Optimal Investment And Consumption With Rough Paths

The optimal investment and consumption problem is mainly to study how to distribute investors’ wealth equitably to investing in various assets and consuming,so as to maximum their utility of consumption and final wealth.Financial asset prices fluctuate irregularly and frequently due to numerous factors,and thus can be regarded as a very rough path.Existing studies almost based on the stochastic processes associated with Brownian motion.However,Brownian motion has some limitations when describing financial time series.Rough path theory provides an effective way to describe such rougher paths,which enables it to contain more information about the irregular fluctuations of financial asset prices.Therefore,we investigate the optimal portfolio and consumption problem in the financial market where asset prices are described as rough paths,and obtain the approximate solution to the optimization problem.Firstly,we use rough path theory to describe the price path of risky assets and obtain the corresponding wealth process of investors.Secondly,we obtain the approximate optimal strategies of investors with power utility function and exponential utility function respectively in both finite time domain and infinite time domain.Finally,we provide both numerical and empirical analyses of our strategy in contrast with the Merton’s classical optimal strategy.Comparing with the Merton model,we find that the rough path is a more suitable description of the real financial market,since it can describe financial asset prices more properly.Therefore,investors can increase their utility by adjusting their optimal consumption and investment strategies more accurately.