# An application of fractional differential equations to risk theory

## Abstract

This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to construct fractional integro-differential equations for the ruin probabilities in collective renewal risk models, with inter-arrival time distributions from the aforementioned family. Gamma-time risk models and fractional Poisson risk models are two specific cases among them, whose ruin probabilities have explicit solutions when claim size distributions exhibit rational Laplace transforms.

## Keywords

Ruin probability Fractional differential operator Collective risk model## Mathematics Subject Classification (2010)

62P05 60K05 26A33## JEL Classification

G22 C02 G33## 1 Introduction

The concept of first passage time is widely used in financial mathematics and actuarial science. It could model various things, from the time to dividend payments of a stock to the exercise date of an American put option or the ruin probability of an insurance company. In this paper, we focus on the *ruin time of an insurance business*, namely the first time in which the business surplus (capital) becomes negative. Our analysis is aimed at solving equations for the probability of ruin expressed as a function of the initial capital (surplus) of the risk process.

*surplus*\(R(t) \) of an insurance company over time as

*initial capital*and \(c>0\) is the

*premium rate*. The claims occur randomly. The positive random variable \(X_{i} \) describes the size of the \(i \)th claim, which happened after waiting \(T_{i} \) units of time since the last claim. The quantity \(N(t) \) gives the number of claims that have happened up to time \(t \). In the classical model (1.1), dating back to Lundberg [25, 26], Cramér [11], all random variables are assumed independent and identically distributed. Moreover, the waiting times are usually assumed to be exponentially distributed, with the resulting counting process \(N\) thus being a Poisson process. The

*ruin probability*of this compound Poisson risk model, for an initial capital \(u\), is defined as

**This notation is used throughout the paper.**

There is a large actuarial literature analysing renewal risk processes. Expressions for the Laplace transform of the ruin probability for risk models with \(\text{Erlang}(2, \beta )\) or mixture of 2-exponential waiting times were derived in Dickson [12] and Dickson and Hipp [13, 14] as solutions of second-order differential equations. Lin and Willmot [24] calculated the joint and marginal moments of the time of ruin, the surplus before ruin and the deficit at ruin, whenever the inter-arrival time distributions have rational Laplace–Stieltjes transforms. Subsequently, Dufresne [16] computed the Laplace transform of the non-ruin probability for inter-arrival time distributions exhibiting rational Laplace transforms. Li and Garrido [23] used a similar approach as Gerber and Shiu [18] to derive a defective renewal equation for the expected discounted penalty due at ruin in a risk model with \(\text{Erlang}(n)\) inter-arrival times. Finally, Chen et al. [8] derived linear ordinary differential equations for ruin probabilities in Poisson jump-diffusion processes with phase-type jumps and obtained explicit results in a few instances. The common thread of these papers consists of deriving the ruin probabilities as solutions of (integro-)differential equations.

In an attempt to develop a general method, Rosenkranz and Regensburger [29, 30] introduced two algebraic structures for treating integral operators in conjunction with derivatives, integro-differential operators and integro-differential polynomials. Their method allows the description of the associated differential equations, boundary conditions and solution operators (Green operator) in a uniform, yet formal language. Their algebraic symbolic structures have immediate applications in ruin theory. For instance, as an extension of the Erlang risk model, Albrecher et al. [2] transformed the integral equation for the expected-discounted-penalty-due-at-ruin function into an integro-differential equation whenever the inter-arrival time distributions have rational Laplace transforms. Rational Laplace transform densities are equivalent to densities that are solutions of ordinary differential equations with constant coefficients. If the claim size distributions also have rational Laplace transforms, these integro-differential equations can be further reduced to linear boundary value problems. Their symbolic computation approach permits extensions to models with premia dependent on reserves (also discussed in Djehiche [15] regarding the upper and lower bounds of finite-time ruin probabilities), the associated boundary problems then involving linear ordinary differential equations with variable coefficients; see Albrecher et al. [1]. A similar duality idea has been studied in Kolokoltsov and Lee [21] and the references therein.

We show that the probability density function of a sum of independent, heterogeneous gamma and Mittag-Leffler random variables satisfies a fractional differential equation, which we write in an operator/symbolic form. As an application, we consider a family of risk models with inter-arrival times from this family of distributions and derive the corresponding fractional integro-differential equations satisfied by the corresponding ruin probabilities. We consider the case of claim sizes described by sums of heterogeneous gamma random variables and show that the corresponding ruin probabilities solve fractional differential equations with constant coefficients. These equations contain both left and right fractional differential operators. We also remark that Eq. (3.6) presented in this paper can be seen as a generalised case of the fractional boundary problems treated by Jin and Liu [20], where critical point theory is used to analyse the fractional differential equations with Dirichlet boundary conditions.

