Viscosity solutions of the integro-differential equation for the Cramér--Lundberg model with annuity payments and investments

1. Problem Statement and Context

The paper addresses the mathematical analysis of the survival probability (the probability that an insurance company never goes bankrupt) within a specific stochastic framework: the Cramér–Lundberg model with annuity payments and investments.

The Model:

• Capital Dynamics ($X_t$): The company pays out a constant annuity rate $c > 0$ and receives incoming premiums modeled by a compound Poisson process ($P_t$) with positive jumps.
• Investments: Reserves are invested in a risky asset following a geometric Brownian motion with drift $a$ and volatility $\sigma$.
• Ruin Definition: Ruin occurs when the capital $X_t$ drops to zero or below. Crucially, because jumps are strictly positive, ruin cannot occur via a jump; it must occur continuously via the diffusion and drift components. Thus, at the moment of ruin $\tau$, $X_\tau = 0$ almost surely.
• The Equation: The survival probability $\Phi(u)$ (where $u$ is initial capital) is expected to satisfy a singular second-order integro-differential equation (IDE):
  $$\frac{\sigma^2 u^2}{2} \Phi''(u) + (au - c)\Phi'(u) - \lambda \Phi(u) + \lambda \int_0^\infty \Phi(u+y) dF(y) = 0$$
  with boundary conditions $\Phi(0)=0$ and $\lim_{u\to\infty}\Phi(u)=1$.

The Challenge:
Standard analytical derivations of this IDE using Itô's formula require the solution $\Phi(u)$ to be twice continuously differentiable ($C^2$) a priori. However, the diffusion coefficient $\frac{\sigma^2 u^2}{2}$ degenerates at zero (it becomes 0 when $u=0$), making the equation singular at the boundary. Previous works often assumed smoothness or used heavy probabilistic machinery to prove it. This paper aims to rigorously justify the existence of a classical solution using the theory of viscosity solutions without assuming smoothness initially.

2. Methodology

The author employs the theory of viscosity solutions, adapted for non-local integro-differential operators, to bridge the gap between probabilistic definitions and classical differential equations.

Key Methodological Steps:

1. Viscosity Definition Adaptation:
   • Standard viscosity definitions require global touching conditions. Due to the one-sided nature of jumps (only positive jumps), the author defines a relaxed touching condition: a test function $\phi$ must touch the solution $w$ from above/below locally near $x_0$ and globally for all $x \ge x_0$. This accounts for the fact that the non-local integral term depends only on the function's values to the right of the current point.
2. Existence Proof (Dynamic Programming):
   • Instead of assuming smoothness, the author uses the Dynamic Programming Principle (DPP) and the Strong Markov Property of the underlying stochastic process.
   • By applying Itô's formula to the semicontinuous envelopes of the survival probability, it is proven that $\Phi(u)$ satisfies the viscosity subsolution and supersolution inequalities.
3. Uniqueness Proof (Comparison Principle):
   • A strict comparison principle is established for bounded viscosity sub- and supersolutions.
   • The proof utilizes the "doubling of variables" technique (Ishii's lemma) adapted for non-local operators.
   • A critical step involves the assumption that the topological support of the jump distribution $F$ is $(0, \infty)$. This ensures that if the difference between a subsolution and supersolution achieves a maximum, the integral term forces a contradiction unless the functions are identical.
4. Regularity Upgrade (Smoothness):
   • Once continuity is established via the comparison principle, the author treats the IDE as a linear non-homogeneous Ordinary Differential Equation (ODE) where the integral term acts as a continuous source term.
   • Since the diffusion coefficient is strictly positive for $u > 0$, local elliptic regularity theory is applied to "upgrade" the continuous viscosity solution to a classical $C^2$ solution on $(0, \infty)$.

3. Key Contributions

• Rigorous Justification of the IDE: The paper proves that the survival probability is the unique classical solution to the singular IDE, removing the need for a priori smoothness assumptions often found in literature.
• Novel Viscosity Definition: It introduces a specific definition of viscosity solutions tailored to operators with one-sided jumps, addressing the unique boundary behavior where ruin is continuous rather than jump-induced.
• Bridging Probability and Analysis: It demonstrates that viscosity solutions can serve as an intermediate tool to prove the existence of classical solutions in financial mathematics, avoiding cumbersome probabilistic proofs of smoothness (unlike previous works relying on exponential functionals of Brownian motion).
• Handling Degeneracy: It successfully handles the singularity at $u=0$ where the diffusion coefficient vanishes, proving that the boundary condition $\Phi(0)=0$ is naturally satisfied by the viscosity solution.

4. Main Results

1. Existence: The survival probability $\Phi(u)$ is a viscosity solution of the singular IDE in the domain $(0, \infty)$.
2. Uniqueness: There is a unique bounded viscosity solution to the boundary value problem.
3. Regularity: The viscosity solution $\Phi(u)$ is continuous on $[0, \infty)$ and belongs to the class $C^2((0, \infty))$.
4. Classical Solution: Consequently, $\Phi(u)$ is the unique classical solution (satisfying the equation pointwise) to the boundary value problem:
   $$\begin{cases} \frac{\sigma^2 u^2}{2} \Phi''(u) + (au - c)\Phi'(u) - \lambda \Phi(u) + \lambda \int_0^\infty \Phi(u+y) dF(y) = 0, & u > 0 \\ \Phi(0) = 0 \\ \lim_{u\to\infty} \Phi(u) = 1 \end{cases}$$

5. Significance

This work is significant for both stochastic control theory and actuarial mathematics:

• Theoretical Robustness: It provides a solid analytical foundation for the Cramér–Lundberg model with investments, confirming that the standard integro-differential equation is mathematically sound without relying on heuristic smoothness assumptions.
• Methodological Advancement: It extends the applicability of viscosity solution theory to models with positive jumps and degenerate diffusion, a scenario where standard non-life insurance models (with negative jumps) do not apply directly.
• Practical Implication: By establishing the existence of a classical $C^2$ solution, the paper validates the use of standard numerical methods and asymptotic analysis techniques that rely on the differentiability of the survival probability, facilitating more accurate risk assessment for insurance companies managing investment portfolios.