Optimal Impulse Control Policy

• Optimal impulse control policy is a decision rule for applying discrete, costly interventions in systems evolving under continuous dynamics to maximize discounted dividends until Parisian ruin occurs.
• The methodology employs a two-threshold (c1, c2) rule alongside analytical Parisian refracted q-scale functions and smooth-fit conditions to determine the optimal intervention points.
• Numerical evidence from models like Brownian motion and the Cramér–Lundberg process demonstrates the policy's robustness and sensitivity to transaction costs and process parameters.

An optimal impulse control policy is a rule for when and how to apply discrete, costly interventions to a system that otherwise evolves according to deterministic or stochastic continuous dynamics. In the setting of surplus management with dividend payments, refracted Lévy processes, fixed transaction costs, and Parisian ruin, the problem is to maximize the expected discounted (or average) dividends paid until ruin, under the constraint that only interventions (impulses) incurring a fixed cost are permitted, and ruin is declared if the process remains below zero for a prescribed duration (the Parisian delay). The optimal impulse policy in this context is characterized by two threshold parameters and relies on new analytical formulas for the so-called Parisian refracted $q$-scale functions.

1. Structure of the Impulse Control Policy

The impulse control policy considered is a two-threshold, or $(c_1, c_2)$, rule:

• Two thresholds $c_1$ (lower) and $c_2$ (upper) are selected so that $c_2 > c_1 + \beta$, where $\beta$ is the fixed transaction cost, and $c_1 \geq 0$.
• The surplus process $R_t$ (modeled as a refracted Lévy process) is monitored continuously.
• At each intervention time (stopping time $\tau_k$), if $R_{\tau_k} > c_2$, a lump-sum dividend is paid to immediately reduce the surplus to $c_1$, i.e., $R_{\tau_k^+} = c_1$ with a payment of $c_2 - c_1 - \beta$ units (the net of transaction costs).
• This rule is applied repeatedly; no dividends are paid as long as $R_t \leq c_2$.

The process continues until the Parisian ruin time: the first epoch when the process stays continuously in the negative half-line for a period of at least $r$ (the Parisian delay parameter).

This $(c_1, c_2)$-policy is a classical form for impulse policies under fixed transaction costs, as continuous (singular) control cannot be optimal due to the presence of the lump-sum cost.

2. Optimality Conditions

To establish optimality of the $(c_1, c_2)$-policy, several analytic and verification steps are required:

• Value Function Construction:
  • Let $v_{c_1, c_2}(x)$ denote the expected discounted sum of dividends under the $(c_1, c_2)$-policy, starting from initial surplus $x$.
  • For $x \leq c_2$, $$v_{c_1, c_2}(x) = (c_2 - c_1 - \beta) \cdot \frac{V^{(q)}(x)}{V^{(q)}(c_2) - V^{(q)}(c_1)}$$.
  • For $x > c_2$, $$v_{c_1, c_2}(x) = (x - c_1 - \beta) + (c_2 - c_1 - \beta) \frac{V^{(q)}(c_1)}{V^{(q)}(c_2) - V^{(q)}(c_1)}$$.
  • Here $V^{(q)}(\cdot)$ is a Parisian refracted $q$-scale function.
• Smooth-fit (Continuous-fit) Conditions:
  • The derivative of the value function at threshold $c_2$ must satisfy:

  $$V^{(q)\prime}(c_2) = \frac{V^{(q)}(c_2) - V^{(q)}(c_1)}{c_2 - c_1 - \beta}.$$ - Alternative or additional conditions involve $V^{(q)\prime}(c_1) = V^{(q)\prime}(c_2)$ or $c_1 = 0$ (natural boundary).

• Verification Lemma: If the candidate value $v_{c_1, c_2}$ satisfies the associated Hamilton–Jacobi–Bellman (HJB) inequalities:

$$(\Gamma - q)v_{c_1, c_2}(x) \leq 0 \quad \text{for all } x,$$

and for $x \geq y$,

$$v_{c_1, c_2}(x) - v_{c_1, c_2}(y) \geq x - y - \beta,$$

then $v_{c_1, c_2}$ is the value function, and the associated $(c_1, c_2)$-policy is optimal.

