AN ABSTRACT OF THE DISSERTATION OF. Sooie-Hoe Loke for the degree of Doctor of Philosophy in Mathematics presented on

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2 AN ABSTRACT OF THE DISSERTATION OF Sooie-Hoe Loke for the degree of Doctor of Philosophy in Mathematics presented on September 1, 215. Title: Ruin Problems with Risky Investments Abstract approved: Enrique A. Thomann In this dissertation, we study two risk models. First, we consider the dual risk process which models the surplus of a company that incurs expenses at a constant rate and earns random positive gains at random times. When the surplus is invested in a risky asset following a geometric Brownian motion, we show that the ruin probability decays algebraically for small volatility and that ruin is certain for large volatility. We use numerical methods to approximate the ruin probability when the surplus is invested in a risk-free asset. When there are no investments, we recover the exact expression for the ruin probability via Wiener-Hopf factorization. Second, we are concerned with incurred but not reported (IBNR) claims, modeled by delaying the settlement of each claim by a random time. When the investments follow a geometric Brownian motion, we derive a parabolic integro-partial-differential equation (IPDE) for the ultimate ruin probability with final value condition given by the ruin probability under risky investments with no delay. Assuming that the delay times are bounded by a constant, we obtain an existence theorem of the final value IPDE in the space of bounded functions, and a uniqueness theorem in the space of square integrable functions. When the delay times are deterministic, we show that delaying the settlement of claims does not reduce the probability of ruin when the volatility is large.

3 c Copyright by Sooie-Hoe Loke September 1, 215 All Rights Reserved

4 Ruin Problems with Risky Investments by Sooie-Hoe Loke A DISSERTATION submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented September 1, 215 Commencement June 216

5 Doctor of Philosophy dissertation of Sooie-Hoe Loke presented on September 1, 215 APPROVED: Major Professor, representing Mathematics Chair of the Department of Mathematics Dean of the Graduate School I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request. Sooie-Hoe Loke, Author

6 ACKNOWLEDGEMENTS Though only my name appears on the cover of this thesis, there have been many great people who contributed to its production. I am most indebted to my advisor, Dr. Enrique Thomann. I have been blessed to have an advisor who gave me the freedom to explore on my own, and the guidance to recover when my steps faltered. Enrique taught me how to question thoughts and express ideas. His patience and support helped me to grow into a better mathematician. Dr. Edward Waymire has been always there to listen and give advice. When Enrique was on sabbatical, Ed guided and helped me exploring ideas in my thesis. He has also provided me numerous opportunities to develop professionally in academia. Dr. Malgorzata Peszynska has given me insightful comments at different stages of my career. I am grateful to her for holding me to a high research standard and strict validations for research results. She inspired me to work on Chapter 3 of this dissertation. I want to thank Dr. Adel Faridani for providing constructive feedback regarding my teaching skills. He has attended and observed many of my classes. I would like to acknowledge Dr. Steven Buccola and Dr. Junjie Wu for their encouragement and practical advice in the field of economics. I am also thankful to them for commenting on my views and helping me understand and enrich my ideas. I want to express my sincere gratitude and appreciation to the Department of Mathematics, Actuarial Science Club, College of Science, Graduate School (the first four organizations are all from OSU), National Science Foundation (NSF), Institute for Mathematics and its Applications (IMA), and Society of Actuary (SOA) for providing me financial support to attend conferences in the past three years. Many friends have helped me stay strong through these five years at OSU. The support and care from friends in the Department of Mathematics and Table Tennis Club

7 helped me overcome setbacks, stay focused on my graduate work, and enhance my experience at OSU. I truly value their friendship and their belief in me. None of this would have been possible without the love, patience, and support of my families in Malaysia and Vietnam. I dedicate this thesis to my late grandparents. I miss them dearly.

8 TABLE OF CONTENTS Page 1. INTRODUCTION Motivation Summary of dissertation CLASSICAL DUAL RISK MODEL The model Lundberg coefficient Expected discounted penalty function DUAL RISK MODEL WITH RISK-FREE INVESTMENTS The model Integral equation approach Numerical scheme The Laplace transform of the time of ruin DUAL RISK MODEL WITH RISKY INVESTMENTS The consumption/investment process An integro-differential equation Small volatility Large volatility RUIN THEORY WITH DELAYED CLAIMS AND INVESTMENTS Delayed risk process without investments Delayed risk process with risky investments Bounded delay times

9 TABLE OF CONTENTS (Continued) 5.4 Deterministic delays Page 6. CONCLUSIONS AND FUTURE RESEARCH Conclusions Ongoing and future research Perturbed dual risk model with constant interest Dependent risk model with risk-free investments APPENDICES A Poisson process B Confluent hypergeometric functions C Stochastic calculus D Karamata-Tauberian theorem

10 LIST OF FIGURES Figure Page 3.1 Exact and approximate ruin probabilities for exponential mean one gains for 1, 1, 1, and 1 subintervals Ruin probabilities for Uniform[, θ] gains for θ = 1, 2, 4, 8, Ruin probabilities for Pareto(.1, α) gains for α =.5, 1, 2, 3, 4, Laplace transform of time of ruin for exponential mean one gains when δ = 1 8, 1 16, 1 32, 1 64,

11 RUIN PROBLEMS WITH RISKY INVESTMENTS 1. INTRODUCTION 1.1 Motivation A natural measure of risk in the insurance business is the probability of bankruptcy. Also known as collective risk theory, ruin theory is a field in actuarial mathematics which uses stochastic processes to model the surplus of an insurance company. Depending on the nature of the company s business, there are various ways to model its surplus. In annuity or pension insurance, the company is liable to pay annuities but receives a lump sum upon death of the policyholders. On the other hand, a non-life insurance company (e.g. fire, auto, product liability, home, etc.) collects premiums but pays claims upon events covered by the insurance policy. The simplest model in non-life insurance was developed by Lundberg [193] and Cramér [193]. This model is known as the Cramér-Lundberg model or the classical risk model. It assumes that the company collects premiums continuously and pays out claims of random amount at random times. Mathematically, the surplus process can be written as N t U t = u + ct X k, (1.1) where u is the initial capital, c is the constant premium rate, X 1, X 2,... are the claim sizes, and N t is the number of claims in the time interval [, t]. In the classical model, N t is assumed to be a homogenous Poisson process, the claim sizes X 1, X 2,... are independent and identically distributed, and the processes {X i } and N t are independent. k=1

