8.1 Classical ruin theory and infinite time horizon

Classical ruin theory is a crucial tool for insurance companies to assess their long-term solvency. It models an insurer's surplus over time, considering premiums and claims, to calculate the probability of going bankrupt or "ruined."

The theory uses mathematical concepts like Poisson processes and compound distributions to analyze claim patterns. Key results include Lundberg's inequality, which provides an upper bound on ruin probability, and the Pollaczek-Khinchine formula for calculating ruin-related quantities.

Classical ruin theory overview

• Classical ruin theory studies the probability of an insurance company's surplus (assets minus liabilities) becoming negative, leading to insolvency or "ruin"
• Provides a mathematical framework to analyze the long-term stability and solvency of an insurer, considering the stochastic nature of claims and premiums

Surplus process in classical ruin theory

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• The surplus process $U(t)$ represents the insurer's surplus at time $t$, starting with an initial surplus $u$
• Defined as $U(t) = u + ct - S(t)$, where $c$ is the premium rate and $S(t)$ is the aggregate claims up to time $t$
• The insurer is considered "ruined" if the surplus becomes negative at any point, i.e., $U(t) < 0$ for some $t > 0$

Probability of ruin in infinite time

• The probability of ruin $\psi(u)$ is the probability that the surplus process $U(t)$ falls below zero at any time, given an initial surplus $u$
• Mathematically, $\psi(u) = P(\inf_{t \geq 0} U(t) < 0 | U(0) = u)$
• The goal is to minimize the probability of ruin by selecting appropriate premium rates, reinsurance arrangements, and investment strategies

Poisson process for claim arrivals

• In classical ruin theory, claim arrivals are often modeled using a Poisson process with rate $\lambda$
• The time between claim arrivals follows an exponential distribution with mean $1/\lambda$
• The number of claims in any time interval of length $t$ follows a Poisson distribution with mean $\lambda t$

Compound Poisson process for aggregate claims

• The aggregate claims process $S(t)$ is modeled as a compound Poisson process
• $S(t) = \sum_{i=1}^{N(t)} X_i$, where $N(t)$ is the number of claims up to time $t$ (a Poisson process) and $X_i$ are the individual claim sizes (independent and identically distributed random variables)
• The distribution of claim sizes $X_i$ is typically assumed to be non-negative and continuous (exponential, gamma, or Pareto distributions)

Lundberg's inequality

• Lundberg's inequality provides an upper bound for the probability of ruin in the classical risk model
• It is a useful tool for quickly assessing the solvency of an insurance company and setting appropriate premium rates

Exponential bound for ruin probability

• Lundberg's inequality states that the probability of ruin $\psi(u)$ is bounded above by an exponential function: $\psi(u) \leq e^{-Ru}$
• $R > 0$ is the adjustment coefficient, which depends on the premium rate, claim arrival rate, and claim size distribution
• The exponential bound becomes tighter as the initial surplus $u$ increases, reflecting the increased stability of the insurer

Adjustment coefficient in Lundberg's inequality

• The adjustment coefficient $R$ is the unique positive root of Lundberg's fundamental equation: $\lambda + cR = \lambda M_X(R)$
• $\lambda$ is the claim arrival rate, $c$ is the premium rate, and $M_X(r)$ is the moment generating function of the claim size distribution
• A larger adjustment coefficient leads to a tighter upper bound on the ruin probability, indicating a more stable insurance portfolio

Pollaczek-Khinchine formula

• The Pollaczek-Khinchine formula is a key result in classical ruin theory that expresses the Laplace transform of the time to ruin in terms of the claim size distribution
• It allows for the derivation of various ruin-related quantities, such as the probability of ruin and the moments of the time to ruin

Laplace transform of time to ruin

• Let $T$ be the time to ruin, defined as $T = \inf\{t > 0: U(t) < 0\}$, and $\phi(u) = E[e^{-sT} | U(0) = u]$ be its Laplace transform
• The Pollaczek-Khinchine formula expresses $\phi(u)$ in terms of the Laplace transform of the claim size distribution and the adjustment coefficient

Pollaczek-Khinchine formula derivation

• The derivation of the Pollaczek-Khinchine formula relies on the concept of ladder heights and the renewal equation
• Ladder heights represent the successive record highs of the surplus process, and their distribution is related to the claim size distribution
• The renewal equation connects the Laplace transform of the time to ruin with the ladder height distribution, leading to the Pollaczek-Khinchine formula

Lundberg's fundamental equation

• Lundberg's fundamental equation is a key equation in classical ruin theory that relates the adjustment coefficient to the claim arrival rate, premium rate, and claim size distribution
• It is used to determine the adjustment coefficient, which appears in Lundberg's inequality and the Pollaczek-Khinchine formula

Roots of Lundberg's fundamental equation

• Lundberg's fundamental equation is given by $\lambda + cR = \lambda M_X(R)$, where $\lambda$ is the claim arrival rate, $c$ is the premium rate, and $M_X(r)$ is the moment generating function of the claim size distribution
• The equation has a unique positive root $R > 0$, known as the adjustment coefficient
• The existence and uniqueness of the adjustment coefficient depend on the net profit condition $c > \lambda E[X]$, which ensures the long-term profitability of the insurance company

