pi }^{-1} \protect \big (\symbfup {\Pi }^-(-x+\mathsf {d}y) \protect \big )^\top \Delta _{\pi } \, \protect \big (\symbfup {\Pi }^+(y,\infty ) - \symbfup {\Psi }^+(0)\protect \big ) + \Delta _{\pi }^{-1} \protect \big (\Delta _{\symbfup {d}}^+\partial \symbfup {\Pi }^-(-x)\protect \big )^\top \Delta _{\pi }, & x < 0, \protect \end {cases} \]

    Two MAP subordinators \((H^\pm ,J^\pm )\) are \(\pi \)-friends if and only if they are \(\pi \)-compatible and \(\Upsilon _{ij}\) is decreasing on \((0,\infty )\) and increasing on \((-\infty ,0)\).

    Then, \(\mathbfup {\Upsilon }\) is a.e. the right/left tail of the matrix Lévy measure of the bonding process.

4 Examples

Examples are hard to come by

  • • Only known MAP factorisation is from the deep factorisation of the stable process

  • • The MAP in question is the Lamperti-Kiu transform of the stable process

  • • The factorisation is obtained using detailed knowledge of the stable process

  • • No simpler proof is known; verifying the conditions of friendship appears difficult

Spectrally negative MAPs

  • • Let \((H^+,J^+)\) be a pure drift, i.e., \(H^+_t = \int _0^t d^+_{J^+_s}\, \dd s\)

  • Theorem 5. A MAP subordinator \((H^-,J^-)\) is \(\pi \)-friends with a pure drift if and only if they are \(\pi \)-compatible and

    \[ -\Delta _\pi ^{-1} \symbfup {\Psi }^+(0)^T \Delta _\pi \symbfup {\Pi }^-(x,\infty ) + \Delta _{\symbfup {d}^+}\partial \symbfup {\Pi }^-(x), \quad x > 0, \]

    is decreasing.

  • • Allows us to construct spectrally negative MAPs

  • • When the \(\Pi _i^-\) have completely monotone density, can make conditions more explicit

  • • Being friends with a drift does not make you friends with anything else: ‘philanthropy’, if it exists, is more complicated

Double exponential MAPs

  • • Let \((H^\pm ,J^\pm )\) be MAP subordinators with exponential jumps in every state

  • • Given some balances between coefficients, such processes can be friends

  • • The bonding MAP has double exponential jumps within and between every state

  • • May be second example of two-sided MAP with known ladder processes?

5 Work in progress | open problems

Uniqueness of Wiener-Hopf factorisation

  • • We have studied the matrix equation \(\mathbfup {\Psi }(\theta ) = -\Delta _\pi ^{-1} \mathbfup {\Psi }^-(-\theta )^T \Delta _\pi \mathbfup {\Psi }^+(\theta )\)

  • • To be sure that \((H^\pm ,J^\pm )\) really are the ladder processes, we need uniqueness

  • • We have partial results, for instance under absolute continuity conditions

  • • Surprisingly, this does not seem to be known in full generality even for Lévy processes

Philanthropy

  • • Is there a notion of ‘philanthropist’ that implies friendship with other philanthropists?

  • • Are there conditions that do not depend on \(\pi \)?

Complex analytic structure

  • • In the ‘meromorphic’ Lévy processes (Lamperti-stable, simple/double hypergeometric), the factorisation can be deduced from the poles and zeroes of the CE

  • • Is there such an approach for MAPs?

  • • This would give an alternative avenue of attack for ‘deep factorization’ type processes

Further reading

  • [1]  L. Döring, L. Trottner and A. R. Watson
    Markov additive friendships
    In preparation (working title)

Thank you!