Optimal Investment And Consumption In The Market With Jump Risk And Capital Tax

Optimal investment and consumption problems have always been one of the focuses of research at home and abroad.Optimal investment and consumption problems mainly study how investors allocate their wealth between consumption and investment under the fixed initial wealth,and maximize the sum of all consumption experience utility and wealth expectation utility.Optimal investment and consumption problems are the fundamental problems that affect the vital interests of all investors,and can guide every rational investor.The government can also speculate on the behavior of most investors based on the solution of optimal investment and consumption problems,and then predict changes in the scale of private investment and consumption,and take measures to maintain a sound and rapid development of the macro economy.With the expansion of China's economic scale,tax evasion has also expanded.Taxation is an important macro-control measure of the government and plays a major role in maintaining economic stability and social equity.The loss of tax revenue,the impaired legal majesty and social inequity which is caused by tax evasion is gradually receiving the attention of relevant authorities recent years.It is especially urgent and important to find effective measures to reduce the scale of tax evasion,maintain the seriousness of laws,and promote social equity.This paper investigates the problem of dynamic investment and consumption in a market,where a risky asset evolves with jumps and capital gains are taxed.Moreover,we explore two scenarios that the jumps of the risky asset are driven by Hawkes process and Possion process,respectively.In addition,the investor's behavior of tax evasion is considered,but this does not mean that this paper considers tax evasion to be legal.Tax evasion is subject to penalty when it is uncovered by audits.And we also explore two scenarios that the jumps of audits are driven by Hawkes process and Possion process respectively.Using dynamic programming approach,this paper solves the optimal investment consumption problem.For investors with logarithmic utility,the following conclusions are obtained:First,jumps in the risky asset do not affect the optimal tax evasion strategy;Second,jump risk lessens the optimal fraction of wealth in the risky asset,which will become more significant when jumps are driven by Hawkes process;Third,both increasing the fine and the frequency of tax audits can reduce tax evasion,but social wealth may be concentrated in risky assets;further more,it is an effective way to increase the frequency of tax audits against investors who had been find evading tax in the past.