12 2 It is vital for the insurance company to maintain its surplus above a certain level. For convenience, this level is set to be zero. One of the most important object of study in ruin theory is the ruin probability, which is given by ψ(u) := P (U s < for some s > U = u). In general, explicit expressions for ψ are hard to get. Hence, one is often interested in approximations of ψ such as inequalities and its asymptotic behavior. There are various mathematical tools used to study the ruin probability such as Wiener-Hopf factorization (see Cramér [1955]), renewal equations (see Grandell [1991]), Pollaczek-Khinchine formula (see Asmussen and Albrecher [21]), large deviation technique (see Konstantinides and Mikosch [25]), integro-differential equation (IDE) approach (see Albrecher et al. [212]), etc. In the models that are presented here, we will take the IDE approach. An important result in the classical model (1.1) is called the Cramér-Lundberg appoximation, which asserts that the probability of ruin behaves asymptotically like an exponential function, that is, lim u ψ(u)e Ru = constant, where R is the so-called Lundberg coefficient. Therefore, the company can now find a minimum capital u so that ψ(u ) is below a tolerable threshold and satisfies the company s risk appetite (see Borch [1967] and Constantinescu and Lo [213]). If it turns out that ψ(u ) is overestimated, the company can pay part of its reserves as dividends. If ψ(u ) is underestimated, the company may seek a reinsurance alternative to reduce its risk. The classical model serves as a toy example because many computations are straightforward here, however the model is highly unrealistic. According to Borch [1967], some impractical assumptions are: 1. The stationary assumption, which means that the nature of the company s business will never change. 2. The assumption that the probability laws governing the risk process are completely known.

13 3 3. The implicit assumption that once a decision has been made, it cannot be changed (e.g. fixed premium rate). One enhancement of the Cramér-Lundberg model is to consider the possibility of the company investing its surplus, e.g. purchasing stocks or investing in bonds and other risk-less assets. In the classical model, the only inflow of money is from collecting premiums from policyholders. Naturally, the company can invest a portion of their capital to maximize profit. Inspired by mathematical finance, we consider models in which the company invests their capital in an asset whose price evolves according to a geometric Brownian motion. There are two main themes in this dissertation. 1. Incurred but not reported (IBNR) claims are claims that have occurred, but are not known to the insurer yet. There are laws requiring insurance companies to set their IBNR reserves above a certain threshold. To model IBNR claims, we first assume that claims arrive according to a homogeneous Poisson process, similar to the classical model (1.1). Once a claim arrives, it will be delayed by a random time until it is settled. The new point process of interest is now called the settlement process. Together with investments modeled by geometric Brownian motion, we study the ruin probability under this risk model. 2. In life annuity insurance or pension insurance, the surplus process of the company follows a different model. A basic model is called the classical dual risk model, where the surplus process is given by U t = u ct + N t k=1 X k. Here, c denotes the expense rate and {X k } represent random gains. Therefore, this models a company with a constant rate of consumption, earning random income at random times. Other examples of such company are non-profit organization and petroleum company where the jumps correspond to random donations (see Chen [21]) and discoveries of oil (see Avanzi et al. [27]), respectively. In this dissertation, we analyze the ruin

14 probability in the dual risk model under no investments, risk-free investments, and risky investments Summary of dissertation In Chapter 2, we are concerned with the classical dual risk model. This is similar to the Cramér-Lundberg model (1.1), but with both signs reversed. Premiums are regarded as expenses whereas claims are interpreted as gains. Using Wiener-Hopf factorization, we study the probability of ruin, as well as a general discounted expected penalty function. The next chapter treats the dual model with risk-free investments. Since the integrodifferential equation (IDE) for the ruin probability is a finite domain problem, Chapter 3 serves as a platform for numerical computations. We provide a numerical scheme that will approximate the ruin probability for any gain distributions. Chapter 4 contains the main results of this dissertation. We investigate the ruin probability in the dual risk model with investments following geometric Brownian motion. We find that ruin probability decays algebraically as initial capital tends to infinity when the volatility is small. On the other hand, using a new approach, we show that the event of ruin is certain when the volatility is large. Going back to insurance models with risky investments, we model the effects of IBNR claims on the ruin probability in Chapter 5. An integro-partial differential equation (IPDE) is obtained, together with existence and uniqueness theorems. Assuming deterministic delay times, we explore an example in which delaying the settlement of claims will not reduce the probability of ruin.

15 5 Finally, we conclude our current findings and propose some new models for future research in Chapter 6. The appendices provide details regarding concepts and theorems heavily used throughout this dissertation such as Poisson processes, confluent hypergeometric functions, stochastic calculus, and Karamata Tauberian theorem.

16 6 2. CLASSICAL DUAL RISK MODEL We consider the surplus process of a company which incurs expenses at a constant rate and earns random positive gains at random times. Examples of such companies can be found in the literature. In life annuity or pension insurance (see Grandell [1991]), continuous payments are made by the company to the policyholder, and upon death of the policyholder, the gross reserve is available to the company as profit. For companies engaging in research and discoveries, such as petroleum or pharmaceutical companies (see Avanzi et al. [27]), random gains correspond to discovery of oil, or development of a new patent. The same model can also be applied to nonprofit organizations (NPO) where the gains represent random donations (see Chen [21]). The term dual risk model was coined by Mazza and Rulliere [24]. They established a link between the hitting times associated with risk processes and random wave motions in physics. Previously, the dual risk model was called the negative claims model due to the fact that it is obtained by negating the signs of premiums and claims in insurance models. Results concerning the ruin probability in the classical dual model can be found in Cramér [1955], Seal [1969], and Grandell [1991]. The dividend problem in the dual model has been an active research area. The optimal dividend problem was posed by de Finetti [1957], and since then, many variations of the problem have been studied by numerous authors including Avanzi et al. [27], Cheung and Drekic [28], and Ng [21]. Zhu and Yang [28] considered a regime switching dual model and obtained explicit expressions of the ruin probabilities as well as some Lundberg-type bounds. Results regarding the dual renewal (Sparre-Anderson) model can be found in Mazza and Rulliere [24] and Dong and Wang [26]. A variant of dual model with tax payments according

