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 integrodifferential equation (u) = in this case. Indeed, the righthand 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(ux) 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 eRu  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) eRu 1+ 1 (R) eRu 1+ 1 R 1 1 c 1+ 1 (R) eRu 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.
 Spring '09
 na

Click to edit the document details