Continuous-time Dynamic Portfolio Optimization With Multiple Risk Measures

Portfolio selection theory is to fnd the best allocation of the wealth to spread risksandraiserevenues. In1952,Markowitzproposedthemean-varianceportfolioselectionmodel to balance the expected return and the variance of the portfolio and eventuallywon a Nobel prize in Economics in1990. The philosophy behind the mean-varianceportfolio selection model tells the investor not only to focus the expected return butalso the risk of the investment. Markowitz’s model has laid the foundation of modernfnance and demonstrated a far-reaching impact. With the study and application ofportfolio selection theory in the last half century, number of risk measures have beendeveloped. The investment environment considered has also developed from completemarket to incomplete market and from discrete time to continuous time. The portfolioselection theory has been studied extensively.On the basis of former works, this paper is devoted to continuous-time dynamicportfolio optimization with multiple risk measures. We take two risk measures intoconsideration, conditional value at risk and safety-frst principle. Combining the t-wo risk measures with mean-variance model, we study the mean-variance-CVaR andmean-variance-SFP model. As we suppose the market is incomplete, we need to trans-form the incomplete market to an complete one. Then we can use martingale methodto study our models.Before the study of our models, we introduce how to use martingale method indynamic problems and compute expectations of some special formulas. Martingalemethodcanhelpusdeveloptheexplicitportfoliopolicyandthecomputationmakesouranalysiseasy. Whensolvingthemodels, wefrstsolveastaticoptimizationproblemfortheoptimalterminalwealthandthenusethereplicatingstrategytoidentifytheportfolio policy which replicates such optimal terminal wealth. In the process of model solving,wecanfndtheformsofsolutionofdiferentriskmeasuresisofgreatdiference,thoughthe idea is the same.At last we conduct simulated analysis. The results show that our dynamic mean-multiple risk portfolio models perform better than mean-variance model and our mod-elsinvestmorewealthinriskassetswhenthemarketconditionisgood. However, thereare still some problems with our models, so further research is needed.