Approximate results for non-linear Cramér-Lundberg type risk model

In this study, a non-linear Cramér-Lundberg risk model is considered, investigated and ruin probabilities, ψ(u), are calculated. In literature, a linear form of this model, also known as classical model, is defined as follows: U(t)=u+ct-S(t) (1) The risk process U(t) in Eq.(1) expresses an amount of capital of an insurance company at a given time t, the constant u is initial capital of the company, c – the premium rate, S(t)= ∑_(i=1)^(N(t))▒X_i is a renewal-reward process which represents the outflow of cash caused by reimbursements for claims occurred in the interval [0,t], N(t) is a renewal process counting the total number of claims within the time frame [0,t] and X_i's are i.i.d. random variables denoting the amount of payment for i^th claim. As seen in (1), the term ct expressing the company's premium income is a linear function of time. However, this assumption is not realistic, because the premium income of an insurance company cannot always increase linearly. This is especially true for the markets saturated with insurance policy holders. Therefore, it is advisable to assume that the premium income is modeled as a function whose rate of growth decreases with time, although this function is monotonically increasing. For this reason, in this work, a more realistic special non-linear mathematical model is constructed and investigated, which is given as follows: V(t)=u+c∑_(i=1)^(N(t))▒〖ln(1+W_i)〗+c ln(1+(〖t-T〗_N(t) ))-S(t) (2) In (2), W_i's (i=1,2,3… ) are positive i.i.d. sequence of random variables describing inter-arrival times of claims; T_N(t) = ∑_(i=1)^(N(t))▒W_i is a renewal-reward process, corresponding to the sequence of random variables W_i's ,i=1,2,3,… , and V(t) defines company's capital balance at any time t which is modelled by so called a Logarithmic Risk Process. The main purpose of this study is to evaluate ruin probability, ψ(u), of non-linear risk model in (2). While establishing the model, stochastic processes, renewal processes, reward-renewal processes and the probabilistic characteristics of these processes were used. In the first stage, the Lundberg type upper bound was found for the ruin probability of our non-linear model. While trying to calculate these probability bounds, numerical solution methods were used when nonlinear equations were encountered. In order to consider various scenarios, different probability distributions and parameters are considered and an approximate solution is found with the regression model. In the second stage, bounds for ruin probability from above and below is found for this non-linear model. This is also known as Cramér-type bound for the ruin probability. For this purpose, the statistical characteristics of the random variable, X ̂, which describes the residual time (limit distribution) of the renewal process produced by the sequence {X_n }, representing the accidents(damages), was exploited. In particular, moment generating function of X ̂ was utilized to determine a constant C, which is an unknown coefficient in the bound expression of the ruin probability. In order to simplify these expressions and transform them into a compact form, calculus methods were used. Similarly, in order to examine and calculate ruin probabilities, various probability distributions and parameters were used to consider different scenarios.