Contents -- Preface -- Notation and conventions -- I Introduction -- 1 The risk process -- 2 Claim size distributions -- 2a Light-tailed distributions -- 2b Heavy-tailed distributions -- 3 The arrival process -- 4 A summary of main results and methods -- 4a Duality with other applied probability models -- 4b Exact solutions -- 4c Numerical methods -- 4d Approximations -- 4e Bounds and inequalities -- 4f Statistical methods -- 4g Simulation -- II Martingales and simple ruin calculations -- 1 Wald martingales -- 2 Gambler's ruin. Two-sided ruin. Brownian motion -- 3 Further simple martingale calculations -- 4 More advanced martingales -- 4a Generators. The Dynkin martingale -- 4b Diffusions and two-sided ruin -- 4c The Kella-Whitt martingale -- III Further general tools and results -- 1 Likelihood ratios and change of measure -- 2 Duality with other applied probability models -- 3 Random walks in discrete or continuous time -- 4 Markov additive processes -- 5 The ladder height distribution -- IV The compound Poisson model -- 1 Introduction -- 2 The Pollaczeck-Khinchine formula -- 3 Special cases of the Pollaczeck-Khinchine formula -- 3a The ruin probability when the initial reserve is zero -- 3b Exponential claims -- 3c Some classical analytical results -- 3d Deterministic claims -- 4 Change of measure via exponential families -- 5 Lundberg conjugation -- 5a Alternative proofs -- 6 Further topics related to the adjustment coefficient -- 6a On the existence of ° -- 6b Bounds and approximations for -- 6c A refinement of Lundberg's inequality -- 7 Various approximations for the ruin probability -- 7a The Beekman-Bowers approximation -- 7b De Vylder's approximation -- 7c The heavy tra±c approximation -- 7d The light tra±c approximation -- 7e Interpolating between light and heavy traffi -- 8 Comparing the risks of different claim size distributions.

9 Sensitivity estimates -- 10 Estimation of the adjustment coefficient -- V The probability of ruin within finite time -- 1 Exponential claims -- 2 The ruin probability with no initial reserve -- 3 Laplace transforms -- 4 When does ruin occur? -- 4a Segerdahl's normal approximation -- 4b Gerber's time-dependent version of Lundberg's inequality -- 4c Arfwedson's saddlepoint approximation -- 5 Diffusion approximations -- 6 Corrected diffusion approximations -- 7 How does ruin occur? -- VI Renewal arrivals -- 1 Introduction -- 2 Exponential claims. The compound Poisson model with negative claims -- 3 Change of measure via exponential families -- 3a The imbedded random walk -- 3b Markov additive representations -- 4 The duality with queueing theory -- VII Risk theory in a Markovian environment -- 1 Model and examples -- 2 The ladder height distribution -- 3 Change of measure via exponential families -- 3a Lundberg conjugation -- 3b Ramifications of Lundberg's inequality -- 4 Comparisons with the compound Poisson model -- 4a Ordering of the ruin functions -- 4b Ordering of adjustment coefficients -- 4c Sensitivity estimates for the adjustment coefficient -- 5 The Markovian arrival process -- 6 Risk theory in a periodic environment -- 6a The model -- 6b Lundberg conjugation -- 6c Markov-modulated approximations -- 7 Dual queueing models -- VIII Level-dependent risk processes -- 1 Introduction -- 1a Two-step premium functions -- 1b Multi-step premium functions -- 2 The model with constant interest -- 3 The local adjustment coefficient. Logarithmic asymptotics -- 3a Examples -- 3b Proof of Theorem 3.2 -- 3c Proof of Theorem 3.3 -- 4 The model with tax -- 5 Discrete-time ruin problems with stochastic investment -- 6 Continuous-time ruin problems with stochastic investment -- IX Matrix-analytic methods.