17 7 to a loss-carry forward policy was studied by Albrecher et al. [28]. Yang and Sendova [214b] computed the Laplace transform of the ruin time when the interarrival times have a generalized Erlang-n distribution. Recently, Yang and Sendova [214a] proposed a Gerber-Shiu type function in the dual model, involving the time of ruin, the time of the last gain before ruin, and the amount of the last gain before ruin. In this chapter, we present the classical dual risk model. We investigate several features of this model and introduce the Lundberg equation. Next, we consider an expected discounted penalty function and show that it satisfies a Wiener-Hopf integral equation. Using Fourier transforms, we derive the explicit representation of the ruin probability. 2.1 The model Let (Ω, F, P ) be a probability space. Define the risk process {U t } t on this probability space by N t U t = u ct + X i. (2.1) Here, U = u > is the initial capital and c > is the constant consumption/expense rate. The counting process N t is called the gain arrival process, describing the number of gains in the interval [, t]. We assume that N = and that N t is a homogeneous Poisson process with intensity λ with arrival times {T k } k N. Basic properties of a Poisson process are discussed in Appendix A. i=1 The gains {X i } i N are assumed to be a sequence of independent and identically distributed (i.i.d.) positive random variables with probability density function (p.d.f.) f, cumulative density function (c.d.f.) F, moment generating function (m.g.f.) m(r) = E[e rx ] which exists in some neighborhood around, and finite mean µ. Moreover, {X i } is assumed to

18 be independent of {N t }. Throughout this thesis, the terms gains, innovations, and donations will be used interchangeably. 8 It is vital for any company to operate above a certain income level. For convenience, we set this level to be zero and define the time of ruin by τ = inf{s > U s }. As a convention, we take the infimum of an empty set to be zero, so that if U s > for all t, then τ = inf =. Define the ruin probability with initial surplus U = u by ψ(u) = P (τ < U = u). Using Wald s identity, we observe that ( ) E[U t ] u ct + λtµ lim = lim = λµ c. t t t t As a consequence of the Strong Law of Large Numbers, we have the following three possibilities: 1. If λµ c <, then U t converges to almost surely (a.s.). 2. If λµ c =, then lim sup t U t = = lim inf t U t a.s.. 3. If λµ c >, then U t converges to a.s. and there is a positive probability that U t for all t. To avoid the trivial case ψ 1, we impose the following condition c < λµ, (2.2) which is called the net profit condition, or the negative loading condition. The interpretation of this condition is that on average, gains should be superior to the expenses, otherwise ruin is certain. As a consequence of this condition, lim ψ(u) =. (2.3) u

19 9 2.2 Lundberg coefficient To motivate this section, we observe that the Laplace transform of the surplus process is given by E[e rut ] = e r(u ct) E[e r N t i=1 X i ] = e r(u ct) e λt e λtm( r) = e ru+θ(r)t, where θ(r) = λ(m( r) 1) + cr for all r such that m( r) exists. This suggests that there is a natural exponential martingale associated with the surplus process. Proposition For r R satisfying m( r) <, the stochastic process {e rut θ(r)t : t } is a martingale. Proof. For t > s, E[e rut θ(r)t F s ] = e rus θ(r)t E[e r(ut Us) F s ] = e rus λ(m( r) 1)t rcs E[e r N t Ns+1 X i ] = e rus λ(m( r) 1)t rcs e λ(t s)(m( r) 1) = e rus θ(r)s, where in the second equality we used the fact that U t is a strong Markov process. This martingale has a constant expectation given by E[e rut θ(r)t ] = e ru. Therefore, the surplus process {U t } generates an exponential martingale iff θ(r) =. The equation = θ(r) = λ(m( r) 1) + cr

20 1 has the trivial solution r =. The first derivative θ (r) = λm ( r) + c, and so θ () = λm () + c = λµ + c < by the negative loading condition (2.2). Moreover, the second derivative θ (r) = λm ( r) >, which means that the function θ(r) is convex. Therefore, besides r =, the equation θ(r) = can have at most one additional solution R >. If this R exists, we call this the adjustment coefficient or the Lundberg exponent of the classical dual model. That is, R > satisfies the Lundberg equation given by = λ(m( R) 1) + cr. (2.4) Example Assuming that {X i } i=1 are exponentially distributed, that is, f(x) = 1 µ e x/µ, we have m( r) = 1 1+rµ. So, (2.4) reduces to = cµr2 + (c λµ)r, 1 + Rµ which has one trivial solution and one nontrivial solution given by R = λµ c cµ >. In general, R is difficult to obtain analytically. Numerical methods are often used to find the Lundberg coefficient. Therefore, we provide a lower bound for R. Proposition Suppose µ 2 E[X 2 ] <. Then, R > 2(λµ c) λµ 2. (2.5) Proof. From Kass et. al. (28), we have θ (r) = λe[x 2 e rx ] λµ 2, θ (r) = θ () + θ(r) = θ() + r r θ (s) ds c λµ + λµ 2 r, θ (s) ds + (c λµ)r + λµ 2 r 2 2.

