Polynomial Decay of Ruin Probabilities

• Polynomial decay of ruin probabilities is defined by a power-law relationship linking initial capital to solvency risk, contrasting the exponential decay seen in light-tailed models.
• The analysis utilizes moment indices and renewal theory to derive decay exponents across discrete-time, stochastic investment, and regime-switching models.
• These insights inform risk model calibration by highlighting how investment dynamics and claim distributions shape ruin probabilities in a non-exponential framework.

The polynomial rate of decay of ruin probabilities describes a fundamental asymptotic behavior in risk processes where, under various structural and distributional assumptions, the probability that the reserve of an insurer falls below zero diminishes as a negative power of the initial capital. Unlike the exponential decay observed in classical light-tailed Cramér–Lundberg models, models incorporating risky investments, heavy-tailed claims, or regime-switching dynamics exhibit power-law (polynomial) decay, characterized by specific decay exponents linked to underlying moment or renewal properties. The precise rate and leading constants are determined by spectral, moment, or implicit renewal indices associated with the model’s random coefficients and claim structures.

1. Discrete-Time and General IID Models: Moment Indices and Ultimate Ruin

In discrete risk models that combine both financial and insurance risks, aggregate losses over $n$ periods are modeled as $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$ with $(A_i, B_i)$ IID, where $A_i$ models the annual stochastic discount (financial risk) and $B_i$ the net insurance loss. The ruin probability is $\psi(u) = P(\sup_n Y_n > u)$, the probability that cumulative (discounted) losses ever exceed initial capital $u$. The polynomial decay is quantified via so-called moment indices. For a real random variable $X$, define

$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$

and a Lundberg-type index $$I^1(X) = \sup\{s \geq 0 : E[X^s] \leq 1\} \leq I(X)$$. Under the assumption that $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$0 (e.g., $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$1, $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$2, $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$3),

$Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$4

i.e., $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$5 for large $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$6, $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$7, with the constant not explicit under general assumptions (Lehtomaa, 2013).

For the finite-horizon probability $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$8, similar asymptotics hold with an explicitly computable decay exponent in terms of joint moment indices across $Y_n = \sum_{i=1}^n A_1\cdots A_{i-1} B_i$9-step sign patterns. Under mild independence or positivity (e.g., $(A_i, B_i)$0), the decay rate simplifies to $(A_i, B_i)$1 for $(A_i, B_i)$2. The existence and calculation of moment indices thus completely determine the polynomial decay regime in such models.

2. Stochastic Investment Models: Geometric Brownian Motion and Lévy Processes

When the insurer’s surplus is invested in a risky asset, often modeled via geometric Brownian motion (gBm), the ruin probability decay is governed by a critical exponent $(A_i, B_i)$3, where $(A_i, B_i)$4 is the mean return, $(A_i, B_i)$5 the variance. For a surplus process

$(A_i, B_i)$6

and under $(A_i, B_i)$7 ($(A_i, B_i)$8), sharp bounds hold:

$(A_i, B_i)$9

with $A_i$0 explicit via renewal formulas. If $A_i$1 with $A_i$2, one has the exact asymptotic $A_i$3 as $A_i$4. If $A_i$5, ruin becomes certain, i.e., $A_i$6 for all $A_i$7 (Pergamenchtchikov et al., 2010).

This polynomial regime is robust to the presence of jumps, alternative sources of randomness, and varying claim arrival intensities. Generalization to reserve evolution driven by independent Lévy processes for asset returns and premium/claims yields analogous tail asymptotics. The critical exponent $A_i$8 becomes the unique positive root of $A_i$9 for

$B_i$0

$B_i$1 being the log-asset process. Explicit asymptotics $B_i$2 hold under non-arithmeticity and finite moments, with $B_i$3 computable via Kesten–Goldie renewal formulas (Kabanov et al., 2016).

3. Markov-Modulated and Switching Investment Models

In models where asset returns switch according to Markovian or randomly resetting regimes, the ruin probability decay exponent $B_i$4 is determined by implicit renewal theory. Under a regime-switching gBm with drift/variance parameters $B_i$5 and transition intensity $B_i$6, the decay exponent $B_i$7 solves an equation of the form

$B_i$8

with $B_i$9, and the ultimate ruin probability satisfies $\psi(u) = P(\sup_n Y_n > u)$0 (Ellanskaya et al., 2020). If drift and volatility coefficients are time-varying and reset at each claim epoch, the ruin probability decays as $\psi(u) = P(\sup_n Y_n > u)$1, where $\psi(u) = P(\sup_n Y_n > u)$2 is the unique solution to $\psi(u) = P(\sup_n Y_n > u)$3 with $\psi(u) = P(\sup_n Y_n > u)$4 encoding the cumulative “dilution” of investment over a random cycle (He et al., 2023).

These results remain valid under only mild conditions: finite higher claim moments, nondegenerate volatility in each regime, and a non-degenerate switching process.