1 Definition and basic properties of phase-type distributions -- 1a Asymptotic exponentiality -- 2 Renewal theory -- 3 The compound Poisson model -- 3a Phase-type claims -- 4 The renewal model -- 5 Markov-modulated input -- 5a Calculations via fluid models. Diagonalization -- 5b Computations via K -- 6 Matrix-exponential distributions -- 7 Reserve-dependent premiums -- 7a Computing (u) via differential equations -- 7b Two-step premium rules -- 8 Erlangization for the finite horizon case -- X Ruin probabilities in the presence of heavy tails -- 1 Subexponential distributions -- 2 The compound Poisson model -- 3 The renewal model -- 4 Finite-horizon ruin probabilities -- 4a Excursion theory for Markov processes -- 4b The time to ruin -- 5 Reserve-dependent premiums -- 6 Tail estimation -- 6a The mean excess plot -- 6b Extreme values and POT -- 6c The Hill estimator -- XI Ruin probabilities for Lévy processes -- 1 Preliminaries -- 1a Special Levy processes -- 1b Exponential change of measure -- 2 One-sided ruin theory -- 3 The scale function and two-sided ruin problems -- 4 Further topics -- 4a Local time at the maximum -- 4b The ladder height process -- 4c Excursions -- 4d The Wiener-Hopf factorization -- 4e A quintuple identity -- 5 The scale function for two-sided phase-type jumps -- XII Gerber-Shiu functions -- 1 Introduction -- 2 The compound Poisson model -- 2a A Laplace transform approach -- 2b Change of measure -- 2c Martingales -- 2d Further ruin-related quantities -- 3 The renewal model -- 3a Change of measure -- 3b A modified random walk -- 3c Integro-differential equations -- 4 Lévy risk models -- 4a Spectrally negative Lévy processes -- 4b The compound Poisson model with two-sided jumps -- XIII Further models with dependence -- 1 Large deviations -- 2 Heavy-tailed risk models with dependent input -- 3 Linear models.

4 Risk processes with shot-noise Cox intensities -- 5 Causal dependency models -- 6 Dependent Sparre Andersen models -- 7 Gaussian models. Fractional Brownian motion -- 8 Ordering of ruin probabilities -- 9 Multi-dimensional risk processes -- XIV Stochastic control -- 1 Introduction -- 2 Stochastic dynamic programming -- 3 The Hamilton-Jacobi-Bellman equation -- XV Simulation methodology -- 1 Generalities -- 1a The crude Monte Carlo method -- 1b Variance reduction techniques -- 1c Rare events simulation -- 2 Simulation via the Pollaczeck-Khinchine formula -- 2a Light tails: importance sampling -- 2b Heavy tails: conditional Monte Carlo -- 2c Heavy tails: importance sampling -- 3 Static importance sampling via Lundberg conjugation -- 4 Static importance sampling for the finite horizon case -- 5 Dynamic importance sampling -- 5a An algorithm by Dupuis, Leder and Wang -- 6 Regenerative simulation -- 7 Sensitivity analysis -- XVI Miscellaneous topics -- 1 More on discrete-time risk models -- 2 The distribution of the aggregate claims -- 2a The saddlepoint approximation -- 2b The NP approximation -- 2c Panjer's recursion -- 2d The distribution of dependent sums -- 3 Principles for premium calculation -- 4 Reinsurance -- Appendix -- A1 Renewal theory -- 1a Renewal processes and the renewal theorem -- 1b Renewal equations and the key renewal theorem -- 1c Regenerative processes -- 1d Cumulative processes -- 1e Residual and past lifetime -- 1f Markov renewal theory -- A2 Wiener-Hopf factorization -- A3 Matrix-exponentials -- A4 Some linear algebra -- 4a Generalized inverses -- 4b The Kronecker product and the Kronecker sum -- 4c The Perron-Frobenius theorem -- A5 Complements on phase-type distributions -- 5a Asymptotic exponentiality -- 5b Discrete phase-type distributions -- 5c Closure properties -- 5d Phase-type approximation -- 5e Phase-type fitting.

A6 Tauberian theorems -- Bibliography -- Index.