The gamma-time risk model considered here is the first generalisation of the case of \(\text{Erlang}( n )\)-distributed waiting times considered in Li and Garrido [23] to that of waiting times distributed as \(\Gamma (r, \lambda )\), \(r \) being now any positive real number. This is of significance since in practice, parameter estimation methods usually yield non-integer-valued shape parameters for the gamma distributions that best fit the available data. It thus becomes necessary to study the ruin theory related to real-valued gamma-distributed random variables. Thorin [33] dealt with a special case of claims with a non-integer shape gamma \(\Gamma (1/b,1/b)\), \(b>1\), distribution, and Constantinescu et al. [10] provided three equivalent expressions for ruin probabilities in a Cramér–Lundberg model with gamma-distributed claims. The fractional Poisson risk model has been previously treated in Beghin and Macci [5] and Biard and Saussereau [6] for exponential claim sizes, but here, via this fractional calculus approach, we are able to derive expressions for the ruin probability for a larger class of claim sizes in fractional Poisson models.

The paper is organised as follows. In Sect. 2, we introduce the concept of fractional integro-differential operators. In Sect. 3, we present the main result, and finally, in Sect. 4, we perform some illustrative numerical calculations and compare the behaviour of the ruin probabilities as a function of the model parameters, for both the gamma-distributed waiting times and the fractional Poisson risk models. Appendix A contains all necessary background on fractional calculus.

## 2 Fractional integro-differential operators

### 2.1 Left and right fractional differential operators

In order to generalise (2.1), it is necessary to explore the world of fractional calculus. Solving fractional differential equations has become an essential issue as fractional-order models appear to be more adequate than previously used integer-order models in various fields. A large number of available analytical methods for solving-fractional order integral and differential equations is discussed in Podlubny [28, Chaps. 5 and 6], including the Mellin transform method, the power series method and the symbolic method.

The symbolic method generalises the Laplace transform method. It uses a specific expansion (e.g. binomial or geometric) on the differential operator and writes it as an infinite sum of fractional derivatives. However, it is always necessary to check the validity of the formal expansion since the interchange of infinite summation and integration requires justification. It is nevertheless a powerful tool for determining the possible form of the solution, and there are numerous examples of the application of this method to heat and mass transfer problems.

### Definition 1

*left fractional differential operator (LFDO)*${}_{a}{}^{\alpha}\mathrm{R}_{x}^{r}$ is defined by

*right fractional differential operator (RFDO)*${}_{x}{}^{\alpha}\mathrm{R}_{b}^{r}$ by

In the case \(a=0 \), integration by parts yields the following characterisation of the formal adjoint of ${}_{0}{}^{\alpha}\mathrm{R}_{x}^{r}$. Along with the integration by parts formula in (A.3), this is the key calculation needed for the proof of our main result.

### Proposition 2

*Let*\(\alpha \in \mathbb{R} \)

*and*\(r>0 \).

*The formal adjoint with respect to integration by parts of the LFDO*${}_{0}{}^{\alpha}\mathrm{R}_{x}^{r}$

*is the RFDO*${}_{x}{}^{\alpha}\mathrm{R}_{\mathrm{\infty}}^{r}$,

*i*.

*e*.,

*for appropriate functions*\(f\)

*and*\(g\);

*see*(A.3).

### 2.2 A generalised family of random variables

The next theorem introduces the family of random variables to which the approach presented in this paper applies. In its full generality, we consider random variables that can be written as finite sums of independent heterogeneous gamma and Mittag-Leffler random variables. At the moment, there is no known explicit formula for the probability density function of such a random variable, but one can always express it in a convolution form. Notice that if only gamma random variables with integer shape parameters are involved in the summation, this random variable is the generalised integer gamma distribution (GIG) [9]. We now characterise the fractional boundary value problem satisfied by the density function of such random variables.

### Theorem 3

*Consider a random variable*\(T\)

*defined by*

*in terms of gamma random variables*\(Y_{i}\sim \Gamma (r_{i},\lambda _{1,i})\)

*and Mittag*-

*Leffler random variables*\(Z_{j}\sim \mathrm{ML}( \mu _{j},\lambda _{2,j})\),

*all independent of each other*,

*where*\(r_{i}\), \(\lambda _{1,i}\), \(\lambda _{2,j}\in \mathbb{R}_{+}\)

*and*\(\mu _{j}\in (0,1]\).

*Then the density function*\(f_{T}^{m,n}(t)\)

*of*\(T\)

*solves the fractional differential equation*

*with boundary conditions*(

*when*\(n\neq 0\))

*for*\(k=2,\dots ,\lceil \sum _{j=1}^{n} \mu _{j} +\sum _{i=1}^{m} r_{i} \rceil \).

