Lévy processes under level-dependent Poissonian switching

This paper derives identities for the exit problems and resolvents of a hybrid process that switches between two Lévy processes based on a level-dependent Poissonian barrier, expressing these results through generalized scale functions and applying them to calculate ruin probabilities in risk processes with delayed dividend payments.

The Gist

Imagine you are managing a very complex, unpredictable river. This river represents the financial health (or "surplus") of an insurance company. Sometimes the water flows smoothly, sometimes it crashes down like a waterfall (claims), and sometimes it rises gently (premiums). In math, we call this a Lévy process.

Now, imagine you want to manage this river by building a dam or a diversion channel. But here's the twist: you can't just build the dam the exact second the water hits a certain height. In the real world, there's always a delay. You need to check the water level, call a meeting, and then decide to open the gate.

This paper introduces a new mathematical model for exactly that kind of "delayed decision-making."

Here is the breakdown of the paper's ideas using simple analogies:

1. The Two Modes of the River

Usually, a river flows one way. But in this paper, the river has two different personalities:

• Mode A (The Calm Flow): When the water level is low (below a certain barrier, let's call it "Level B"), the river flows according to its natural, wild rules (Process X).
• Mode B (The Diverted Flow): When the water level is high (above Level B), the river wants to change its behavior. Maybe it starts paying out dividends (like water flowing out to shareholders), which slows the river down.

2. The "Poisson" Check-In (The Delay)

In older models, the moment the water crossed Level B, the river would instantly switch to Mode B. But that's unrealistic. In reality, you only check the water level at specific times.

The authors introduce a "Poisson arrival" concept. Imagine a security guard who checks the water level at random intervals (like a clock that rings unpredictably).

• If the guard checks the level and sees it is above Level B, the river immediately switches to the "Diverted Flow" (Mode B).
• If the guard checks and it's below Level B, the river stays in "Calm Flow" (Mode A).
• If the river crosses Level B between guard checks, nothing happens until the next check.

This creates a hybrid system: The river flows naturally, but its rules change only when a random "check-in" happens while it's in a specific zone.

3. The Mathematical "Map" (Scale Functions)

Mathematicians love to predict the future of these rivers. They want to know:

• The Exit Problem: What is the chance the river will flood (go above a high limit) or dry up (go below zero/ruin) before the next check-in?
• The Potential Measure: How much time does the river spend in a specific area before it dries up?

To answer these, the authors invented a new set of "Maps" (called generalized scale functions).

• Think of the old maps as simple road maps for a straight highway.
• The new maps are like GPS navigation for a car that changes its engine type depending on which neighborhood it's in and when the traffic light turns green. These new maps allow the authors to calculate the odds of the river flooding or drying up, even with all the delays and random switches.

4. The Real-World Application: Insurance Ruin

Why does this matter? The paper applies this to Insurance Risk.

Imagine an insurance company that promises to pay dividends to shareholders when they are profitable (above Level B).

• The Problem: In the real world, you don't pay dividends the second you make a profit. There is a delay. You have to wait for the board meeting (the Poisson check-in).
• The Risk: If the company has a sudden crash in profits, but the board hasn't met yet to stop the dividend payments, the company might run out of money (Ruin) faster than expected.

The authors use their new "GPS maps" to calculate the exact probability of ruin for this delayed system. They show that accounting for these delays changes the math significantly compared to models that assume instant reactions.

Summary

• The Old Way: A river changes its flow instantly the moment it crosses a line.
• The New Way: The river changes its flow only when a random "inspector" checks it and sees it's crossed the line.
• The Result: The authors created new mathematical tools to predict exactly how likely this river is to flood or dry up, which helps insurance companies manage their money more safely in a world where decisions take time.

In short, this paper is about mathematically modeling the "lag" in decision-making for financial systems, ensuring we don't get caught off guard by the time it takes to switch strategies.

Technical Summary

Here is a detailed technical summary of the paper "Lévy processes under level-dependent Poissonian switching" by Beelders, Ramsden, and Papaioannou.

1. Problem Statement

The paper addresses the fluctuation theory of a specific class of stochastic processes that generalize the refracted Lévy process.

• Context: Standard refracted Lévy processes switch dynamics immediately when the process crosses a barrier $b$ continuously. Generalized refracted processes allow the dynamics to switch between two different Lévy processes ($X$ and $Y$) based on whether the process is above or below $b$.
• The Gap: Existing literature often assumes continuous switching or bounded variation. The authors consider a scenario where the switch between two Spectrally Negative Lévy Processes (SNLPs), $X$ and $Y$, does not occur at the exact moment the barrier $b$ is crossed. Instead, the switch is triggered only when the process is on the "correct" side of the barrier (above or below) and coincides with an arrival epoch of an independent Poisson process $N$ with rate $\lambda$.
• Objective: To derive explicit identities for the two-sided exit problems, one-sided exit problems, and potential measures (both killed and non-killed) for this process $U$. Additionally, the paper aims to prove the existence of a strong solution to the governing Stochastic Differential Equation (SDE) and apply these results to an insurance risk model with delayed dividend payments.

2. Methodology

The authors employ a combination of pathwise construction, fluctuation theory of Lévy processes, and the theory of scale functions.

