Advances in mathematical economics. Vol.03, Edition: Reprint by Charles Castaing (ed.)

By Charles Castaing (ed.)

Loads of financial difficulties can formulated as restricted optimizations and equilibration in their options. a variety of mathematical theories were providing economists with vital machineries for those difficulties coming up in financial concept. Conversely, mathematicians were prompted by means of numerous mathematical problems raised by means of financial theories. The sequence is designed to collect these mathematicians who have been heavily drawn to getting new demanding stimuli from fiscal theories with these economists who're looking for potent mathematical instruments for his or her researchers. individuals of the editorial board of this sequence involves following popular economists and mathematicians: handling Editors: S. Kusuoka (Univ. Tokyo), T. Maruyama (Keio Univ.) Editors: R. Anderson (U.C.Berkeley), C. Castaing (Univ. Montpellier), F. H. Clarke (Univ. Lyon I), G. Debreu (U.C. Berkeleyer), E. Dierker (Univ. Vienna), D. Duffie (Stanford Univ.), L.C. Evans (U.C. Berkeley), T. Fujimoto (Okayama Univ.), J. -M. Grandmont (CREST-CNRS), N. Hirano (Yokohama nationwide Univ.), L. Hurwicz (Univ. of Minnesota), T. Ichiishi (Ohio country Univ.), A. Ioffe (Israel Institute of Technology), S. Iwamoto (Kyushu Univ.), ok. Kamiya (Univ. Tokyo), ok. Kawamata (Keio Univ.), N. Kikuchi (Keio Univ.), H. Matano (Univ. Tokyo), ok. Nishimura (Kyoto Univ.), M. okay. Richter (Univ. Minnesota), Y. Takahashi (Kyoto Univ.), M. Valadier (Univ. Montpellier II), M. Yano (Keio Univ)

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A(e) set of nodes of the scenario tree representing possible events at time t. e. set of all interior nodes of the scenario tree. e. path from root to leaf in the scenario tree. index denoting an event (node of the scenario tree), which can be identified by an ordered pair of scenario and time period. ancestor of event e E '1{ (parent in the scenario tree). Pe branching probability of event e: Pe Pe probability of event e: E[-] Expectation with respect to p. A E IR nxn stochastic vector of return values for the n assets, t 1, ...

16 COMPUTATIONAL METHODS IN ECONOMICS AND FINANCE conditions without the burden of tax impacts. Second, importantly, they are able to accept greater short-term risks in order to attain greater long-term returns. In a sense to be shown later, long-term investors are able to capitalize on the inherent volatility often associated with higher-return assets. Economic growth theory suggests that a multi-stage investor should maximize the expected log of wealth at each time period ( [18]). For a given set of scenarios, s E S, we optimize the investment decision as follows: Maximize E U(w) = ~lts X log(w) s where U(w) is a von Neumann Morgenstern utility function, log(·), lts = probability of scenario s E S, and = investor's wealth under scenario s E S.

For any pair of asset classes, covariance is also calculated from historical data. A ten-year planning horizon is studied. It is assumed that portfolio rebalancing is done quarterly. Thus, the problem is a multi-stage portfolio optimization over 40 periods. The goal is to maximize the risk and rewards for total asset wealth (and surplus) at the end of the 40th period. One thousand scenarios are generated for each asset class using a system of stochastic differential equations. Based on these scenarios, the mix that maximizes the final total wealth is found.

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