*Here and subsequently*, \(\Lambda _{m,n}\)

*denotes*

### Remark 4

We further assume that all the \(\lambda _{1,i}\) are different, i.e., \(\lambda _{1,i}\neq \lambda _{1,k}\) for all \(i\neq k\). In other words, each variable \(Y_{i}\) has a gamma distribution with a different rate parameter. The uniqueness of the \(\lambda _{1,i}\) can be realised without any loss of generality. Whenever we have \(\lambda _{1,i}=\lambda _{1,k}\) for some \(i\neq k\), we can consider the sum of their corresponding random variables, which is still a gamma random variable with the same rate parameter.

### Remark 5

### Remark 6

Equation (2.7) along with its boundary conditions can be regarded as the generalisation of a pair of boundary problems discussed in Rosenkranz and Regensburger [30]. When the fractional differential algebra is properly defined, these fractional-order boundary problems can be factorised and further solved by obtaining their corresponding Green operators.

## 3 Main results

The LFDO and RFDO give us the ability to study a very general family of distributions that may find applications in various areas, e.g. queuing theory, risk theory and control theory. Although many of the available techniques for the analysis of the associated equations are numerical or asymptotic, the fractional differential approach still offers analytic insights to the related problems. In this section, we aim at accomplishing this with particular problems arising in risk theory. A special family of renewal risk models of the form (1.1) is considered, including the \(\text{Erlang}(n)\) and fractional Poisson risk models. We shall show that the ruin probabilities in these models solve fractional integro-differential equations involving the LFDO and RFDO operators.

Before moving on to the main result, we introduce a lemma that allows us to change the argument of our operators on a bivariate function under certain circumstances.

### Lemma 1

*For positive real numbers*\(\alpha \), \(r\)

*and*\(c\),

*we have the identity*

*where*\(x\)

*and*\(y\)

*are real numbers and*${}_{x}{}^{\alpha}\mathrm{R}_{\mathrm{\infty}}^{r}$

*is defined in*(2.3).

### Proof

Now we are able to generalise the results from [23, 2, 6] to a risk model with inter-arrival times of the form (2.6). The main result of this paper is the following.

### Theorem 2

*Consider a renewal risk model*

*where the inter*-

*arrival times*\(T_{k}\)

*are assumed to be a finite sum of independent gamma random variables*\(Y_{i}\sim \Gamma (r_{i},\lambda _{1,i})\)

*and Mittag*-

*Leffler random variables*\(Z_{j}\sim \mathrm{ML}( \mu _{j},\lambda _{2,j})\)

*as in*(2.6).

*Then the ruin probability*\(\psi (u)\)

*under the model*\(R_{m,n} \)

*satisfies the fractional integro*-

*differential equation*

*with the universal boundary condition*\(\lim _{u\rightarrow \infty } \psi (u)=0\).

*Here*,

*the constant*\(\Lambda _{m,n} \)

*is given by*(2.8),

*and*\(\mathcal{A}_{m,n}^{*} \)

*is the formal adjoint of*\(\mathcal{A}_{m,n} \) (

*see*(2.7))

*and is given by*

### Proof

### Corollary 3

*The non*-

*ruin probability*\(\phi (u) = 1-\psi (u)\)

*for the risk model in Theorem*3.2

*satisfies the fractional integro*-

*differential equation*

*with the universal boundary condition*\(\lim _{u\rightarrow \infty } \phi (u)=1\) (

*see*(2.8)

*and*(3.3)

*for the definitions of the constant*\(\Lambda _{m,n} \)

*and the operator*\(\mathcal{A}_{m,n}^{*}(c\frac{\mathrm{d}}{\mathrm{d}u}) \)).

Theorem 3.2 characterises a fractional integro-differential equation satisfied by the ruin probability \(\psi \) for a large class of waiting time distributions. The solvability of this fractional integro-differential equation depends on the particular form of the claim size distribution function \(F_{X} \).

We now restrict the rest of the analysis to claim sizes \(X_{i}\) distributed as a sum of an arbitrary number of independent gamma random variables. The next theorem shows that under this assumption, (3.2) can be written as a boundary value problem with only fractional derivatives. It is important to note that if the claim sizes include any Mittag-Leffler components, as is the case of \(T\) in Theorem 3.2, we have \(\mathbb{E}[X_{i}] = \infty \) and ruin happens with probability one since the net profit condition is violated.

### Theorem 4

*Consider the renewal risk model in Theorem*3.2.

*Assume further that the claim sizes*\(X_{i}\)

*are each distributed as a sum of*\(\ell \)

*independent*\(\Gamma (s_{k},\alpha _{k})\)-

*distributed random variables for some*\(s_{k},\alpha _{k} >0 \), \(k=1,\dots ,\ell \),

*i*.