21 11 Therefore, which gives the desired inequality (2.5). = θ(r) (c λµ)r + λµ 2 R 2 2, Example Suppose that λ = 4 and c = 2. Assume that the gains are given by 1 with probability.6, X = 2 with probability.4. The lower bound of R is given by 2(4(1.4) 2) 4(2.2) =.8182, whereas the exact solution to 4(.6e R +.4e 2R 1) + 2R = is given by R using numerical root solver. 2.3 Expected discounted penalty function In the classical insurance model, the seminal work by Gerber and Shiu [1998] builds around the study of the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The so-called Gerber-Shiu function Φ(u) = E[w(U τ, U τ )e δτ 1 [τ< ] U = u] is shown to satisfy a renewal equation. Here, w is interpreted as some kind of penalty when ruin occurs and δ represents force of interest. Various quantities are obtained by choosing suitable penalty functions. When w 1, Φ(u) is the Laplace transform of the time of ruin, and setting δ =, one obtains the ruin probability ψ(u). Setting w(x, y) = 1 e ρy δ where ρ is the Lundberg coefficient, Φ(u) is the expected present value of some deferred annuity. Another interesting example also arises in the context of option pricing. In the classical dual model, two of the three random variables above are identical. Ruin is caused by the continuous consumption at rate c. Hence, the surplus immediately before

22 12 ruin and the deficit at ruin are both equal to zero. Cheung [212] proposed to study the function m(u) = E[e δt w( U T )1 [τ< ] U = u], where T is the time of the first jump after ruin and U T represents the absolute value of the shortfall immediately before the first gain after ruin. Recently, another variant of the Gerber-Shiu function was proposed by Yang and Sendova [214a]. They considered a function of the time of ruin (τ), the time of the last jump before ruin (denoted by T ), and the amount of the surplus after the last jump (U T = U T + ). Specifically, they defined the expected discounted penalty function by Φ(u) = E[e δt w(u T )1 [τ< ] U = u], where δ is the discount factor and the penalty w : [, ) R is a measurable function. All three random variables are linked by the equation τ = T + U T c. Choosing w(x) = e δx/c, one obtains Φ(u) = E[e δτ 1 [τ< ] U = u] which is the Laplace transform of the time of ruin. Choosing w(x) = x k, one recovers Φ(u) = E[e δt U k T 1 [τ< ] U = u] which is the discounted moments of the surplus at the time of the last jump. We first show that Φ(u) satisfies an integro-differential equation (IDE). Theorem The expected discounted penalty function satisfies the following IDE: cφ (u) + (λ + δ)φ(u) = λ Φ(u + x)f(x) dx + (δw(u) + cw (u))e λu/c. (2.6) Proof. To obtain the IDE, we first condition on the time and the amount of the first jump. The first gain can occur before or after time u c. If the first gain happens after u c,

23 13 the company is ruined and so T = here. Therefore, Φ(u) = u/c λe λs Φ(u cs + x)f(x) dx e δs ds + u/c λe λs w(u ) ds = λ u c e (λ+δ)u/c e (λ+δ)v/c Φ(v + x)f(x) dx dv + w(u)e λu/c. Applying the operator ( d du + λ+δ ) c to the equation above, we get Φ (u) + λ + δ Φ(u) = λ c c which is the desired IDE (2.6). Φ(u + x)f(x) dx + w (u)e λu/c + δ c w(u)e λu/c, Remark Yang and Sendova [214a] derived the above IDE with specific penalties w(x) = e δx/c and w(x) = x k. The IDE above can be written as an integral equation (IE). Denoting Ξ(u) = Φ (u), and utilizing the limiting property lim u Φ(u) =, we can rewrite the IDE as c Ξ(u) δ or equivalently u c Ξ(u) = Ξ(s) ds = λ u u+x u Ξ(s) ds f(x) dx + (δw(u) + cw (u))e λu/c, (λ F (s u) + δ)ξ(s) ds + (δw(u) + cw (u))e λu/c, where F (x) = 1 F (x) is the tail distribution of X. Letting K(t) = 1 c (λ F ( t) + δ)1 [t<] and g(u) = ( δ c w(u)+w (u))e λu/c, we obtain the following non-homogeneous Wiener-Hopf integral equation Ξ(u) = K(u s)ξ(s) ds + g(u). (2.7) Using Fourier transform, one can solve the above IE for Ξ(u). Finally, the boundary condition Φ() = w() (provided that lim u + w(u) = w() exists) is used to obtain Φ(u). Now, we apply this technique to find an expression for the ruin probability. In the Cramér- Lundberg model (1.1), an explicit expression for the ruin probability cannot be obtained

24 14 in general and has to be computed in a case-by-case basis depending on the p.d.f. of the claims. However, in the classical dual model (2.1), the ruin probability has an explicit form given by the following theorem. This theorem can be found in most papers regarding dual risk models. Cramér [1955], Seal [1969], and Takacs [1969] have derived this ruin probability result by exploiting the fact that the homogeneous Poisson process has independent increments, or the connection between the classical and dual risk models. Ng [29] used martingale arguments to prove the same result. We present a new approach to obtain this well-known result. Although a more general version of the following theorem is valid (see e.g. Cramér [1955], Grandell [1991], and Ng [29]), we assume that the gains are exponentially distributed so that Fourier inversion can be done easily. Theorem Assume that the gains are exponentially distributed with mean µ. Let R be the Lundberg coefficient satisfying (2.4). Then, ψ(u) = e Ru. (2.8) Proof. When w(y) 1 and δ =, we have Φ(u) = ψ(u). The IDE (2.6) reduces to a homogeneous Wiener-Hopf integral equation of second kind Ξ(u) = K(u s)ξ(s) ds, < u <, (2.9) where Ξ(u) = ψ (u) and K(t) = λ F c ( t)1 [t<]. In solving the above equation, we follow closely with the notation provided by Masujima [25]. First, the Fourier transform of the kernel K(t) is given by ˆK(z) = λ c = λ c = e izt K(t) dt F ( t)e itz dt [ F ( t) e itz iz + e itz iz λf( t) dt λ + λm(iz). (2.1) ciz ]

25 Although Ξ(u) is known only for positive u, we extend the definition of Ξ(u) for negative 15 u. Let Ξ + Ξ(u), for u >, (u) =, for u <, so that (2.9) can be written as Ξ + (u) = For u <, define the function Ξ (u) by Ξ (u) = K(u s)ξ + (s) ds, u >. (2.11) K(u s)ξ + (s) ds, u <, (2.12) and define Ξ (u) = for u >. We can now rewrite (2.11) and (2.12) as Ξ + (u) + Ξ (u) = K(u s)ξ + (s) ds, < u <. (2.13) Taking the Fourier transform of the above equation and rearranging terms, we obtain (1 ˆK(z)) ˆΞ + (z) = ˆΞ (z). The next step is essential in solving the equation. We perform Wiener-Hopf factorization to the function (1 ˆK(z)), that is, by rewriting it as (1 ˆK(z)) = Y + (z) Y (z), where Y + (z) is analytic in the lower half plane and Y (z) is analytic in the upper half plane. This factorization can be analytically obtained by the Cauchy integral formula. By Masujima [25], this factorization can also be done via inspection, by assigning poles or zeros in the lower half plane to Y and poles or zeros in the upper half plane to Y +.

