Log-Optimal Asset Portfolio Theory

Marking the start of the analysis of quantitative finance Markowitz (H.Markowitz) proposed in 1952, portfolio theory, can be used as the starting point for the study of mathematical finance, portfolio theory, quantitative finance and investment research laid the theoretical foundation. The basic idea of the probability theory and method to study the limit theory in the field of economic problems, the formation of a major financial mathematical theory branch. Currently the best choice theory under uncertainty Stochastic portfolio, is a hot issue financial mathematics research. Its main purpose is given in the next income level so that the investment risk is minimized or at a given level of investment risk under the so-income investments maximized find an optimal portfolio.That is the classic portfolio theory:mean-variance model, many scholars have done a lot of research, access to a wealth of research results. For example, some scholars in the study of the optimal portfolio problem under the influence of inflation, given the specific definition of VaR and CVaR and related properties, and on this basis, proposes a single-cycle based on VaR and CVaR risk control Log-optimal portfolio model, in the United States well-known experts information theory Markovitz discuss optimal solution for this two models existence and uniqueness of the problem, a genetic algorithm to solve the model and the Numerical simulation to compare the differences between different models, and further proposes a Log optimal portfolio model with CVaR risk control, research its optimal solution existence, uniqueness and related properties of the model, a numerical calculation.In the literature of the random variable dependent conditions are too harsh, difficult to meet the needs of practical application. This article will continue to thoroughly investigate this problem, we use a sample asymptotic logarithmic likelihood ratio concept to characterize the dependence between random variables, studied in the case of appropriate weakening dependent conditions, and the use of Borel-Cantelli lemma and other properties has been strong convergence of random sequence. After this discussion when there is a deviation between the investors about its true distribution of income edge vector product distribution limit behavior of investors over a period of time, average earnings, and to obtain an estimate of the deviation of the upper and lower bounds, making the model more meet the needs of practical application.In this paper, sample asymptotic logarithmic likelihood ratio sequence to characterize the dependence of investment, the actual conditions of the independent variables were reduced using sequence analysis of randomized strong limit of the the relevant results to the case of dependent and so that only the information required period of time can be calculated yield any sequence portfolio. And access to the nature of any portfolio growth and some limit theorems under normal market conditions.