4. Ruin Under Heavy-Tailed Claims and Non-Existence of Lundberg Exponents

In portfolios subjected to heavy-tailed claims (e.g., regularly varying tails with index $\psi(u) = P(\sup_n Y_n > u)$5), even in the absence of risky investments, ruin probabilities decay only polynomially. For i.i.d. claim sizes $\psi(u) = P(\sup_n Y_n > u)$6 with $\psi(u) = P(\sup_n Y_n > u)$7 and capital allocation scaling with $\psi(u) = P(\sup_n Y_n > u)$8, one finds

$\psi(u) = P(\sup_n Y_n > u)$9

for large $u$0 (Zapata, 30 Dec 2025). In the case of discrete-time processes with proportional reinsurance and investment, and Pareto-claims ($u$1), ruin probability decays as $u$2, where $u$3 solves a matrix renewal equation reflecting both reinsurance retention and interest-rate dynamics (Jasiulewicz et al., 2013).

Notably, in heavy-tailed models, the adjustment (Lundberg) coefficient does not exist, and the decay exponent is dictated by the tail index of the claim size distribution and systemically by the risk-sharing or investment mechanism.

5. Integro-Differential Equations and Polynomial Tails in Risky Investment/Annuity Models

In surplus models with mixed risky and riskless investment and Cramér–Lundberg-type jumps, the survival probability $u$4 satisfies a second-order integro-differential equation:

$u$5

On integrating and reducing to a Volterra equation, the solution’s derivative exhibits $u$6, leading to

$u$7

with $u$8 and $u$9 (Promyslov, 4 Jan 2026). The leading constant is given explicitly in terms of the integrated solution. Such results confirm that the presence of risky investments universally replaces the classical exponential decay by a polynomial law, even when the jump mechanism admits only minimal regularity.

6. Polynomial Approximation Rates in Scaled Classical Models

When analyzing the rate at which the ruin probability in the scaled classical Cramér–Lundberg risk process converges to its diffusion approximation, the error decays polynomially with respect to the scaling parameter $X$0. Specifically, for the scaled process with claim arrival intensity $X$1, claim size $X$2, and initial capital $X$3,

$X$4

uniformly in $X$5, where $X$6 is the diffusion-limit solution (Cohen et al., 2019). In the exponential claim case, higher-order expansions yield

$X$7

These polynomial rates describe the asymptotic convergence speed to the continuous approximation and highlight the robustness of polynomial error bounds in risk models.

7. Structural Mechanisms and Universality of Power-Law Decay

The universal mechanism underlying polynomial decay of ruin probabilities is the emergence of random recurrent affine equations for the surplus process, either at claim epochs or through embedded Markov chains. Renewal and implicit renewal theory (notably Kesten–Goldie-type results) provide the analytic foundation, linking decay exponents to roots of spectral equations for random multipliers $X$8 (often $X$9). The value of the polynomial exponent is sensitive to the balance between mean investment return and volatility, as well as the heaviness of claim tails. Risky investments, time-heterogeneous regime-switching, and heavy-tail phenomena each suppress exponential rates and replace them with explicit power laws, determined by moment or spectral criteria across all models considered (Pergamenchtchikov et al., 2010, Ellanskaya et al., 2020, Kabanov et al., 2016, Promyslov, 4 Jan 2026, Zapata, 30 Dec 2025, Lehtomaa, 2013).

Model class	Ruin probability decay	Exponent formula / regime
Discrete risk model	$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$0	$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$1
Risky investment (gBm)	$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$2	$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$3
Lévy-driven asset returns	$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$4	$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$5, $$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$6
Markov-modulated returns	$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$7	$$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$8 solves $$I(X) = \sup\{s \geq 0 : E[(X^+)^s] < \infty\}, \quad X^+ := \max(X,0),$$9 (see above)
Heavy-tailed claims	$$I^1(X) = \sup\{s \geq 0 : E[X^s] \leq 1\} \leq I(X)$$0	$$I^1(X) = \sup\{s \geq 0 : E[X^s] \leq 1\} \leq I(X)$$1 from tail/renewal equations

• "Asymptotic behaviour of ruin probabilities in a general discrete risk model using moment indices" (Lehtomaa, 2013)
• "Ruin probability in the presence of risky investments" (Pergamenchtchikov et al., 2010)
• "The ruin problem for Lévy-driven linear stochastic equations with applications to actuarial models with negative risk sums" (Kabanov et al., 2016)
• "On ruin probabilities with risky investments" (Ellanskaya et al., 2020)
• "On ruin probabilities in the presence of risky investments and random switching" (He et al., 2023)
• "Heavy-tailed distributions; extreme value theory; large deviations; ruin probabilities; solvency risk" (Zapata, 30 Dec 2025)
• "Ruin probability of a discrete-time risk process with proportional reinsurance and investment for exponential and Pareto distributions" (Jasiulewicz et al., 2013)
• "On the integro-differential equation arising in the ruin problem for annuity payment models" (Promyslov, 4 Jan 2026)
• "Rate of Convergence of the Probability of Ruin in the Cramér-Lundberg Model to its Diffusion Approximation" (Cohen et al., 2019)