Ruin probability in the presence of risky investments

2010/11/05 by Pergamenchtchikov, Serguei, Omar, Zeitouny
#Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Probability (math.PR) #Risk Management (q-fin.RM)

paper · doi:10.48550/arxiv.1011.1329

Abstract

We consider an insurance company in the case when the premium rate is a bounded non-negative random function c_\zst and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a and volatility σ>0. If β:=2a/σ²-1>0 we find exact the asymptotic upper and lower bounds for the ruin probability Ψ(u) as the initial endowment u tends to infinity, i.e. we show that C_*u^-β≤Ψ(u)≤ C^*u^-β for sufficiently large u. Moreover if c_\zst=c^*e^γt with γ≤ 0 we find the exact asymptotics of the ruin probability, namely Ψ(u)∼ u^-β. If β≤ 0, we show that Ψ(u)=1 for any u≥ 0.

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