CS11: Spectrally negative Lévy models with level-dependent features

Organizer: Jean-François Renaud (UQAM)

Optimality of a barrier strategy in a spectrally negative Lévy model with a level-dependent intensity of bankruptcy

Jean-Francois Renaud

The stochastic control problem concerned with the maximization of dividend payments in a model based on a spectrally negative Lévy process (SNLP) has attracted a lot of research interest since the papers of Avram, Palmowski & Pistorius (2007) and Loeffen (2008). In that problem, a dividend strategy is said to be optimal if it maximises the expected present value of dividend payments made up to the time of ruin, which is a standard first-passage time below zero. In this talk, I will consider a version of this stochastic control problem in which the (controlled) process is allowed to spend time under zero, but is then subject to a level-dependent intensity of bankruptcy. In a joint paper with Dante Mata (UQAM & CRM), we were able to prove that there exists a barrier strategy that is optimal for this control problem, under a mild assumption on the Lévy measure.

Lévy processes under level-dependent Poissonian switching

Noah Daniel Beelders

The applicability of spectrally negative Lévy processes (i.e. Lévy processes with only downward jumps) in ruin theory dates back to the classical Cramér-Lundberg model which utilised a compound Poisson process with positive drift as a model of the underlying surplus process. Since then, research in this field has grown rapidly and the literature on such models has continually expanded (see \([1]\) for an extensive monograph on ruin theory and related concepts).

\(\;\;\) Further extensions of these models have received a lot of attention in recent years. For instance, the refracted Lévy process, first introduced in \([5]\), extends the classical Lévy risk model by allowing for a reduction in its drift immediately once it moves above some threshold \(b\). This may hence be used to model the dividend payments of an insurer once their surplus is greater than some threshold \(b\) (see also \([4, 7, 9]\) for further contributions relating to refracted Lévy processes). The work of \([8]\) further generalised the refracted Lévy process by allowing it to immediately change dynamics between two independent spectrally negative Lévy processes (SNLPs) \(X := \{X_t\}_{t \geq 0}\) and \(Y:= \{Y_t\}_{t \geq 0}\) when it moves below or above a threshold \(b\), respectively, provided that there is no Gaussian component in the unbounded variation case.

\(\;\;\) For this presentation (based on \([2]\)), a further extension of the aforementioned models is considered in which the switch between \(X\) and \(Y\) does not occur immediately when \(b\) is crossed, but instead when it is above (below) \(b\) and coincides with an arrival epoch of an independent Poisson process. Under this extension, the corresponding process \(U := \{U_t\}_{t \geq 0}\) is a solution to the SDE with level-dependent switching \[\tag{1} U_t = U_0 + \int^t_0 \mathbf{1}_{\{U_{T_{N(s)}} \leq b\}} \mathrm{d} X_s +\int^t_0 \mathbf{1}_{\{U_{T_{N(s)}} > b\}} \mathrm{d} Y_s ,\] where \(X\) and \(Y\) are defined as above, and are independent of the Poisson process \(N\) with arrival times \(T_0 = 0\) and \(\{T_i\}_{i\geq 1}\).

\(\;\;\) Utilising the Poisson arrival epochs, we show that a pathwise solution exists to Eq. \((1)\) even in the unbounded variation case with a Gaussian component. It will then be shown that a set of identities for the two sided exit problems and the potential measures (killed and non-killed) of \(U\) can be established in terms of generalisations of the classical scale functions from Chapter 8 of \([6]\) (see also Chapter VII.2 in \([3]\)). Then, to illustrate the applicability of these results, the probability of ruin is derived for a risk process with delays in the dividend payments.

Bibliography

\([1]\) S. Asmussen and H. Albrecher. , volume 14 of Advanced Series on Statistical Science & Applied Probability. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, second edition, 2010.

\([2]\) N. Beelders, L. Ramsden, and A. D. Papaioannou. Lévy processes under level-dependent poissonian switching. Avaliable at arXiv:2505.00453, 2025.

\([3]\) J. Bertoin. , volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.

\([4]\) I. Czarna, J.-L. Pérez, T. Rolski, and K. Yamazaki. Fluctuation theory for level-dependent Lévy risk processes. , 129(12):5406–5449, 2019.

\([5]\) A. E. Kyprianou and R. L. Loeffen. Refracted Lévy processes. , 46(1):24–44, 2010.

\([6]\) A. E. Kyprianou. . Universitext. Springer, Heidelberg, second edition, 2014.

\([7]\) A. E. Kyprianou, J. C. Pardo, and J. L. Pérez. Occupation times of refracted Lévy processes. , 27(4):1292–1315, 2014.

\([8]\) K. Noba and K. Yano. Generalized refracted Lévy process and its application to exit problem. , 129(5):1697–1725, 2019.

\([9]\) J.-F. c. Renaud. On the time spent in the red by a refracted Lévy risk process. , 51(4):1171–1188, 2014.

Fluctuations of Omega-Killed Level-Dependent Spectrally Negative Levy Processes

Meral Şimşek

A reflected level-dependent Lévy process solves the stochastic differential equation (SDE) \[\tag{1} \mathrm{d} V(t) = \mathrm{d} X(t)+ \mathrm{d} R_t - \phi(V(t))\mathrm{d}t, \quad t \geq 0,\] where \(X(t)\) is a spectrally negative Lévy process and \(R(t)\) is a non-decreasing and right continuous process that pushes upward the process given in \((1)\), when it attempts to go below \(0\). In this study, we solve exit problems and potential measure for a omega-killed reflected level-dependent Lévy process which is exponentially killed with a killing intensity \(\omega(\cdot)\), that depends on the present state of the process. All identities are given in terms of new generalizations of scale functions, which are solution to Volterra equations. We also discuss the existence of solution of the SDE given in \((1)\).