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Elements of Utility Theory
Published in Craig Friedman, Sven Sandow, Utility-Based Learning from Data, 2016
In this section, we provide examples of several popular utility functions. All are special (sometimes limiting) cases of the HARA (hyperbolic absolute risk aversion) utility function: U(W)=1-γγaW1-γ+bγ.
A study on optimal consumption and portfolio with labour income under inflation
Published in Systems Science & Control Engineering, 2019
Fanhong Zhang, Weiyin Fei, Mingxuan Shen, Kui Jiang
For various reasons the investors will face the different inflation volatilities at different times. Thus, investors should adjust the consumption and portfolio strategies to reduce risky exposure and get more benefits according to the changes of the volatility of inflation. Based on the framework of Moos (2011), This paper studies the consumption and portfolio problem of an investor with labour income under inflation. The innovation of this paper are as follows. First, we divide the investor's cycle life into two phases of retirement and employment and focus on the phase of employment, then the mathematical model under inflation and real financial process are built. Next, we obtain the HJB equation of investor's terminal utility maximization problem by using the method of stochastic control, and an explicit expression in the case of hyperbolic absolute risk aversion (HARA) utility is obtained. Finally, the range of inflation volatility is given, and according to the numerical simulation method we analysed the effects of inflation volatility for the optimal consumption and portfolio of the investor with labour income.
Management Strategies for the Defined Benefit Pension Fund Under Stochastic Framework
Published in American Journal of Mathematical and Management Sciences, 2020
Patrick Kandege Mwanakatwe, Lixing Song, Xiaoguang Wang
Substituting Equation (11) into Equation (1) results in: By considering logarithmic utility function and substituting (12) into (13) gives the following after simplification: Thus, the stochastic optimal control problem has been transformed into a nonlinear second order PDE given by (14). Due to the complexity of Hyperbolic Absolute Risk Aversion (HARA) utility and boundary conditions, it is very challenging to conjecture directly the form of the solution of (14). To overcome the difficulties, we apply the Legendre transform technique to change the equation into its dual equation, whose solution is easy to conjecture.