Example_Dec_3

# Example_Dec_3 - During the final class on 3rd December it...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: During the final class on 3rd December, it was mentioned that the Cramer's asymptotic expression corresponds to the exact ruin probability when claim sizes are exponential with mean . Here we give the example that the expression 1 -Ru (u) = e 1+ solves the integro-differential equation (u) = in this case. Indeed, the right-hand side of (1) is given by c c c u (u) - c c u 0 (u - x) fX (x) dx - c u fX (x) dx (1) (u) - 0 (u - x) fX (x) dx - u fX (x) dx = = = u x 1 -R(u-x) 1 - 1 -Ru -u e - e e dx - e 1+ 0 1+ u 1 -Ru 1 -Ru 1 - 1 -R x -u e - e e dx - e 1+ 1+ 0 1 - -R u 1 -Ru 1 1 - e 1 -Ru -u e - e -e 1 c 1+ 1+ -R = = Given that R = 1 1+ , c c 1 -Ru 1 1 e-Ru - e e - 1 1+ 1+ -R 1 -Ru e 1+ 1- 1 1 u - -e u - -R +e u - 1 1 1+ 1 1 -1 -R . 1 1 1+ c = = = = c u 1 1 - 1 = 0, -R which implies that (u) - 0 (u - x) fX (x) dx - -R -R c 1 u fX (x) dx 1 -Ru e 1+ 1 1 (-R) e-Ru 1+ 1 (-R) e-Ru 1+ 1 -R 1 1 c 1+ 1 (-R) e-Ru 1+ = (u) , which is what we want to prove. 1 ...
View Full Document

## This note was uploaded on 11/23/2010 for the course ACTSC act431 taught by Professor Na during the Spring '09 term at Waterloo.

Ask a homework question - tutors are online