Parisian Refracted Scale Function

• The paper demonstrates that the Parisian refracted scale function facilitates explicit formulas for exit identities and optimal impulse control strategies.
• Parisian refracted scale function is defined as an extension of classical scale functions, capturing both refraction effects and Parisian ruin with grace periods.
• The framework uses analytical techniques, explicit examples, and sensitivity analysis to optimize dividend policies in risk theory models.

The Parisian refracted scale function is a technical extension of the classical scale function for spectrally negative Lévy processes. It is specifically designed to encode the joint impact of two advanced structural features in risk theory models: (i) refraction, that is, a change in process dynamics (such as a lowered premium rate) when the surplus exceeds a certain threshold; and (ii) Parisian ruin, wherein ruin is declared only if the surplus process remains below a critical level for an exponential “grace period,” with ultimate bankruptcy triggered if the process crosses a lower barrier. The Parisian refracted scale function enables explicit formulas for exit identities, optimal value functions under impulse dividend strategies with transaction costs, and sensitivity analysis for control parameters in such models (Gao et al., 3 Sep 2025).

1. Analytical Formulation and Definition

Let $X = (X_t)_{t \geq 0}$ be a spectrally negative Lévy process, and consider the refracted process $U$ satisfying the SDE

$$dU_t = dX_t - \delta \mathbf{1}_{\{U_t > b\}} \, dt,$$

with threshold $b$ and refraction rate $\delta > 0$. For a Parisian ruin model with exponentially distributed implementation delay of parameter $m$, and an ultimate bankruptcy barrier at $-l$, the Parisian refracted $(q, m)$-scale function is denoted by $\theta^{(q+m, q)}(x, -l)$. It is defined by: $$\theta^{(q+m, q)}(x, -l) := w^{(q)}(x; -l) + m \int_0^{-l} w^{(q)}(x; -l + y) W^{(q+m)}(y) \, dy,$$ where:

• $w^{(q)}(x; -l)$ is a refracted $q$-scale function:

$$w^{(q)}(x; -l) = W^{(q)}(x + l) + \delta \int_0^x \mathcal{W}^{(q)}(x - y) W^{(q)\prime}(y + l) \, dy$$

for $x \geq 0$, and $W^{(q)}$ is the classical $q$-scale function,

• $W^{(q+m)}(\cdot)$ corresponds to the scale function with an increased discount rate,
• $\mathcal{W}^{(q)}$ is the scale function associated with $Y_t = X_t - \delta t$.

The function $\theta^{(q+m, q)}$ represents the fundamental building block in all derived fluctuation identities for the optimal impulse control problem. For instance, for first passage times,

$$\mathbb{E}_x [e^{-q \kappa_c^+} \mathbf{1}_{\{\kappa_c^+ < T\}}] = \frac{\theta^{(q+m, q)}(x, -l)}{\theta^{(q+m, q)}(c, -l)}$$

where $\kappa_c^+$ is the first upcrossing above $c$, and $T$ is the Parisian ruin time or the lower barrier crossing.

2. Optimal Impulse Control and Value Function Expression

In the optimal dividend problem with transaction costs, dividend decisions are implemented by a double-threshold impulse control policy: whenever the controlled surplus exceeds $c_2$, it is reduced to $c_1$ (with $c_2 > c_1 + \beta$, where $\beta$ is the transaction cost).

The expected discounted reward (value) function for such a $(c_1, c_2)$-policy is given explicitly in terms of the Parisian refracted scale function: $$v_{(c_1, c_2)}(x) = \begin{cases} (c_2 - c_1 - \beta) \frac{\theta^{(q+m, q)}(x, -l)}{\theta^{(q+m, q)}(c_2, -l) - \theta^{(q+m, q)}(c_1, -l)}, & x \leq c_2, \ x - c_1 - \beta + (c_2 - c_1 - \beta) \frac{\theta^{(q+m, q)}(c_1, -l)}{\theta^{(q+m, q)}(c_2, -l) - \theta^{(q+m, q)}(c_1, -l)}, & x > c_2. \end{cases}$$ This representation allows for direct computation of the expected discounted utility for any given set of thresholds, transaction cost, and model parameters (Gao et al., 3 Sep 2025).

