By Jacques Janssen (auth.), Jacques Janssen, Christos H. Skiadas, Constantin Zopounidis (eds.)

*Advances in Stochastic Modelling and knowledge Analysis* provides the newest advancements within the box, including their purposes, generally within the components of assurance, finance, forecasting and advertising. furthermore, the potential interactions among information research, synthetic intelligence, choice aid platforms and multicriteria research are tested by way of most sensible researchers. *Audience:* a large readership drawn from theoretical and utilized mathematicians, akin to operations researchers, administration scientists, statisticians, laptop scientists, bankers, advertising and marketing managers, forecasters, and clinical societies comparable to EURO and TIMS.

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5) where: e(t) = [m(t) + Jl- r)/cr. 7) 1<$

23 T A= f e-Dt{IA(t), 0 T B = Je-Dt{IB( t ). expa'Z'(t). 2 Consequently, for every t, the random variable A(t) A(O) has a lognormal distribution so that E[A(t)] = A(O)ellt. 29) is a necessary condition for mean profitability up to time T. 30) Let us c' A = define now the following quantities ~ Ai (0) Jl· [T + (Jl· _ 8 ) T 2 ] i = 1 A(O) 1 1 2 ' ( 4. 2 A necessary condition for balance sheet is given by I mean profitability I A(O)CA ~ B(O)CB. 32) immunization Let us consider now the immunization problem.

Suppose that the 36 hypotheses of the optional sampling theorem are satisfied. 3) =min{Tb, t}. 2; let Th be a stopping time. Suppose that the hypotheses of the optimal sampling theorem are satisfied. If: i) A. , lim (sup E[ fV