26 From this point on, we will work with exponentially distributed gains with mean µ. From 16 (2.1), (1 ˆK(z)) = 1 λ ( ) 1 ciz 1 izµ 1 = 1 izµ λµ c 1 izµ = iz + R iz 1/µ, where R = λµ c cµ as defined in Example The poles and zeros of (1 ˆK(z)) are z = 1 µ i and z = Ri, respectively. So we assign We now have Y = iz 1/µ and Y + = iz + R. Y + (z)ˆξ + (z) = Y (z)ˆξ (z), (2.14) where the left and right hand sides of (2.14) are analytic in the lower and upper half plane, respectively. Since there is a common strip of analyticity and since analytic continuation is unique, it follows that there is a unique entire function that coincides with (2.14). Moreover, Liouville s theorem (see Polyanin and Manzhirov [28]) implies that where A is a constant. Hence, and by Fourier inversion formula, Y + (z)ˆξ + (z) = Y (z)ˆξ (z) = A ˆΞ + (z) = A Y + (z) = A iz + R Ξ(u) = Ae Ru, u. Finally, we observe that if we begin the surplus process with zero initial capital, we will be instantly ruined, that is, τ =. Therefore, we obtain the initial condition Φ() = ψ() = 1, which yields Φ(u) = ψ(u) = e Ru.

27 17 Remark In the spirit of this thesis, we take a slightly different route to prove Theorem 2.3.2, that is, by working directly with the integral equation and solving it via Wiener-Hopf factorization. The same technique was employed by Cramér [1955] to obtain similar result. We point out that Cramér took the Fourier transform in the time variable of the finite time ruin probability, whereas in this chapter, we take the Fourier transform in the space variable of the derivative of the infinite time horizon ruin probability. Remark In the proof of Theorem 2.3.2, we assume that gains are exponentially distributed so that the Wiener-Hopf factorization for (1 ˆK(z)) can be done via inspection. In general, depending on the m.g.f. of the gains, we can write (1 ˆK(z)) = j (z z+ j )h+ j (z) k (z z k )h k (z), where {z j + } are poles and zeros in the upper half plane, {z k } are poles and zeros in the lower half plane, and h + j (z) and h k (z) are holomorphic functions on the lower and upper half planes, respectively. The Wiener-Hopf factorization can analytically be obtained by the Cauchy integral formula (see Masujima [25]). As a concluding remark, Yang and Sendova [214a] provided an explicit expression for Φ when the penalty function is a power function. When w(u) = u k, (2.7) becomes Ξ(u) = K(u s)ξ(s) ds + (δu k + cku k 1 )e λu/c. Using the method of undetermined coefficient, a particular solution is given by Ξ p = k a i u k i e λu/c, i= where a,..., a k are some constants. Although not done here, we can obtain similar results by solving the non-homogeneous Wiener-Hopf integral equation.

28 18 3. DUAL RISK MODEL WITH RISK-FREE INVESTMENTS The classical model in non-life insurance (1.1) assumes that the only source of income is from collecting premiums. In the past, many models incorporated investments with constant force of interest, for example, investing all (or part) of the surplus in bonds or time accounts. The study of these risk models dated back to Segerdahl [1942], who provided an explicit expression for the ruin probability when the claims are exponentially distributed. Sundt and Teugels [1995] gave an extensive treatment of the ruin probability with constant interest force. Cai et al. [29] considered the well-known Gerber-Shiu function under this risk model with interest rates, and more recently, Schmidli [215] studied a variant of the discounted penalty function where a penalty applies when the surplus process leaves a finite interval. Yang and Wang [21] investigated the asymptotic behavior of the ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims. Renewal risk models with constant interest was well studied by Konstantinides et al. [21]. Contrary to the vast literature on the insurance models, there were very few results published in the dual model with constant force of interest. Dong and Wang [28] studied the ruin probability under the renewal model. Zeng and Xu [213] considered the perturbed dual risk model with constant interest and a threshold dividend strategy. They used the sinc method to approximate the expected present value of total dividends. In this chapter, we examine the ruin probability numerically. We first rewrite the IDE as an IE. Upon discretization, the IE reduces to a linear matrix equation. The ruin

29 19 probability can then be obtained numerically for any jump distributions. For numerical illustrations, we consider exponential, uniform, and Pareto gains. As a concluding remark, we show that the same method can be applied to other functionals of the time of ruin, e.g. the Laplace transform of the time of ruin. 3.1 The model We retain all assumptions in the previous classical model (2.1) and further assume that the company invests all of its surplus in a risk-free asset with constant force of interest a >. The risk process can now be written as U t = u ct + a t N t U s ds + X i. (3.1) For convenience, we define the consumption/investment process {Z t } by Z t = u ct + a or equivalently, in differential form, t i=1 Z s ds dz = (az c)dt, Z = u. Observe that {Z t } is a deterministic process and that the above is an ODE with solution Z t = ( u c ) e at + c a a. Since Z t for all t whenever u c a, we have that ψ(u) = for all u c a. (3.2) In other words, if the initial capital is greater than c a, the contribution from the risk-free investments always offsets the expenses, and so the company can never be ruined. Hence, in this chapter, we only consider ψ(u) for values of u in the interval (, c a).

