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Keywords
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Claim process, Maximum surplus distribution, Ruin probability, Standard Brownian motion, Wiener process
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Abstract
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We mainly adapt the well-known classical Cramér-Lundberg model with a homogeneous Poisson process to the perturbed
risk model by adding a Wiener process and interest rates which may be caused by a claim amount or by fluctuation. The
dynamic equation forms of risk model with having the standard Brownian motion and exponential stochastic interest
rate are proposed and the dynamic value function of infinite time ruin probability is obtained using the Hamilton-Jacobi-
Bellman equation. We give a barrier strategy to obtain the exponential integro-differential equations for the distribution
of maximum surplus before ruin. Applying the Itô’s formula together with some mathematical analysis methods and a
general form of quadratic polynomial, we obtain some integro-differential equations for evaluating the infinite time ruin
probabilities. Furthermore, we deduce and simplify the closed-form results of these equations through the exponential
claim amounts. Finally, to better illustrate the derived formulas, we shall study several examples in details and investigate
the effect of parameters of models on the ruin probabilities
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