| Optimal consumption and investment is a most fundamental problem in Financial Mathematics, the research of which originates from Merton (1969, 1971). Investors may choose freely his consumption and investment to except to maximum the consumption and final wealthutility in[0,T]or[0,∞]. Merton and other scholars all consider, in theiroptimum consumption and investment model, that the investor's assets can choose between consumption and securities portfolio. However, with the economic development, changes in people's financial assets structure have taken place, which include not only the savings deposit in bank and negotiable securities, but also the ever increasing insurance purchasing. Taking the above into consideration, investors no long confine the distribution of their individual assets to only two aspects between consumption and securities investment, and they would consider whether to purchase various kinds of insurances or not; that is, in order that the expected utility can achieve its maximum, the investors should think over how to reasonably distribute their assets among consumption, securities investment and insurance purchasing.This dissertation focuses on the improvement of the optimum consumption and investment model by Merton and other scholars and makes a thorough consideration of the reasonable assets distribution among consumption, securities portfolio and insurance purchasing. Besides, this dissertation also discusses the influence insurance purchasing upon the consumption and investment after insurance is introduced. The insurances discussed in this dissertation consist of individual periodical insurance payable at death for life, insurance of property and annuity insurance.By establishing a dynamic mathematic model, this dissertation works out a reasonable solution to the distribution of individual assets among consumption, investment and insurance purchasing in order to achieve the maximum expected utility of consumption and final wealth endowment. Except for the introduction of insurance model into the optimum consumption and investment model by Merton and other scholars, there isone more improvement, that is, besides the fixed investment time horizon, limited or unlimited, discussed by Merton and other scholars, part of this dissertation discusses the unfixed decisive time horizon as well, for the time for the investors' death is indefinite.Methods adopted in this dissertation are the dynamic programming principle and the stochastic analysis theory, through which the HJB equation corresponding to the control question can be worked out, and therefore, the optimal strategy with feedback from can be obtained.This dissertation can be chiefly divided into three parts. The first part covers from Chapter Two to Chapter Five, mainly concerning the optimum investment, consumption and periodical insurance payable at death for life model; the second part contains Chapter Six and Chapter Seven, the major topic of which is the optimum investment, consumption and insurance of property; the third part is Chapter Eight, which concentrates on the optimum investment, consumption and annuity insurance.In Chapter Two, the general model of the optimum investment, consumption and periodical insurance payable at death for life is discussed and its corresponding optimum control question is solved .The optimum strategy can be got through the corresponding HIB (Hamilton-Jacobi-Bellman) equation. As to the CRRA (constant relative risk aversion), a sort of utility function, indicatively, the optimum investment process, consumption process and the periodical insurance payable at death for life purchasing process can be gained with the feedback form.On what has been discussed in Chapter Two, Chapter Three takes the varied situations when the deposit and loan interest rate changes into consideration. By employing the dynamic programming principle and the stochastic analysis theory, the optimum strategy can be achieved through HJB equation corresponding to the control question. As to the sp...
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