• Pathwise Construction (Existence Proof):

  • The process $U$ is defined by the hybrid SDE:
    $$U_t = U_0 + \int_0^t \mathbb{1}_{\{U_{N(s)} \le b\}} dX_s + \int_0^t \mathbb{1}_{\{U_{N(s)} > b\}} dY_s$$
  • Unlike classical refracted processes where switching can happen infinitely often in finite time (if the process oscillates around $b$), the Poissonian mechanism ensures that switches occur only at discrete random times $T_i$.
  • The authors construct the solution pathwise by defining recursive stopping times $K^+_{b,n}$ and $K^-_{b,n}$ based on the Poisson arrival times. This construction proves the existence of a strong solution and the strong Markov property for the augmented process $(U_t, Q_t)$, where $Q_t$ indicates the current regime.

• Generalized Scale Functions:

  • The core analytical tool is the introduction of new generalizations of scale functions.
  • Standard scale functions ($W^{(q)}$ and $Z^{(q)}$) are used for single SNLPs.
  • The authors define "second-generation" scale functions (e.g., $W^{(p,q)}_u$) and further extend them to handle the Poissonian switching. These new functions (denoted with superscripts like $(q, \lambda)$) incorporate the Laplace transform of the Poisson arrival times and the interaction between the two regimes $X$ and $Y$.
  • Key auxiliary functions ($\gamma, \alpha, G, A, U, V$) are defined via integrals involving these generalized scale functions to formulate the final exit identities concisely.

• Fluctuation Theory:

  • The derivation relies heavily on conditioning arguments, the Strong Markov Property, and known identities for SNLPs observed at Poisson times (referencing works by Albrecher, Ivanovs, Zhou, and Loeffen).
  • The authors derive identities for the Laplace transforms of exit times by conditioning on the first Poisson arrival that triggers a regime switch or an exit.

3. Key Contributions and Results

A. Existence and Strong Markov Property

The paper establishes that a strong solution exists for the hybrid SDE (1) even in the unbounded variation case. This is a significant theoretical advancement over previous generalized refracted models where existence was often an open question for unbounded variation. The Poissonian switching mechanism simplifies the problem by ensuring a finite number of switches in any compact time interval.

B. Two-Sided Exit Identities

The authors derive explicit formulas for the Laplace transforms of the first passage times to an upper level $a$ and a lower level $0$:

• Upward Exit: $E_x[e^{-q\tau^+_{a,U}} \mathbb{1}_{\{\tau^+_{a,U} < \tau^-_{0,U}\}}]$
• Downward Exit: $E_x[e^{-q\tau^-_{0,U}} \mathbb{1}_{\{\tau^-_{0,U} < \tau^+_{a,U}\}}]$
  These are expressed in terms of the new generalized functions $U^{(q,\lambda)}_{b,a}(x)$ and $V^{(q,\lambda)}_{b,a}(x)$, which are constructed from the underlying scale functions of $X$ and $Y$.

C. One-Sided Exit and Potential Measures

• One-Sided Limits: By taking limits as $a \to \infty$, the authors derive identities for one-sided exit probabilities (ruin or reaching infinity).
• Potential Measures: The paper provides explicit expressions for the resolvent (potential measure) of the process $U$, both for the process killed upon exiting $[0, a]$ and for the process killed only upon hitting $0$or$\infty$. These are crucial for calculating occupation times and expected discounted dividends.

D. Application to Risk Theory (Ruin Probability)

The paper applies the theoretical results to a specific insurance risk model:

• Model: An insurer's surplus follows a Lévy process $X$. Dividends are paid at a constant rate $\delta$ when the surplus is above $b$. However, due to administrative delays, the switch to the "dividend-paying" regime (process $Y_t = X_t - \delta t$) and the switch back to the "no-dividend" regime only occur at Poisson arrival times.
• Result: An explicit expression for the probability of ruin is derived (Corollary 15). The formula accounts for the delay parameter $\lambda$ and the dividend rate $\delta$, showing how the delay in switching regimes affects the insurer's solvency.

4. Significance

• Theoretical Rigor: The paper resolves the existence issue for a class of hybrid SDEs with discontinuous coefficients in the unbounded variation setting, a problem previously noted as open in related literature.
• Generalization: It unifies and extends existing results on refracted Lévy processes and Lévy processes observed at Poisson times. The derived identities reduce to classical results when the switching rate $\lambda \to \infty$ (continuous switching) or when $X=Y$ (no switching).
• Practical Application: The model offers a more realistic representation of insurance risk management. In reality, dividend payments are not triggered instantaneously by a surplus crossing a threshold; there are processing delays. The Poissonian switching mechanism mathematically captures this "delay," providing actuaries with a tool to quantify the increased risk of ruin caused by such administrative lags.
• New Mathematical Tools: The introduction of the new generalized scale functions ($U^{(q,\lambda)}$, $V^{(q,\lambda)}$, etc.) provides a framework for solving future problems involving regime-switching Lévy processes with observation delays.

In summary, this paper provides a comprehensive fluctuation theory for Lévy processes with level-dependent Poissonian switching, offering rigorous existence proofs, explicit analytical formulas for exit problems, and a direct application to calculating ruin probabilities in delayed dividend models.