30 Theorem Assume that the ruin probability ψ(u) is differentiable. For < u < c a, the ruin probability satisfies the following integro-differential equation: with boundary conditions (au c)ψ (u) λψ(u) + λ c/a u ( c ψ() = 1 and ψ =. a) 2 f(x)ψ(u + x) dx = (3.3) Proof. We consider the risk process U t in the time interval (, h) where h < 1 a So, By the chain rule, ψ(u) = E(1 [τ< ] 1 [N(h)=] U = u) + E(1 [τ< ] 1 [N(h)=1] U = u) + E(1 [τ< ] 1 N(h)>1] U = u) = e λh ψ(z h ) + h ln c c au c/a Zs λe λs ψ(z s + x) df (x) ds + o(h). (3.4) ψ(z h ) ψ(u) lim = ψ (u) a(u c/a). h h Dividing (3.4) by h and taking the limit as h, we obtain c/a u = (au c)ψ (u)e + ψ(u) ( λ) + λe ψ(u + x)df (x).. The IDE (3.3) can be solved explicitly when the gains follow certain distributions, e.g. exponential or mixture of exponentials, as exemplified in Dong and Wang [28]. We present a different method in obtaining the ruin probability in these two examples in the next section. 3.2 Integral equation approach In this section, we obtain a simple integral equation for the derivative of ψ. This provides a framework for numerical approximations of the ruin probability, which is shown

31 21 in the subsequent section. For u, define ψ (u), u c a χ(u),, u > c a. By (3.2), for u c a, we have λψ(u) λ = λ = λ = λ = λ c/a u c/a u f(x)ψ(u + x) dx u+x u u f(x)(ψ(u) ψ(u + x)) dx + u f(x) t u χ(t) dt dx f(x) dx χ(t) dt F (t u) χ(t) dt. c/a u f(x)(ψ(u) ) dx Since we are only concerned with the values of u between and c a, the IDE (3.3) becomes an IE given by c/a (au c)χ(u) = λ u F (t u) χ(t) dt, < u < c a. Therefore, with c a = b and λ a = d, the IE of interest can be rewritten as χ(u) = with the integrability condition d b b u u b F (t u) χ(t) dt, < u < b, (3.5) χ(u) du = ψ(b) ψ() = 1. (3.6) Performing the change of variable χ(u) = (b u) d 1 χ(u), (3.5) becomes (b u) d 1 χ(u) = d b b u u F (t u) (b t) d 1 χ(t) dt, < u < b.

32 22 Therefore, and so we get b (b u) d χ(u) = F (t u) χ(t) d u dt (b t)d dt b = F (t u) χ(t)(b t) d u b + (b t) d d [ ] F (t u) χ(t) dt dt = b u (b t) d d dt u [ F (t u) χ(t) ] dt, < u < b. (3.7) We now illustrate two examples in which explicit expressions of ψ are obtained using (3.7). The solutions are identical with the ones found in Dong and Wang [28]. Example Assume that the gains are exponentially distributed with mean µ, that is, f(x) = e x/µ µ. Then, for u c/a, where is the incomplete Gamma function. ψ(u) = Γ( λ a, ) Γ( λ a, c au aµ ) Γ( λ a, ) Γ( λ a, c aµ ), Γ(b, x) := Proof. Since F (x) = e x/µ, (3.7) reduces to b = e u/µ (b t) d d dt u x t b 1 e t dt [ ] e t/µ χ(t) dt. Since the above holds for all u, the integral term must be equal to zero. Differentiating this equation leads to which reduces to the differential equation (b u) d d [ ] e u/µ χ(u) =, du χ (u) 1 χ(u) =, µ with solution given by χ(u) = e u/µ.

33 23 Therefore, χ(u) = (b u) λ/a 1 e u/µ. We now use the boundary conditions to determine ψ(u). For some (negative) constant K, we get ψ(u) = u = K = K Ke s/µ (c/a s) λ/a 1 ds + ψ() c/a c/a u c/(aµ) (c au)/(aµ) exp((c/a t)/µ) t λ/a 1 dt + 1 e y (µy) λ/a 1 dy + 1 = ˆK(Γ(λ/a, (c au)/(aµ)) Γ(λ/a, c/(aµ))) + 1, where K and ˆK are some constants. We know that ψ(c/a) =, so ˆK = This gives the desired expression 1 Γ(λ/a, c/(aµ)) Γ(λ/a, ). ψ(u) = Γ( λ a, ) Γ( λ a, c au aµ ) Γ( λ a, ) Γ( λ a, c aµ ). We observe that as a, we obtain the ruin probability in the classical case, since lim[1 ψ(u)] = lim a a = u (1 ax/c)λ/a 1 e x/µ dx c/a (1 ax/c) λ/a 1 e x/µ dx u e λx/c e x/µ dx e λx/c e x/µ dx = 1 e ( λ/c+1/µ)u, and so where R = λµ c cµ lim ψ(u) = a e Ru, > is the Lundberg coefficient, as seen in Example

34 Example Assume that the gains are distributed as a mixture of exponentials, i.e. f(x) = pαe αx + (1 p)βe βx where α > β. The derivative of the ruin probability is given by where Ψ 1 (u) = e αu ( c a u ) λ/a 1 Ψ 2 (u) = e αu ( c a u ) λ/a 1 U χ(u) = c 1 Ψ 1 (u) + c 2 Ψ 2 (u), ( (1 p)λ 1F 1, λ ( c ) ) a a, (α β) a u, ( (1 p)λ, λ ( c ) ) a a, (α β) a u. The functions 1 F 1 and U are known as confluent hypergeometric functions (see Appendix B). The boundary conditions are then used to obtain the exact expression for the ruin probability, as seen in Dong and Wang [28]. 24 Proof. First, (3.7) can be written as b = pe αu (b t) d d [ e αt χ(t) ] b [ ] + (1 p)e βu (b t) d d e βt χ(t) u pe αu I 1 (u) + (1 p)e βu I 2 (u). (3.8) u Observe that I 1(u) = (b u) d [e αu χ (u) αe αu χ(u)], I 1 (u) = d(b u) d 1 e αu [ χ (u) α χ(u)] (b u) d e αu [ χ (u) 2α χ (u) + α 2 χ(u)], and that similar computations hold for I 2 (u). Now, differentiating equation (3.8) once and substituting (3.8) again, we have = (α β)pe αu I 1 (u) + pe αu I 1(u) + (1 p)e βu I 2(u), or equivalently, (β α)pi 1 (u) = pi 1(u) + (1 p)e (β α)u I 2(u).

