| The issue of household asset allocation has been studied for many decades,and there has been plenty of research on it in the literature.Although it is often overlooked by ordinary families in real life,the need for research on this issue has gradually emerged as families’ income and wealth grow.Research on household asset allocation did not only provide households with reasonable advice to optimize household welfare,but also help the government to fully understand the response mechanism of household demand to the economic environment,so as to appropriately develop policies and plans.A commonly adopted pattern is to build a mathematical optimization model based on some features of the actual data or a type of preference that households may have,and then observe whether the optimal strategy fits the actual features through numerical experiments.For dynamic optimization of intertemporal decisions,stochastic control is a powerful frameworks,under which the household’s investment-consumption decision is a classical linear control problem.The mainstream methods for investment-consumption problem in the stochastic control framework include dynamic programming,martingale method,and stochastic maximum principle.For dynamic programming approach,with the help of Bellman’s optimality principle,one shall derive the Hamilton-Jacobi-Bellman(HJB)equation that the optimal value function satisfies under the smoothness assumption,and thereby establish a sufficient optimality condition to verify that the solution provided by the HJB equation is exactly the value function and the feedback function of the optimal investment-consumption strategy.For martingale method,one shall derive the optimality condition as that the consumption and terminal wealth are almost surely equal to some shadow-price-related random variables under the risk-neutral measure,and then derive the investment by using the martingale representation theorem.For stochastic maximum principle,one shall adopt a spike modification to perturb the investment-consumption strategy over a sufficiently small period,and compare the differences in the state processes and objective functions before and after the perturbation.Then,two backward stochastic differential equations as adjoint equations are introduced,such that the sufficient/necessary optimality condition can be expressed as that the optimal control must maximize the Hamiltonian.Thanks to the abovementioned methods or their improvements and extensions,the thesis has studied several specific investment-consumption problems.As the main body of this thesis,they will be presented in Chapters 3-5,which are divided into the following three categories: classical investment-consumption problem in maximizing the utility of consumption and terminal wealth,seeking time-consistent strategies to overcome the timeinconsistency of the utility maximizer,and seeking time-consistent strategies under the criterion of the higher moments of terminal wealth.More precisely,in the first category there are three investment-consumption problems: the problem with random income and the income-dependent borrowing constraint,the problem with multiple habit formation of interaction,and the problem with regime-switching-modulated habit formation and jumpdiffusion.The second category includes two time-inconsistent problems: the consumptionportfolio control problem with regime-switching-modulated habit formation,and the investment-consumption-insurance control problem with multiple habit formation and nonexponential discounting.The third category includes a portfolio selection problem and a general stochastic control problem,which are corresponding to higher moment criteria.Apart from the three main chapters,the rest of this thesis is organized as follows.Chapter 1 of the thesis is an introductory chapter for providing the background,content,ideas and methodology of the study,giving a detailed introduction to the current state of research and literature,and organizing the mathematical notation used in the text for ease of reference.In Chapter 2,the introduction and solution of Merton’s investment-consumption problem demonstrates how this problem can be solved by the three mainstream methods of stochastic control.This can be regarded as mathematical preliminaries of the whole thesis.The final chapter provides some concluding remarks,technical breakthroughs,results and economic insights in the thesis,and lists some relevant follow-up research questions.As a challenging attempt and innovation,this thesis uses general utility functions in the problems.After deriving the analytical form or sufficient conditions for optimal control or time-consistent control,the form of utility functions is specified to calculate the corresponding results.Thus,the problem is not always solved in a way that the form of the solution is first constructed and then substituted into the HJB equation or the forwardbackward stochastic differential equation(FBSDE)for verification,but rather the properties or even the expression of the solution are characterized with necessity.This makes it possible that the solution process itself may yield inspiration for solving other problems in mathematical finance.For the investment-consumption problem with random income and income-dependent borrowing constraint in Section 3.1,the case in which the household liabilities is limited to the human capital represented by the present value of future income is easily solved by the dynamic programming principle with a duality transformation for the HJB equation.However,when household liabilities are limited to a certain proportion of human capital,the optimality condition at this boundary becomes quite complicated and the duality transformation is no longer applicable.Since that household income is not a controlled process,parameterizing it allows the value function associated with the problem to be expressed in terms of a random field.The dynamic programming principle gives the stochastic HJB equation associated with this value random field.By the duality transformation,especially for boundary conditions,the stochastic HJB equation can be transformed into another backward stochastic partial differential equation(BSPDE)with constraints,which is associated with a singular control problem.Following the steps of the martingale convex duality method,one can obtain the max-min duality between the primal problem and the dual singular control problem.Finally,in the case of power utility,infinite time-horizon,fixed income rate and constant parameters,an explicit expression for the optimal feedback control of the primal problem is obtained based on the closed-form solution of the dual singular control problem.For utility maximization problem with multiple habit formation of interaction in Section3.2,the household’s