*e*., \(f_{X}\)

*satisfies*

*with boundary conditions*(

*see Theorem*2.3)

*for*\(q=2,\dots ,\lceil \sum _{k=1}^{\ell }s_{k} \rceil \).

*Let*\(\mathcal{A}_{m,n}^{*}(c\frac{\mathrm{d}}{\mathrm{d}u}) \)

*and*\(\Lambda _{m,n} \)

*be as in*(3.3)

*and*(2.8),

*respectively*.

*Then the non*-

*ruin probability*\(\phi (u)\)

*satisfies*

*with the universal boundary condition*\(\lim _{u\rightarrow \infty } \phi (u)=1\)

*and initial values*

*for*\(k'=1,\,\dots ,\,\lceil \sum _{k=1}^{\ell }s_{k} \rceil -1\).

### Proof

### 3.1 The characteristic equation method

### Proposition 5

*Suppose*\(s_{k} \in \mathbb{N}\), \(k=1,\dots ,\ell \),

*in Theorem*3.4.

*The number of initial*-

*value boundary conditions of*\(\phi (u)\)

*is*\(N=\sum _{k=1}^{\ell }s_{k}\),

*and they are given explicitly by*

*where the values of*\(s_{p,k}\)

*are to be computed as follows*:

*let*

*and define*

### Proof

## 4 Explicit expressions for ruin probabilities in gamma-time and fractional Poisson risk models

The class of models considered in Theorem 3.2 is very general. In this section, we thus focus on two specific models which might be of interest to applications, and where explicit forms of ruin (non-ruin) probabilities can be derived.

### Remark 1

It has been shown in Asmussen and Albrecher [4, Theorem 2.1] that for any renewal risk model, the ruin probability always has an exponential form when the claim distribution is exponential. However, the fractional differential equation approach bridges a solid connection between the classical risk model and a class of renewal models which might be applied in a more sophisticated model.

### 4.1 Gamma-time risk model

A gamma-time risk model describes the reserve process \(R_{r}\) of an insurance company by replacing the Poisson process \(N\) in the classical model (1.1) with a renewal counting process \(N_{r}\) with \(\Gamma (r,\lambda _{1})\)-distributed waiting times. This is a natural extension of the \(\text{Erlang}(n)\) risk model considered by Li and Garrido [23].

### Example 2

### Remark 3

Note the substantial impact on \(\psi _{r}(u)\) when changing the Poisson assumption (\(r = 1\)). Ruin is more likely to happen in the gamma-time risk model with a larger shape parameter \(r\) of inter-arrival times, and vice versa. The reason is that in this case, the expected inter-arrival time \(r/\lambda _{1}\) is fixed whereas the variance \(r/\lambda _{1}^{2}\) of the inter-arrival time decreases as \(r\) increases, which means that the chance of having shorter waiting periods between claims will decrease. Since ruin is usually caused by not enough capital, the model with a larger shape parameter \(r\) is more likely to survive. Figure 1(a) coincides with the findings from [23], which focuses on \(\text{Erlang}(n)\) risk models.

The next example goes a step further and assumes gamma distributions for both the inter-arrival times and the claim sizes. This case is simple enough that the two positive roots of the characteristic equation can be bounded.

### Example 4

### 4.2 Fractional Poisson risk model

### Example 5

Note the substantial impact on \(\psi _{\mu }(u)\) when the classical Poisson assumption (\(\mu = 1\)) is changed. Increasing either \(\lambda _{2}\) or \(\mu \) increases the chances for ruin to happen. The reason is that for large enough \(t \), the expected number of jumps before time \(t\) in the fractional Poisson process (see (C.2)) is an increasing function of both \(\lambda _{2}\) and \(\mu \). Figure 2(b) shows the values of the natural logarithm of \(u_{5}\) as a function of \(\mu \) and \(\lambda _{2}\). Note that the contour lines in this plot are not parallel to each other. As the values of \(\mu \) decrease, the parameter \(\lambda _{2}\) plays a less significant role in the ruin probability function.

Notice that the operator ${}_{u}{}^{\mathrm{C}}\mathrm{D}_{\mathrm{\infty}}^{\mu}$ tends to the identity operator when \(\mu \rightarrow 0+\). Thus we obtain the following result.

### Corollary 6

*In the fractional Poisson risk model*,

*the ruin probability*\(\psi _{ \mu }(u)\)

*converges to a function*\(\psi _{0}(u)\)

*as*\(\mu \rightarrow 0\).

*Moreover*,

*the function*\(\psi _{0}(u)\)

*satisfies the integral equation*

*with the universal boundary condition*\(\lim _{u\rightarrow \infty }\psi _{0}(u)=0\).

## Notes

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