Stochastic Modeling in Insurance – Some general introduction in addition to this narrative is provided in wikipedia, Stochastic Modeling (insurance).

Recent cataclysmic events like the tsunami, torrential downpour, floods, cyclones, earthquakes, etc. underscore the fact that everyone would like to be assured that there is some (non-supernatural) agency to bank upon in times of grave need.

If the affected parties are too poor, then it is the responsibility of the government and the “haves” to come to the rescue. However, there are also sizeable sections of the population who are willing to pay a regular premium to suitable agencies during normal times to be assured of insurance cover to tide over crisis.

Insurance has thus become an important aspect of modern society. In such a set-up, a significant proportion of the financial risk is shifted to the insurance company. The implicit trust between the insured and the insurance company is at the core of the interaction. A reasonable mathematical theory of insurance can possibly provide a scientific basis for this trust.

Certain types of insurance policies have been prevalent in Europe since the latter half of the 17th century. But the foundations of modern actuarial mathematics were laid only in 1903 by the Swedish mathematician Filip Lundberg, and later in the 1930’s by the famous Swedish probabilist Harald Cramer. Insurance mathematics today is considered a part of applied probability theory, and a major portion of it is described in terms of continuous time stochastic processes. This article should be accessible to anyone who has taken a course in probability theory.

**Collective Risk Model**

We shall mainly look at one model, known as the Cramer-Lundberg model; it is the oldest and the most important model in actuarial mathematics. This model is a particular type of a collective risk model. In a collective risk model there are a number of anonymous but very similar contracts or policies for similar risks, like insurance against fire, theft, accidents, floods or crop damage/failure, etc. The main objectives are modelling of claims that arrive in an insurance business, and decide how premiums are to be charged to avoid ruin of the insurance company. Study of probability of ruin and obtaining estimates for such probabilities are also some of the interesting aspects of the model.

There are three main assumptions in a collective risk model:

- The total number of claims, say N occurring in a given period is random. Claims happen at times {T
_{i}} satisfying 0 ≤ T_{1}≤ T_{2}≤ We call them*claim arrival times*(or just*arrival times*). - The
_{i}_{-t}h claim arriving at time T_{i}causes a payment T_{i}. The sequence {X_{i}} is assumed to be an independent and identically distributed sequence of nonnegative random variables. These random variables are called*claim sizes*. - The claim size process {X
_{i}} and the claim arrival times {T_{j}} are assumed to be independent. So {X_{i}} and N are independent.

The first two assumptions are fairly natural, whereas the third one is more of a mathematical convenience.

Take T_{}. Define the *claim number process* by

N (t) = max {i ≥ o : T_{i }≤ t}

= number of claims occurring bt time t, t ≥ 0. **(1)**

Also define the *total claim amount process* by

S (t) = X_{1} + X_{2} + ….. + X_{N (t)} = ≥ 0 **(2)**

These two stochastic processes will be central to our discussions. Note that a sample path of N and the corresponding sample path of S have jumps at the same times T_{i}, by 1 for N and by X_{i} for S.

A function ƒ(·) is said to be o(h) if = 0; that is, if ƒ decays at a faster rate than h.

**Poisson Processes**

We first consider the claim number process {N (t) : t ≥ 0}. For each t ≥ 0, note that N(t, ) is a random variable on the same probability space (Ω Ƒ, Ƥ). We list below some of the obvious/ desired properties of N (rather postulates for N), which may be considered in formulating a model for the claim number process.

**(N1):**N(0) = 0. For each t ≥ 0, N(t) is a non-negative integer-valued random variable.**(N2):**If 0 ≤ s < t then N(s) ≤ N(t). Note that N(t) – N(s) denotes the number of claims in the time interval (s,t). So N is a non-decreasing process.**(N3):**The process {N(t) : t ≤ 0} has*independent increments*; that is, if 0 < t_{1}< t_{2}< < t_{n}< ꝏ then N(t_{1}), N(t_{2}) – N(t_{1}), , N(t_{n}) – N(t_{n-1}) are independent random variables, for any n = 1, 2, In other words, claims that arrive in disjoint time intervals are independent**(N4):**The process {N (t)} has*stationary increments*; that is, if 0 ≤ s < t, h > 0 then the random variables N(t) – N(s) and N(t+h) – N(s+h) have the same distribution (probability law). This means that the probability law of the number of claim arrivals in any interval of time depends only on the length of the interval.**(N5):**Probability of two or more claim arrivals in a very short span of time is negligible; that is,**(3)**

**(N6):**There exists A > 0 such that P(N(h) = 1) = λh + o(h), as h ↓ 0.**(4)**The number λ is called the*claim arrival rate*. That is, in very short time interval the probability of exactly one claim arrival is roughly proportional to the length of the interval.

Remark 1. The first two postulates are self-evident. The hypothesis (N3) is quite intuitive; it is very reasonable at least as a first stage approximation to many real situations. (N5), (N6) are indicative of the fact that between two arrivals there will be a gap but may be very small; (note that bulk arrivals are not considered here). (N4) is a time homogeneity assumption; it is not very crucial.

Remark 2. In formulating a model, it is desirable that the hypotheses are realistic and simple. Here ‘realistic’ means that the postulates should capture the essential features of the phenomenon/problem under study. And ‘simple’ refers to the mathematical amenability of the assumptions; once a model is chosen, theoretical properties and their implications should be considerably rich and obtainable with reasonable ease. These two aspects can be somewhat conflicting; so success of a mathematical model depends very much on the optimal balance between the two.

To see what our postulates (N1)-(N6) lead to, put

P_{n}(t) = P(N9t)=n). t ≥ 0, n = 0, 1, 2, **(5)**

Observe that

P_{}(t+h)

= P(N(t) = 0), N(t+h) – N9t) = 0) (by (N1), (N2))

= P(N(t) = 0) P(N(t+h) – N(t) = 0) (by (N3))

= P_{}(t) P(N(h) = 0) (by (N4), N(1))

= P_{}(t) [1 – λh + 0(h)] (by (N5), (N6))

Whence we get (as 0 ≤ P_{}(t) ≤ 1),

**(6)**

By (N1), note that P0(0) = P(N(0) = 0) = 1. So the differential equation (6) and the above initial value give

**(7)**

Similarly for n ≥ 1, using (N3) – (N6), we get

P_{n}(t+h) = P(N(t+h) = n) = I_{1} + I_{2} + I_{3}.

Where

I_{1} = P(N(t) = n, N(t+h) – N(t) = 0),

I_{2} = P(N(t) = n – 1, N(t+h) – N(t) = 1),

I_{3} = P(N(t) ≤ n – 2, N(t+h) – N(t) ≥ 2),

And hence

P_{n}(t+h) – P_{n})t) [1 – λh + 0(h)] + P_{n-1}(t)[λh + 0(h)] + 0(h).

We now get as before

**(8)**

Using the initial values Pn(0) = P(N(0) = n) = 0, n ≥ 1 it is fairly easy to inductively solve (8) and get

### Stochastic Modeling in Insurance

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