Mathematical Approach to the Ruin Problem with Compounding Assets

: This study considered the Ruin problem with an income process with stationary independent increments. The characterization is obtained which is general for the probability of r ( y ), that the asset of a firm will never be zero whenever the initial asset level of the firm is y . The aim of this study is also to determine r ( ŷ ) = P { T <  | Y (0) = y }, If we let T = in f { t ≥ 0; Y ( t ) < 0}, A condition that is necessary and sufficient is studied for a distribution that is one – dimensional of X N which coverages to X * . The result that is obtained concerning the probability, is of ruin before time t . Riemann-Stieltjes integral, two functions f and  with symbol as ( ) ( ) b a f x d x   was used and is a special case in which  () = x , where  has a continuous derivative. It is defined such that the Stieltjes integral ( ) ( ) b a f x d x   becomes the Riemann integral ( ) ( ) | . b a f x d x  


Introduction
This research considers the generation of the collective risk theory of the classical model, where the assumption is that the cumulative income of the firm is given by a process X*. The process has stationary independent increments and the interest is continuously earned on the firm's assets. The assets of the firm of the first-time t, which is Y(t) are represented by a simple path-wise integral concerning the income process X (Harrison, 1977). A general characterization can be obtained for the probability r(y), which implies that assets will not fall to zero when the initial asset level is y. By the action of the force of interest, a classical surplus process is modified, which can be considered as recursive algorithms for the calculation of the probability of ruin in finite time. This is in line with (Dickson and Waters, 1999) who worked on two numerical algorithms, such as the stable recursive algorithm and the method of product integration that was used to obtain numerically an approximation to this probability of ultimate ruin. Also, the survey of the problem of ruin in a risk model when assets earn investment income is treated (Paulsen, 2008), but the exact recursive expressions for ruin probabilities will also be looked into.

Income Process
Let X = {X(t), t ≥ 0} represent an independent increment of a stochastic process with stationary and, finite variance where X(0) = 0, which is called the income process. Let y also represent a positive level of initial asset and a positive interest rate. Therefore, the asset process: An investment during a period t becomes e (t) at the end, then, Eq. (1) becomes a compounding asset (De Kok, 2003): If t ≥ 0, we define: Riemann-Stieltjes integral on Eq.
(2) is given as: will exist and is finite for all t ≥ 0 and almost all sample paths of X will be a well-defined path-wise functional of the income process (Feller, 2008;Eisen, 1969):

Proposition
The Riemann-Stieltjes integral: Is finite, exists, and satisfies:

Example
Let the income process of a compound Poisson form i.e., Let c in Eq. (7) be positive and that: Feller, 1971). Thus, for the parts of the income process slope, there is an absolute jump size with an exponential distribution of mean m. The exponent function is given as: By c.f. of gamma distribution: It was observed that the jumps of Y and X are the same, which implies that the property of the exponential distribution gives the amount the assets will fall below zero upon ruin.
Ruin will occur if a general jump of the same income process has the same exponential distribution. Therefore: Combining (9) and (11), with T <  a gamma distribution there is the convolution of H with the conditional distribution of Y(T), then: Now suppose in Eq. (7) that c is negative and that: Eq. (9) i.e.: for, f(x) = 0 and x = 0. Thus, there is an absolute jump size with an exponential distribution of mean m and the exponent function is given as the paths of the income process.
Then we obtain: The gamma distribution: There are no negative jumps in the income process.