• Technical Assumptions: The Lévy measure should have no atoms at $0$ in the bounded variation case, and $V^{(q)}$ should be sufficiently smooth.

3. Parisian Refracted $q$-Scale Functions

A central technical tool is the development of explicit formulas for the Parisian refracted $q$-scale function $V^{(q)}(x)$:

• Classical $q$-Scale Function: For a spectrally negative Lévy process $X$, with Laplace exponent $\psi(\theta)$, the $q$-scale function $W^{(q)}(x)$ satisfies:

$$\int_0^{\infty} e^{-\theta x} W^{(q)}(x) dx = \frac{1}{\psi(\theta) - q}, \quad \theta > \Phi(q).$$

• Refracted Scale Function: For a refracted process,

$$w^{(q)}(x; a) = \begin{cases} W^{(q)}(x - a), & x < 0, \ W^{(q)}(x - a) + \delta \int_{0}^{x} \mathcal{W}^{(q)}(x - y) W^{(q)\prime}(y - a) dy, & x \geq 0, \end{cases}$$

where $\delta$ is the refraction (payout) rate for $x \geq 0$.

• Parisian Refracted $q$-Scale Function:

$$V^{(q)}(x) = \int_0^{\infty} w^{(q)}(x; -z) \frac{z}{r} \mathbb{P}(X_r \in dz),$$

where $r$ is the Parisian delay and $X_r$ is the value of the underlying Lévy process at time $r$.

Specific explicit formulas are computed for two canonical cases:

• Linear Brownian Motion: $X_t = \mu t + \sigma B_t$; the scale functions $W^{(q)}(x)$ and $V^{(q)}(x)$ are given in terms of exponentials parameterized by model coefficients and the refraction parameters.
• Cramér–Lundberg Model with Exponential Claims: Here, $X_t = pt - \sum_{i=1}^{N_t} U_i$ with $U_i$ exponential, and the $q$-scale functions admit series and closed-form representations involving incomplete gamma functions and related expressions.

4. Numerical Evidence and Sensitivity

Numerical illustrations are provided for both the linear Brownian motion and Cramér–Lundberg examples. Main observations:

• The Parisian refracted $q$-scale functions $V^{(q)}$ and their derivatives determine the optimal thresholds $(c_1^*, c_2^*)$ via the smooth-fit conditions.
• In the Brownian case, parameter choices (e.g., low vs. high $\beta$) shift the thresholds and can even drive optimal doors to the boundary ($c_1^*=0)$ for large transaction costs.
• In the Cramér–Lundberg case, $V^{(q)}$ displays discontinuity at $x=-pr$ (the minimal value reachable after no claims in delay $r$). The location and existence of the optimal thresholds are sensitive to both process and policy parameters.

5. Implications and Applications

• Extension of Classical Problems: The analysis extends classical dividend/impulse control models by incorporating refracted dynamics (downward drift when positive), Parisian ruin features (delayed default), and fixed transaction costs.
• Explicit Analytic Tools: The new Parisian refracted scale function provides a practical and theoretically sound toolkit for actuaries and financial engineers.
• Policy Structure: The two-threshold $(c_1, c_2)$ structure is robust for the considered models and can be directly computed once $V^{(q)}$ and its derivative are available.
• Applicability: Relevant to insurance risk, dividend optimization, capital management with delays (regulatory or operational), and any context where interventions incur fixed costs and instantaneous ruin does not occur.

Model	Scale/Value Function	Threshold Policy Form
Brownian Motion	Explicit (exponential form)	$(c_1, c_2)$ uniquely determined by $V^{(q)}$
Cramér–Lundberg	Series/closed form	$(c_1, c_2)$ threshold as for Brownian, adjusted for claim process

In summary, the refracted Lévy/Parisian ruin framework with transaction costs retains the fundamental $(c_1, c_2)$ threshold impulse control structure, enriches it with rigorous new analytical scale function formulas, and provides an explicit, numerically implementable methodology for identifying the unique optimal impulse control policy across a wide class of models (Czarna et al., 2019).