35 25 Differentiating the above yields (β α)pi 1(u) = pi 1 (u) + (1 p)(β α)e (β α)u I 2(u) + (1 p)e (β α)u I 2 (u), which gives a second order linear ODE (b u) χ (u) + ((b u)(α + β) + d) χ (u) + ( αβ(b u) d(αp + β(1 p))) χ(u) =. From Polyanin and Zaitsev [22], the solution to the above ODE is given by e αu J ((1 p)d, d; (α β)(b u)), where J (p, q; z) is an arbitrary solution of the confluent hypergeometric equation zy (z)+ (q z)y (z) py(z) =. Hence, χ(u) = (b u) d 1 χ(u) = (b u) d 1 e αu J ((1 p)d, d; (α β)(b u)), and we are done. In general, an explicit expression for ψ can be obtained if the p.d.f. of the gains satisfies an ODE with constant coefficients, such as exponential or gamma distributions (see Examples and 3.2.2). If the p.d.f. is not of this form, for example, uniform or Pareto gains, explicit formulas are difficult to get. This leads us to the next section, which aims to provide numerical approximations for ψ. 3.3 Numerical scheme Here, we present a numerical framework for the ruin probability under any gain distributions. The numerical scheme requires solving a simple linear system A χ = b where A and b are to be determined and the vector χ consists of values of the function χ(u) evaluated at some discrete points.

36 26 Consider the discretization = u < u 1 < < u N 1 < u N = b. For each j = 1,..., N 1, (3.7) can be written as = b u j (b t) d d dt [ F (t u j ) χ(t)] dt. For each fixed u j, we further discretize in the t variable where the length of t coincides with the length of u. We approximate the derivative term using forward difference method. Defining χ i χ(u i ), we arrive at the following system of equations N 1 = (b u k ) d [ F (uk+1 u j ) χ k+1 F ] (u k u j ) χ k, j = 1,..., N 1. k=j We rewrite the above as = (b u j ) d χ j + N k=j+1 F (u k u j ) χ k [(b u k 1 ) d (b u k ) d ], j = 1,..., N 1. The integrability condition (3.6) can be approximated by N 1 (b u i ) d 1 χ i = 1. i=1 Therefore, we obtain the matrix equation A χ = b given by (b u 1 ) d a a 1,N 1 a 1N (b u 2 ) d... a 2,N 1 a 2N (b u N 1 ) d a N 1,N (b u 1 ) d 1 (b u 2 ) d 1... (b u N 1 ) d 1 χ 1 χ 2. χ N 1 χ N =.. (3.9) 1 That is, χ = [ χ 1,, χ N ] T, b = [,,, 1] T, and A = (a jk ) where for each j = 1,..., N 1, the entry a jk = F (u k u j )[(b u k 1 ) d (b u k ) d ], k = j + 1,..., N. Moreover, a jj = (b u j ) d, j = 1,..., N 1

37 27 and a Nk = (b u k ) d 1, k = 1,..., N 1. It is vital to check that the matrix A is invertible. To see this, we first perform row operations R i R i (b u i )R N for each i = 1,..., N 1, to get a matrix with zero diagonal entries but non-zero off-diagonal entries. We then perform further row operations to reduce the matrix to the form... c 1... c c N 1 which is a companion matrix, hence possessing characteristic polynomial p(t) = c + c 1 t + c N 1 t N 1 + t N. Since c, the determinant of this companion matrix is not zero. The above row operations affect the determinant of A by a constant multiple, hence we conclude that det(a). Thus, χ = A 1 b. To recover φ, for each j = 1,..., N 1, we multiply the jth component of χ by (b u j ) d 1. Since χ = ψ, we perform numerical integration ψ j = N k=j χ i to recover ψ. Finally, we normalizing the vector ψ in order to impose the boundary condition ψ() = 1. We emphasize that this numerical scheme can be used for any gain distributions. We now present three examples with different gain distributions. Naturally, our first example concerns the exponentially sized gains.

38 28 Subintervals Maximum Error TABLE 3.1: Maximum error for exponential mean one gains for 1, 1, 1, and 1 subintervals. Example Suppose that F (x) = e x. Let c a = b = 4 and λ a = d = 3.5. In the previous section, we obtained the exact expression ψ(u) = Γ(1.5, ) Γ(1.5, 4 u). Γ(1.5, ) Γ(1.5, 4) Figure 3.1 shows that the numerical approximation of ψ approaches the exact solution as we increase the number of subintervals. For error analysis, we use the maximum error which is the largest absolute difference between the approximation and the true value of ψ. Table 3.1 suggests that this numerical scheme has a first order of accuracy, which is unsurprising since we are using first order discretization for the integral term as well as for the derivative term.

39 FIGURE 3.1: Exact and approximate ruin probabilities for exponential mean one gains for 1, 1, 1, and 1 subintervals. 29

40 3 We present two more examples in which the probability distribution functions of the gains do not satisfy any ODE with constant coefficients and the exact expression for the ruin probability is not known. In both examples, varying one of the parameters in the model leads to some interesting analytic results. Our next example is a simple distribution with a piecewise continuous p.d.f., namely the uniform distribution. Example Consider uniformly distributed gains on the interval [, θ]. Then, 1, x F (x) = 1 x θ, < x θ, x > θ. For the numerical experiment, we set c a = b = 4, λ a = d = 1.5, and N = 1. Figure 3.2 shows that the numerical approximation of ψ approaches a limiting curve as we increase the parameter θ. This limiting curve can be found by investigating the IDE (3.3). For each parameter θ, denote the associated ruin probability by ψ θ. Then, the IDE is given by Now, (au c)ψ θ (u) λψ θ(u) + λ λ θ c/a u c/a u 1 [<x<θ] ψ θ (u + x) dx = λ θ 1 θ 1 [<x<θ]ψ θ (u + x) dx =. min{c/a,θ} u ψ θ (y) dy as θ by Dominated Convergence Theorem since < ψ θ < 1. Hence, as θ, the IDE becomes (au c)ψ (u) λψ (u) =, and so the solution of the above which satisfies the boundary conditions ψ() = 1 and ψ(b) = is given by ψ (u) = ( 1 u b ) λ/a, < u < b. We observe that there is a natural monotonicity in this case. If θ 1 < θ 2, then ψ θ1 (u) > ψ θ2 (u) for all < u < b. This property agrees with the model, since larger gains will reduce the probability of ruin.

