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 $r(y) = P \{ T < \infty | Y(0) = y \}$, If we let $T = inf \{ t \ge 0; Y(t) < 0 \}$, 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 $\int_a^b f(x)da(x)$ was used and is a special case in which $a() = x$, where $a$ has a continuous derivative. It is defined such that the Stieltjes integral $\int_a^b f(x)da(x)$ becomes the Riemann integral $\int_a^b f(x)a^| d(x)$.