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

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

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