The Ruin Problem
Suppose that the asset of a firm will not fall to zero when the initial asset level is taken as y.
We assume that an insurance company received a N(t), number of claims at time t and this is a Poisson Process with parameter . If the insurance company received cash at a constant rate of 1 per unit time t. Then, the cash balance at time t can be expressed as: where the initial capital of the company is x and Yi, ≥ 1, are the claims. Suppose that the company invested the risk reserve of the company with a continuous interest rate of  (Davidson, 1969) and if the account of the company ever falls to zero, money can be borrowed from the bank at the same interest rate . Then, the content of the account at time t is given as: where, y and  are already defined as an initial asset and interest rate through the investment of a firm's assets respectively (Harrison et al., 1975) X(t) is the firm's net operating profit during the interval [0, t]. Let: We want to determine: With the time of ruin and ruin function as T and r(.) respectively.
The ruin function: Let X be defined on (, , P) probability space and has a stationary, independent increment with E[X(t)] = t and Var[X(t)] =  2 t where - <  <  and 0 < 2 < . We assume that X has a continuous probability distribution. The characteristic function (c.f.) of X(t) we then have: where, G is a probability distribution on . X is strong Markov (Apostol, 1974). From Eq.

The Compound Brownian Approximation
The Brownian motion with drift approximates the income process, then the assets process Y* and r(y) is also approximated by a corresponding probability r* (t) (Karlin and Taylor, 1981). The diffusion Y* is called compounding Brownian motion.
Let y be the initial asset level and  the interest rate be fixed, then.
Let X1, X2,…, be income process defined on a common probability space , , P. and.
Let X* be a Brownian motion with mean * and variance  2 , also defined on (, , P), we assume that: Thus, X* be each of the natural diffusion approximation for the income process Xn.
Let  denote weak convergence for a sequence of a probability measure on a metric space. In the case of a random variable (random elements of ), weak convergence is equivalent to the convergence of distribution function at continuity points of the limit (Bartle, 1974) The restriction to [0, t] at any process above is a random element of D[0, t].

Example
We assume now that X(t) = W(t) + t.
where, W is the standard (zero drift and unit variance) Weiner process.
Our basic representation is given by: with mean of (t) =  + y and variance of  2 (y) =  2 y  .

Conclusion
We consider the financial situation of an insurance company at discrete times, say t = 0,1,2,… and we assume that there is an initial surplus X0 = x, which is a non-negative integer.
The surplus Xt+1 at time t + 1 results from Xt as follows (t = 0,1,2,…): where, Zt is the dividends paid to the shareholders at time t; Yt+1 represents the income (possible negative) of the company during the time interval from t to t + 1. Both Zt and Yt are integer-valued and Y't s are independent identically distributed random variables. By Pj, we denote the probability that the income resulting from a certain time interval (t, t + 1) will be equal to j(i = 0, 1, 2,…).
Assume the expectation of t0 exists and is positive then: Zi'os are decision variables. w = dividend strategy which determines how dividends will be paid out to the shareholders. In any event, Zt will assume one of the values 0,1,2,…, Xt (Xt be surplus at time t). Ruin occurs as soon as the surplus gets negative epoch  such that X<0.
We are only interested in t ≥ . Accordingly, we put We defined for each w and initial surplus x: If V(x, w) = V(x) for all x = 0,1,2,…,w is called dividend strategy optimal.

Basic Difference Equation Factorization
Let a be a non-negative integer and V(x, a) be the expected sum of the discounted payments (initial surplus x) concerning the barrier strategy belonging to a.
Assuming Yi = j and applying a rule of total probability, we obtain the equations: The system (24) has a unique solution W(x) and can be seen by expressing W(x+1) in terms of W(0), W(1),…, W(x).
Also, from (22) The maximization of V(x, a) amounts to minimizing the difference w(a+1)w(a).
Relation with the Probability of Ruin We want to show a close connection between the fundamental function W(x) defined in (24) and the probability of ruin of a certain conjugate random walk (Beekman, 1974).
For s > 0, we define Ws(x) = S -x W(x), x = 0,1,2,… Thus, Eq. (24) is equivalent to: The random walk belonging to   sj jj P ds P = is called a conjugate random walk of {Pj} from s > 1 and (19) we have that the expectation of   s j P positive. From Eq. (26), we have that Ws(x) may be known as the probability of survival of the conjugate random walk (given initial surplus x and absorbing barrier at -1). Thus,