Risk Processes with Normal Inverse Gaussian Claims and Premiums

Abstract

We study risk processes where claims and premiums are modeled by independent normal inverse Gaussian (NIG) Lévy processes; claims by a spectrally positive NIG Lévy process. Using martingale technique, the Lundberg inequality for ruin probability is proved.

1 Introduction

Boykov [1] studies risk processes where claims and premiums are modeled by independent compound Poisson processes. In [4], the difference of premiums and claims are modeled by different Lévy processes; capitalizing on the NIG. Of recent, Stanojevic and Levajkovic [6] proposed modeling premiums with a time changed subordinated Lévy process. We implement this proposal using NIG-Lévy process since it can be represented as an inverse Gaussian time changed Brownian motion with drift [5]. Further, we model aggregate claims by a spectrally positive NIG Lévy process. Using martingale technique, we prove the Lundberg inequality for ruin probability.

2 Model Considerations

2.1 Modified Premium Process

Generally, premiums are determinate, discrete, independent, non-negative stationary increments [2] and we consider an infinite number collected within the time period. Modified premium incorporates the effects of inflation, dividend payouts, tax, interest rate fluctuations, claims processing costs and other administrative expenses by the insurance company but claims. It can take on negative or positive values and can be represented by a function with support on the entire real line. Our proposal is to use a finite mean and finite variance NIG Lévy process; with the mean representing constant loaded premium and variance the variation in this modified premium process i.e. stochastic premiums. This is because NIG Lévy process has paths composed of an infinite number of small jumps and exhibit diffusion-like feature with a jump driven structure [3]. NIG¹ Lévy process² has its Laplace exponent through Lévy-Khintchine theorem [4] as follows:

$$ \varPsi_1(\lambda) = a\lambda+ \int^{+\infty}_{-\infty} \bigl[e^{\lambda x}-1-\lambda xI_{\{|x|\leq1\}} \bigr]\nu(dx) $$

(1)

where a, λ are real constants and ν(dx) is a measure on R∖{0} such that $\int^{+\infty}_{-\infty} (1\wedge X^{2} )\nu(dx) < \infty$.

2.2 Claims

Claims also are generally independent,³ indeterminate, stationary, non-negative increments with an infinite number collected within considered finite time interval [2]. We model these with a spectrally positive NIG Lévy process i.e. an NIG Lévy process with no negative jumps and chosen to have finite mean, finite variance and support on the positive real line (see Fig. 1). It is not the negative of a subordinator.⁴ Generally, if X is spectrally positive, then −X is spectrally negative. Spectrally negative Lévy processes are well studied in the literature. NIG spectrally negative has its Laplace exponent through Lévy-Khintchine theorem as follows:

$$ \varPsi_2(\lambda) = -a_1\lambda+ \int ^{0}_{-\infty} \bigl[e^{\lambda y}-1-\lambda yI_{\{y>-1\}} \bigr]\nu_1(dy) $$

(2)

where a ₁ is a real constant representing drift, λ also real and ν ₁(dy) is a measure on R∖{0} such that $\int_{-\infty}^{\infty} (1\wedge Y^{2})\nu_{1}(dy) < \infty$.

Fig. 1

Spectrally positive NIG(50,−10,1,0) process depicting aggregate claims

3 Proposed Model

We propose a Cramer-Lundberg risk model with modified stochastic premiums and stochastic claims both modeled by different NIG-Lévy processes. The risk process U(t),t≥0 is defined as

$$ U(t) = u + X_1(t) - X_2(t) $$

(3)

where u is the initial capital, X ₁(t)∼NIG(α,β,δ ₁ t,μ ₁ t) the modified premium process and X ₂(t)∼NIG(α,β,δ ₂ t,μ ₂ t) the aggregate claims process. The risk process U(t)−u is distributed as NIG(α,β,(δ ₁−δ ₂)t,(μ ₁−μ ₂)t) with its Laplace exponent through Lévy-Khintchine theorem:

$$ \varPsi_T(\lambda)=\varPsi_1(\lambda)+ \varPsi_2(\lambda) = (a-a_1)\lambda+ \int ^{+\infty}_{0} \bigl[e^{\lambda x}-1-\lambda xI_{\{|x|\leq1\}} \bigr]\nu(dx) $$

(4)

taking in to account the chosen approximation $\int^{0}_{-\infty} [e^{\lambda x}-1-\lambda xI_{\{|x|\leq1\}} ]\nu(dx)\approx\int^{0}_{-\infty} [e^{\lambda y}-1-\lambda yI_{\{y>-1\}} ]\nu_{1}(dy)$. This approximation basically means the company has controls in such a way that most claims can be settled, but ultimate ruin or profitability is determined by (4). Ψ _T (λ) of (4) has both a constant part (a−a ₁) and a stochastic part $\int^{+\infty }_{0}[e^{\lambda x}-1-\lambda xI_{\{|x|\leq1\}}]\nu(dx)$ representing in a sense the modified risk process. Therefore, net profitability condition simply translates to E(X(1))>E(Y(1)), where X(1) represents modified premium and Y(1) aggregate claims.⁵ Considering the Martingale⁶ approach to ruin probability, if we can find a value r=R in the domain of the definition of Ψ(λ) such that Ψ(λ)=0 and τ<∞, then we could simply write

$$ \psi(u)=E_Q\bigl[e^{RU(\tau)}\bigr]e^{-Ru}, \quad u\geq0. $$

(7)

We calculate such an $R = \frac{2 (d\sqrt{\alpha^{2} -\beta^{2}}+\beta )}{d^{2} +1}$ where $d=\frac{\mu_{1} - \mu_{2}}{\delta_{2} - \delta_{1}}$, similar to [4] where their c=μ ₁−μ ₂ and δ=δ ₂−δ ₁; employing similar analysis. Simulated graph (Fig. 2) demonstrates how the proposed model looks like.

Fig. 2

Risk process which is the difference of NIG(50,−10,1,0) and spectrally positive NIG(50,−10,2,0)