41 FIGURE 3.2: Ruin probabilities for Uniform[, θ] gains for θ = 1, 2, 4, 8,

42 32 FIGURE 3.3: Ruin probabilities for Pareto(.1, α) gains for α =.5, 1, 2, 3, 4, 5. Our last example features a heavy tail gain distribution, which is a distribution whose tail is not exponentially bounded. Recall that a Pareto distribution has two parameters, namely a scale parameter x m and a shape parameter α >, which is sometimes known as the tail index. Its tail distribution is given by 1, x x m F (x) = ( xmx ) α, x > xm. Example For the numerical illustration, we set c a = b = 4, λ a = d = 3.5, and N = 1. Figure 3.3 shows that the numerical approximation of ψ approaches a limiting curve as we increase the parameter α. As α, X converges to a constant random variable x m, that is, the p.d.f. f(x) δ xm. Hence, from the IDE (3.3), we obtain the

43 33 following delay differential equation (au c)ψ (u) λψ(u) + λψ(u + x m ) =, < x m < b u, (3.1) (au c)ψ (u) λψ(u) =, x m > b u. (3.11) For u > b x m, the solution is given by ψ(u) = ( 1 u b ) λ/a. Once this is known, (3.1) become a linear nonhomogenous first order ODE which is then solvable for b 2x m < u < b x m. We can then proceed inductively to obtain the solution to (3.1). 3.4 The Laplace transform of the time of ruin The same procedure can be applied to other functionals of the time of ruin. Denote the Laplace transform of the time of ruin by Φ(u) = E[e δτ 1 [τ< ] U = u]. We first find an IDE for Φ. The proof that is used here is different than the proof in the Theorem From the IDE of Φ, one recovers the IDE for the ruin probability simply by setting δ =. Theorem The Laplace transform of the time of ruin satisfies the following IDE: = (au c)φ (u) (λ + δ)φ(u) + λ b u Φ(u + y) f(y) dy. Proof. Let b = c/a. Recall that the investment process is given by Z t = (u b)e at + b and ( ) that the solution to the equation Z t = is given by t = 1 a ln b b u. Conditioning on the time and amount of the first gain, we get Φ(u) = = λ a t u b Zt λe (λ+δ)t Φ(Z t + y)f(y) dy dt + e (λ+δ)t ( b u b v ) λ+δ a b v Φ(v + y)f(y) dy 1 b v dv + ( b u b ) λ+δ a.

44 34 Applying the operator ( (b u) d du + λ+δ ) a to the above equation, we obtain (b u)φ (u) + λ + δ a Φ(u) = λ a b u Φ(u + y) f(y) dy. Letting Ξ(u) = Φ (u) and using the fact that lim u Φ(u) =, we have (au c)ξ(u) = δφ(u) + λ = δ u u Ξ(t) dt + λ u+y u Ξ(t) dt f(y) dy F (t u) Ξ(t) dt. Therefore, following the steps in the previous section, with Ξ(u) = (b u) d 1 Ξ(u), we obtain = b u (b t) d d dt [ F (t u) Ξ(t) ] with the integrability condition b b δ dt + u a (b t)d 1 Ξ(t) dt, < u < b (3.12) (b u) d 1 Ξ(u) du = 1. Similar discretization yields the matrix equation B Ξ = b where the matrix B = A + Ā. Here, A is the matrix given in the previous section, (b u 1 ) d 1 (b u 2 ) d 1... (b u N 1 ) d 1 (b u 2 ) d 1... (b u N 1 ) d 1 Ā = ,... (b u N 1 ) d 1... Ξ = [ Ξ 1, Ξ 2,..., Ξ N ] T, and b = [,,, 1] T. For a numerical example, we consider exponentially distributed gains, where F (x) = e x. Let c a = b = 4, λ a = d = 3.5, and n = 1. Figure 3.4 shows that the numerical approximation of Φ approaches the ruin probability as we let the Laplace transform parameter δ.

45 35 FIGURE 3.4: Laplace transform of time of ruin for exponential mean one gains when δ = 1 8, 1 16, 1 32, 1 64,

46 In summary, this chapter provides a numerical framework to study numerical approximations of the ruin probability as well as the Laplace transform of the time of ruin when the gain distribution is arbitrary. Although minimizing numerical error is not the focus here, we point out that other numerical schemes can possibly yield better results instead of the current left Riemann sum. Finally, another straightforward example which is not done here is when the gains have probability mass function given by a, with probability p, P (X = x) = b, with probability 1 p, where a << b and p >>. This example has a meaningful economic interpretation since it represents a firm with frequent small donations with sparse large gifts. 36

47 37 4. DUAL RISK MODEL WITH RISKY INVESTMENTS For a complete survey of ruin problems with general investment strategies, see Paulsen [28] and Paulsen [1998], and references therein. We are specifically interested in investments modeled by geometric Brownian motion. Ruin problems with Poisson claim arrival process were studied by Frolova et al. [22] and Pergamenshchikov and Zeitouny [26]. Albrecher et al. [212] and Constantinescu and Thomann [213] established a general IDE and investigated asymptotic behavior of the ruin probability under renewal processes. In this chapter, the analysis of the ruin probability is divided into two cases: small volatility and large volatility. When the volatility is small, we show that ruin probability decays algebraically using analytic techniques such as Laplace transform, Frobenius method, and Karamata-Tauberian theorem. In the case of large volatility, we show that ruin is inevitable by treating the integral equation as an eigenproblem. 4.1 The consumption/investment process Here, we assume that the entire surplus is invested in a risky asset which follows a Geometric Brownian motion, that is, the price of the asset satisfies dr(t) = ar(t) dt + σr(t) dw (t), where a and σ are constants and {W (t), t > } is a standard Brownian motion independent of N t. Then, the surplus can be written as N t U t = u ct + X i + a i=1 t U s ds + σ t U s dw (s). (4.1)