current preference depends not only on current consumption but also on past consumption.Past consumption would generate an addictive pattern,which makes the household derive no satisfaction from this habitual consumption.Moreover,the consumption habits generated within the decision period also have a potential impact on preference after the time-horizon,which is included in the terminal wealth utility.Also,for the problem this thesis studies households’ categorization of consumption(such as daily living expenses,educational inputs,and health care costs),and assumes without loss of generality that habit formation for all categories of consumption has an interactive effect.Habit formation is assumed to be a linear system based on the consumption process.Then,the functional fixed point of this linear system stands for a minimal future consumption,in comparison with all the feasible consumption strategy of the problem.Accordingly,the present value of this minimum consumption is subtracted from the household wealth to construct a new state process.And then the optimal control is obtained by the dynamic programming principle and the duality transformation for the HJB equation.The results show that the optimal consumption process is clearly expressed as the sum of the habitual and non-habitual components,and the non-habitual component is proportional to the newly constructed state process.The optimal investment process is not only proportional to the new state process,but also serves as a hedging strategy such that the terminal value of the state process is almost surely equal to some random variable related to the utility function of terminal wealth.The optimal investment-consumption strategy is in a closed form for both cases of power utility summation across consumption categories and power utility of a constant elasticity of substitution function for all consumption..For the problem with regime-switching-modulated habit formation and jump-diffusion in Section 3.3,the stock price process is modelled by a jump-diffusion stochastic differential equation incorporated with a Markov chain that simulates macro-economic systematic transitions,by which both household preferences and habit formation are also influenced.In order to obtain the analytic form of optimal control,it is necessary to introduce additional assets for market completion.The Poisson jump and Markov jump assets are introduced to hedge the randomness of the minimum consumption level generated by the habit formation,while their optimal positions can be considered as household demand for derivatives such as catastrophe insurance,index-linked insurance,annuities,and reverse mortgage bonds.The problem is formulated based on the state process constructed from abstracting the present value of the minimum consumption from the household wealth,and is solved via a Markov regime-switching-modulated stochastic HJB equation arising from the dynamic programming principle.With the help of shadow price,solving the stochastic HJB equation is transformed into solving a flow of linear FBSDE,which can be further transformed into a parabolic BSPDE.By advantage of the conditional expectation form of the BSPDE solution,the optimal investment-consumption strategy can be explicitly represented.The chapter illustrates the closed-form optimal investment-consumption for a two-state regime-switching model and a power utility function.The problem of incomplete markets based on an artificial model with continuous stock price,specific habit formation and power utility is also considered to show the technical differences arising from whether the market is complete or not in solving for the value function.For the time-inconsistent consumption-portfolio control problem with regime-switchingmodulated habit formation in Section 4.1,the household’s habit formation is completely dependent on the current macro-economic state.Intuitively speaking,households review all the past consumption and then re-calculate a habit level based on the current situation,while predicting future habit formation in view of the current macro-economic state.Thus,systematic shift in macro-economic states leads to time-inconsistency in the optimal control,i.e.,once the macro-economic state changes systematically,the investment-consumption strategy that maximizes utility at the current moment will also deviate from the past optimal planning.To overcome the time-inconsistency,households can insist on maximizing utility at a given moment or state,and keep the optimal strategy unchanging even if the macroeconomic state changes.Such a utility maximizer is called a pre-commitment strategy,and can be derived from the main steps of the previous problem with Markovian regime switching.Another approach to overcome the time-inconsistency is to regard the current and future decisions as multi-person sequential games,where the equilibrium strategy can be adopted and implemented automatically over the entire decision period,without the actual execution of investment-consumption decisions deviating from the expected plan.In particular,the sequential games are essentially cooperative,i.e.,each controller takes the utility of the subsequent controllers as its own terminal utility,rather than re-evaluating their strategy with his/her own preferences as non-cooperative games.This approach is justified by the fact that there is no opponent in the games for the individual household’s decision problem,where the subsequent controllers in the sequential games are still the future households themselves.Through the essentially cooperative approach,the problem of seeking a time-consistent strategy can be simply described as a utility optimization problem with time-consistency,and the corresponding optimal control is explicit..In Section 4.2,the investment-consumption-insurance control problem with multiple habit formation and non-exponential discounting involves the mortality risk of the head of household and general intertemporal discounting for utility functions of consumption,bequest and terminal wealth.Moreover,with reference to the consideration of the multiple habit formation,the bequest considered here also has an endogenous reference level and interacts with consumption habit.Since bequest is directly related to the death benefit of life insurance,the interaction between consumption habits and bequest reference levels can also be regarded as a modelling of the relationship between household demand in terms of consumption and lift insurance.Given the generality and heterogeneity of the discount factor,the investmentconsumption-insurance strategy that maximizes utility exhibits time-inconsistency.In addition to a pre-commitment strategy,we seek an open-loop Nash equilibrium control to overcome the time-inconsistency.Based on the spike variation