A weak approximation for the Extrema s distributions of Levy processes

‎Suppose that Xt is a one-dimensional and real-valued L evy‎ ‎process started from X0=0‎, ‎which (1) its nonnegative‎ ‎jumps measure ν satisfying ∫‎‎Rmin{1,x²}ν(dx) ∞ and (2) its stopping time‎ ‎τ(q) is either a geometric or an exponential‎ ‎distribution with parameter q independent of Xt and‎ ‎τ(0)=∞. This article employs the Wiener-Hopf‎ ‎Factorization (WHF) to find‎, ‎an Lp∗(R) (where‎ ‎1/p∗+1/p=1 and 1 p≤2)‎, ‎approximation for the extrema s‎ ‎distributions of Xt. Approximating the finite (infinite)-time‎ ‎ruin probability as a direct application of our findings has been‎ ‎given‎. ‎Estimation bounds‎, ‎for such approximation method‎, ‎along‎ ‎with two approximation procedures and‎ ‎several examples are explored.