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 ( y ) = P { T <  | Y (0) = y }, If we let T = inf { 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 x dx   .