By Pablo Azcue, Nora Muler (auth.)
ISBN-10: 1493909940
ISBN-13: 9781493909940
ISBN-10: 1493909959
ISBN-13: 9781493909957
The major goal of the publication is to teach how a viscosity strategy can be utilized to take on regulate difficulties in assurance. the issues lined are the maximization of survival chance in addition to the maximization of dividends within the classical collective probability version. The authors reflect on the opportunity of controlling the danger strategy by means of reinsurance in addition to via investments. They convey that optimum price capabilities are characterised as both the original or the smallest viscosity resolution of the linked Hamilton-Jacobi-Bellman equation; additionally they learn the constitution of the optimum concepts and convey how to define them.
The viscosity procedure was once widespread up to the mark difficulties regarding mathematical finance yet till relatively lately it was once now not used to unravel keep an eye on difficulties concerning actuarial mathematical technological know-how. This ebook is designed to familiarize the reader on the best way to use this procedure. The meant viewers is graduate scholars in addition to researchers during this area.
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Additional resources for Stochastic Optimization in Insurance: A Dynamic Programming Approach
Sample text
12) Let us call Ai D Œxi ; xi C1 /. xi / Ä "=4 for all i . Rt /t 0 in the following way: For t Ä T , take Rt D Rt , so T ^ R D T ^ R . In the case that R > T , take i0 such that X R R 2 Ai0 and follow strategy RT Cs D Rsi0 for all s 0; note that R T C T^ Ri0 . x/ Â Â D E E In Ex . X R T^ i Ex . x; T / // " 2 "; t u and so we have the result. We want now to derive heuristically the HJB equation. Ui /: i D1 with ruin time R . 13) R2R In Chap. 3, we will show that ı is a viscosity solution of this equation, and in Chap.
X/ D 0, for x < 1 P. Azcue and N. 1; 1/. 1 p ˇ /Ifx p XŒx 1g i D1 e ˇ pi . x p i /i . 3) We will show in Chap. 4 that this weak solution u is indeed the survival probability function ı; the graph of u can be seen in Sect. 1. 9) can be non-differentiable. a/ < 1. a/, and so FR is discontinuous at a. Assume that pR > ˇ R and consider the problem of optimal survival probability with reinsurance family R D fRg. The value function ı of this problem is just the survival probability of the classical risk model with premium rate pR and claim-size distribution FR .
1a, b, respectively (the identity function is shown in dotted line). 2) The case Á1 D Á is called cheap reinsurance. There are other criteria for computing the premium rate of the reinsurance company; see for instance Teugels [62]. We assume that the premium rate left to the insurance company is pR D p qR > 0. 1. Let us call RA the family of all the retained loss functions with positive pR ; RP RA the subfamily of proportional retained loss functions, and RXL RA the subfamily of the excess-of-loss retained loss functions.