method and a united procedure,precommitment strategy and open-loop Nash equilibrium control can be characterized by a stochastic maxima principle.By reversing the direction of their corresponding coupled stochastic differential equations and embedding them into a flow of standard linear FBSDEs.Then,their solutions can be explicitly represented.In the case of power utility functions and the hyperbolic discounting,numerical simulations are presented for the closed-form solution,including· a comparison between the consumption-premium-investment-wealth processes corresponding to the pre-commitment strategy and those corresponding to the open-loop Nash equilibrium control,· the effect of the presence or absence of consumption habits and bequest reference levels on the consumption-premium-investment-wealth processes corresponding to open-loop Nash equilibrium control,· and the sensitivity of the consumption-premium-investment-wealth processes corresponding to open-loop Nash equilibrium control to parameters in hyperbolic discounting and multiple habit formation.For the time-consistent portfolio selection under higher-order moment criterion in Section 5.1,this thesis simplifies the dynamics of household wealth and focus on the portfolio selection.The mean-variance problem is revisited,with the pre-commitment strategy for maximizing the objective function and the time-consistent strategy arising from multi-person sequential games.As an extension of the mean-variance case,the control problem,whose the objective function is a linear combination of the expectation and higher central moments of the terminal wealth,is studied.In deriving a time-consistent strategy,this thesis uses a sequence of multipliers to connect the Feynman-Kac representation of the origin moments of the terminal wealth,and impose an optimality condition on portfolio,so as to establish candidate extended HJB systems for a preliminary screening of feasible feedback controls.Sufficiency conditions for closed-loop Nash equilibrium control can be established by setting a specific form of the abovementioned multipliers.In solving the corresponding extended HJB system,deriving the equilibrium value function from its Bellman equation can be avoid.On the other hand,the candidate extended HJB systems are also adopted in the derivation of a time-consistent strategy through the sequential game approach.The optimization of the investment strategy over each period is transformed into the choice of multipliers.As the mesh size is sent to zero,the extended HJB system obtained by the sequential games is line with that for the closed-loop Nash equilibrium control.As a result,when the coefficients in the linear combination of central moments in the objective function are independent of the current wealth and satisfy certain concavity conditions,the problem admits a time-consistent investment strategy independent of the wealth level and the coefficients on the odd-order central moments.As a generalization of the previous time-inconsistent problem,the final stochastic control problem with higher central moment criterion in Section 5.2 is built on the basis of a more general theoretical model,in which the objective is a state-dependent function of the expectation and higher central moments of the terminal value of state process.The sufficient condition for closed-loop Nash equilibrium controls is constructed with the extended HJB system,which is established by specifying the multipliers for connecting the Feynman-Kac representation of the original moments.To verify that the extended HJB system is in line with the extended HJB system proposed in the literature,which is constructed around the Bellman equation of the equilibrium value function,this thesis derives an expansion of the objective function with respect to the spike variation in the sense of feedback control.Then the Bellman equation of the equilibrium value function,with the required consistency,is obtained.In terms of open-loop Nash equilibrium controls,this thesis introduces a flow of FBSDEs based on the expansion of the objective function with respect to the spike variation in the sense of control process,and then immediately obtain the sufficient maximum principle.It is interesting that in the case where the following two conditions are both satisfied:· the objective function does not depend on the initial state,but is linear in the expectation of the terminal value of state,· the state dynamics are linear,but its diffusion term does not contain the state itself,if a certain concavity condition is satisfied,then not only the closed-loop and open-loop Nash equilibrium controls exist,but also they can be independent of the state and the preference on odd-order central moments.Moreover,the control processes corresponding to these two Nash equilibrium controls are identical.Finally,this thesis shows two special examples,where the objective functions are linear combinations of expectation and central moment with the highest order central moment tending to infinity.In each example,the limit of the result arising from finite-order problems also provides the closed-loop and open-loop Nash equilibrium control associated to the objective function arising from the convergent series argument.While focusing on the use of stochastic control techniques to solve the abovementioned problems,this thesis derives some intuition from the analytical solutions or numerical results for dynamic decisions on household asset allocation.For example,households should stop investing in risky assets as soon as their liabilities hit the constraint and use their surplus income to reduce their debts with keeping consumption below the level of immediate income,instead of benefiting from investments to pay off.When facing with different types of risks,households can hedge their risks while also enhancing utility welfare by searching the market for derivatives related to these risks.For obtaining a time-consistent strategy,households can consider a cooperative perspective on current and future decisions,at least to the extent that this has the benefit of effectively reducing the mental or physical costs of decision making.If households appropriately include historical behavior in their current utility preferences,it is possible that their financial situation may improve slightly provided that the time-consistent strategy is implemented.Moreover,the results for the higher order moment portfolio decision problem are slightly counterintuitive.They suggest that under a fairly broad class of higher order moment criteria,households’ time-consistent portfolio decisions should not be driven by their wealth and income levels,nor should they be influenced by preference/aversion to the odd-order central moments. |
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