3. Characterization and Existence of the Optimal Impulse Strategy

To identify the optimal thresholds $(c_1^\star, c_2^\star)$, the paper establishes necessary and sufficient conditions using differentiability and monotonicity properties of the Parisian refracted scale function. Specifically, the function

$$H(c_1, c_2) := \frac{\theta^{(q+m, q)}(c_2, -l) - \theta^{(q+m, q)}(c_1, -l)}{c_2 - c_1 - \beta}$$

must be minimized. The optimal pair is characterized via first-order conditions on the derivatives of $\theta^{(q+m, q)}$, which are verified by martingale or Hamilton–Jacobi–Bellman (HJB) arguments.

The verification lemma (see Lemma 3.4) uses the infinitesimal generator $\mathcal{A}$ of the process to show that if the candidate value function satisfies the HJB inequalities (including constraints at the refraction and bankruptcy barriers), the corresponding impulse strategy $\pi_{(c_1^\star, c_2^\star)}$ is optimal.

4. Explicit Examples and Computation

The analytical framework admits closed-form expressions for $\theta^{(q+m, q)}$ in significant special cases:

• Refracted Brownian Motion: Expressions for the scale functions are constructed from exponentials with parameters determined by drift, variance, refraction, and Parisian delay. Plots illustrate sensitivity to the Parisian delay $m$, ultimate bankruptcy barrier $-l$, and refraction intensity $\delta$.
• Refracted Cramér–Lundberg Model with Exponential Claims: All required functions reduce to simple exponentials and integrals involving the adjustment coefficient and claim parameters. The explicit form of the Parisian refracted scale function enables direct numerical computation of the optimal $(c_1^\star, c_2^\star)$ pair and value function.

These explicit examples underpin numerical results for policy optimization and stress testing, including comparative statics with respect to model and control parameters (Gao et al., 3 Sep 2025).

5. Sensitivity Analysis and Monotonicity Criteria

A key technical result is the criterion on the first derivative of the Parisian refracted scale function: for the verified optimality of the $(c_1^\star, c_2^\star)$-policy, the function $\theta^{(q+m, q)}(\cdot, -l)$ and its derivative must satisfy a monotonicity property on an appropriate interval. For instance, Theorem 4.7 posits that monotonic increase of $\theta^{(q+m, q)\prime}$ above $c_2^\star$ ensures admissibility and uniqueness of the solution.

Sensitivity analysis (as illustrated in the figures and parameter studies) shows that:

• Increasing the Parisian delay intensity $m$ (corresponding to shorter allowable excursions in deficit) leads to higher values of the scale function and more conservative optimal thresholds.
• Decreasing the bankruptcy barrier $-l$ increases resilience to ruin but can delay or reduce optimal dividend payouts.

6. Theoretical Implications and Applications

The Parisian refracted scale function generalizes the classical scale-function-based methodology used throughout Lévy risk process literature (e.g., de Finetti dividend problems, two-sided exit problems) by simultaneously capturing both Parisian implementation delays and state-dependent process modifications (refraction). As such, it provides a unified explicit tool for:

• Solving singular and impulse control (dividend optimization) problems in the presence of Parisian delays with transaction costs,
• Expressing joint fluctuation identities (e.g., first passage, occupation times, and bankruptcy probabilities) in models with realistic regulatory grace periods and dynamic surplus management,
• Bridging mathematical rigor (through use of HJB inequalities, scale function theory, and stochastic calculus) with tractable, implementable strategies relevant to actuarial practice and quantitative finance.

The approach is robust to extensions: the technical structure allows for modifications to include multi-layer refraction, time-dependent delays, and non-trivial lower (bankruptcy) barriers (Gao et al., 3 Sep 2025).

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Summary Table: Parisian Refracted Scale Function—Key Objects

Object	Definition/Formula	Context of Use
$w^{(q)}(x; a)$	$$W^{(q)}(x-a) + \delta \int_0^x \mathcal{W}^{(q)}(x-y) W^{(q)\prime}(y-a) dy$$	Refracted scale function
$\theta^{(q+m, q)}(x, -l)$	$$w^{(q)}(x; -l) + m \int_0^{-l} w^{(q)}(x; -l+y) W^{(q+m)}(y) dy$$	Parisian refracted scale function
Value function $v_{(c_1, c_2)}(x)$	See Section 2 above (piecewise in terms of $\theta$)	Expected discounted dividend computation
$H(c_1, c_2)$	$$[\theta^{(q+m, q)}(c_2, -l) - \theta^{(q+m, q)}(c_1, -l)] / (c_2 - c_1 - \beta)$$	Policy optimization criterion

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This analytical and computational framework establishes the Parisian refracted scale function as a central instrument in modern risk-control theory for Lévy models with realistic operational constraints (Gao et al., 3 Sep 2025).