Lundberg coefficient vs adjustment coefficient

• The Lundberg coefficient $\alpha$ is defined as the unique positive root of the equation $\lambda M_X(r) = 1$
• The adjustment coefficient $R$ is related to the Lundberg coefficient by $R = (1 - \lambda/c)\alpha$
• Both coefficients play important roles in ruin theory, with the adjustment coefficient being more directly related to the probability of ruin through Lundberg's inequality

Exact ruin probabilities

• While Lundberg's inequality provides an upper bound for the probability of ruin, exact ruin probabilities can be calculated in certain cases
• Exact ruin probabilities are particularly useful when the claim size distribution has a special structure or when the surplus process is modified

Ruin probability as a geometric compound

• When the claim size distribution is a combination of exponential distributions (hyper-exponential), the ruin probability can be expressed as a geometric compound
• $\psi(u) = \sum_{i=1}^{n} A_i e^{-R_i u}$, where $A_i$ are constants and $R_i$ are the roots of Lundberg's fundamental equation
• This representation allows for efficient computation of ruin probabilities and related quantities

Beekman's convolution formula

• Beekman's convolution formula is an integral equation that relates the ruin probability to the claim size distribution and the adjustment coefficient
• It is given by $\psi(u) = \frac{\lambda}{c} \int_0^u \psi(u-x) (1 - F_X(x)) dx + \frac{\lambda}{c} \int_u^\infty (1 - F_X(x)) dx$, where $F_X(x)$ is the cumulative distribution function of the claim size distribution
• The formula can be solved numerically or, in some cases, analytically to obtain exact ruin probabilities

Heavy-tailed claim size distributions

• Heavy-tailed claim size distributions, such as Pareto or lognormal distributions, are characterized by a slower decay of the tail probability compared to light-tailed distributions like exponential or gamma
• In the presence of heavy-tailed claims, the ruin probability is significantly higher, and the classical ruin theory results may not apply directly

Subexponential distributions in ruin theory

• Subexponential distributions are a class of heavy-tailed distributions that satisfy $\lim_{x \to \infty} \frac{\overline{F^{*2}}(x)}{\overline{F}(x)} = 2$, where $\overline{F}(x) = 1 - F(x)$ is the tail probability and $F^{*2}$ is the convolution of the distribution with itself
• Examples of subexponential distributions include Pareto, lognormal, and Weibull with shape parameter less than 1
• Subexponential distributions have a significant impact on ruin probabilities and require special treatment in ruin theory

Ruin probabilities for heavy-tailed claims

• For subexponential claim size distributions, the ruin probability has an asymptotic form $\psi(u) \sim \frac{\lambda}{c} \int_u^\infty \overline{F}(x) dx$ as $u \to \infty$
• This asymptotic form suggests that the ruin probability decreases more slowly with the initial surplus compared to light-tailed distributions
• Simulation techniques or more advanced analytical methods, such as the single big jump approximation, are often used to estimate ruin probabilities for heavy-tailed claims

Surplus process modifications

• The classical surplus process can be modified to incorporate additional features, such as investment income, dividend payments, or reinsurance
• These modifications lead to more realistic models and provide insights into the impact of various management strategies on the ruin probability

Brownian motion approximation

• The surplus process can be approximated by a Brownian motion with drift when the claim sizes are small relative to the initial surplus and the premium income
• The approximation is given by $U(t) \approx u + (c - \lambda E[X])t + \sqrt{\lambda E[X^2]} B(t)$, where $B(t)$ is a standard Brownian motion
• The ruin probability under the Brownian motion approximation has an explicit formula involving the standard normal distribution function

Diffusion approximation for ruin probability

• The diffusion approximation is a more general approach that approximates the surplus process by a diffusion process
• The approximation is based on the functional central limit theorem and is valid when the claim sizes and inter-arrival times satisfy certain conditions
• The ruin probability under the diffusion approximation can be obtained by solving a boundary value problem for the corresponding partial differential equation

Ruin theory applications

• Ruin theory has numerous applications in insurance and risk management, including setting premium rates, determining optimal reinsurance strategies, and managing dividend payments
• The insights provided by ruin theory help insurers maintain long-term stability and solvency while meeting the needs of policyholders

Optimal surplus and dividend problems

• Ruin theory can be used to determine the optimal initial surplus and dividend payment strategies that maximize the expected total discounted dividends until ruin
• The optimization problem involves finding the optimal dividend barrier, above which all surplus is paid out as dividends
• Dynamic programming techniques and Hamilton-Jacobi-Bellman equations are used to solve the optimization problem and obtain the optimal strategies

Reinsurance and ruin probability impact

• Reinsurance is a risk management tool that allows insurers to transfer a portion of their risk to another insurer (the reinsurer)
• Different reinsurance contracts, such as proportional (quota-share) or non-proportional (excess-of-loss) reinsurance, have different impacts on the ruin probability
• Ruin theory can be used to analyze the effect of reinsurance on the insurer's stability and to determine the optimal reinsurance strategy that balances risk reduction and cost