Hamilton-Jacobi-Bellman Equation of an Optimal Consumption Problem Shuenn-Jyi Sheu Institute of Mathematics, Academia Sinica WSAF, CityU HK June 29-July 3, 2009
1. Introduction X c,π is the wealth with the consumption policy ( c, π ) . t π t is a trading strategy, c t X c,π is the rate of consumption. t We consider the optimal consumption problem � ∞ e − ρt 1 γ ( c t X c,π t ) γ dt ] , (1 . 1) sup E [ ( c,π ) ∈A 0 A is a family of admissible strategies. ρ is the discount factor.
γ ( x ) γ is the HARA utility with parameter γ , 1 U ( x ) = γ < 1 , γ � = 0 . Purpose: find an optimal strategy. • We consider a model for the market (factor model). Some economic factors affect the returns and volatilities of the stocks. • Dynamic programming approach is used. The Hamilton-Jacobi-Bellman (HJB) equation is derived.
• We study the solution of HJB equation. A general existence result for the solutions of HJB equation will be proved from the existence of a pair of sub/super-solution (of HJB equation). • We construct a suitable pair of sub/supersolution. • We give the verification theorem. The policy constructed from the solution is shown to be optimal. • We show that the solutions have different behaviors for γ > 0 and γ < 0 .
References H.Hata and S.J.Sheu (2009), Hamilton-Jacobi-Bellman equation for an optimal consumption problem, preprint. H.Hata and S.J.Sheu (2009), An optimal consumption and investment problem with linear Gaussian model, preprint.
2. A Brief History There are two kinds of investment problem that we can find many discussions in the literature. • Optimal consumption problem discussed in this paper. • Optimization problem of expected utility of final wealth, E [1 γ ( X π T ) γ ] , (2 . 1) sup π ∈ ˆ A
We show an interesting relation of these two problems following a recent result of Hata-Sheu. Fleming-Hernandez (2005) is one of few examples discussing such relation. We start a brief review of the studies of these two problems in the literature. In Merton(1969), the following problem is discussed, � T e − ρt U ( C ( t )) dt + B ( X ( T ) , T )] . sup E [ 0 C ( t ) is the consumption rate, X ( t ) is the wealth process.
U, B are utility functions. R.C. Merton (1969), Lifetime portfolio selection under uncertainty: the continuous time case, The Review of Economics and Statistics, Vol 51, 246-257 ( with 1475 citations). In Cox-Huang (1986), a similar problem to Merton (1969) is considered for general complete markets. J. C. Cox and C.F. Huang (1989), Optimal consumption and portfolio policies when the asset prices follow a diffusion process, J. Economic Theory, Vol 49, 33-83(with 690 citations). In Pliska(1986), the following problem is considered for
the complete market with general diffusion model, π E [ U ( X π ( T ))] . sup S. Pliska(1986), A stochastic calculus model of continuous trading: optimal portfolios, Math. Operation Research Vol.11, 371-382 (with 218 citations) The developments in Pliska(1986) and Cox-Huang (1989) start the use of stochastic calculus, martingale representation theorem and duality argument. Using their approach, the optimal solution can be explicitly
calculated without solving PDE. This is very interesting and is also a reason for receiving so many citations. Their method is different from the approach in Merton (1969). In Merton (1969), the solution is obtained by solving the HJB (Hamilton-Jacobi-Bellman) equation. HJB is a PDE (partial differential equation). For a simple model, the equation can be solved explicitly. In general, it is difficult to calculate the solution.
The idea in Pliska, Cox-Huang can not be applied to incomplete markets. Previous studies have suggested some possible directions for research: • Obtain an explicit solution by generalizing the idea of Pliska and Cox-Huang. • Study HJB equation. In practice, it is also important to solve the equation numerically, if an analytical solution is not possible.
In addition to these, there are also a huge number of possible applications can suggest many interesting problems. The following show some models which have analytical solution, Wachter(2002),Chacko-Viceira(2005),Jun Liu (2007) An initial attempt to use HJB equation in more complicated models is first proposed in Fleming (1995). The idea is to reformulate the investment problem as a stochastic control problem. Then the dynamic programming approach can be used to derive the HJB
equation. Solving HJB equation will provide a candidate of optimal investment strategy. This suggests several interesting questions for the solutions of such HJB equations. • Study the regularity, growth conditions of the solutions. • Obtain suitable estimates for the solutions. This is needed when we want to prove the candidate of optimal portfolio derived from a solution is indeed optimal (verification theorem).
To show how this works, Fleming-Sheu(1999) study a simple model different from that of Merton(1969) and provides a detailed analysis. The model can be briefly described as follows There is one stock and one banking account that an investor can trade. The interest rate for the banking account is constant r > 0 . The price of stock is given by P t = exp( L t ) ,
dL t = c ( µt + α 0 − L t ) dt + σdW t . It turns out that the model in Fleming-Sheu (1999), after reformulation, becomes a special case of factor model. y ( t ) = L t − µt − ( α 0 − µ c ) plays the role of factor. Using this approach, the risk-sensitive portfolio optimization problem for more general factor models have been considered in a list of papers: Fleming-Sheu (1999a, b), Fleming-Sheu(2002), Kuroda-Nagai(2002), Nagai-Peng(2002), Nagai(2003),
Kaise-Sheu(2004), Hata-Sekine(2005), Bilecki-Pliska- Sheu(2005) An useful theorem about the structure of the solutions of HJB equation is given in Kaise-Sheu(2006). The studies mentioned above have interesting applications to the minimization of down-side risk probabilities, min P (log X π T (2 . 2) ≤ k ) . T There is a duality relation between (2.2) and the risk sensitive portfolio optimization problem (2.1) with a particular risk sensitive parameter γ = γ ( k ) < 0 .
The following is a list of papers, Hata-Nagai-Sheu(2009), Hata-Sheu(2008), Hata(2008), Nagai(2008, 2009) The result is interesting because of the following reasons. • The problem (2.2) is not a conventional optimization problem and a direct solution is not available. The result shows that we can solve (2.2) using a solution of (2.1). • HARA utility appears naturally from (2.2). Although
(2.2) seems to not have relation with utilty function at the first look. For this connection, see also works of Pham (2003) on the maximization problem of up-side chance probabilities, max P (log X π T ≥ c ) . T We remark that a recent work of Follmer-Schachermayer (2008) seems to also relate to our study, In this talk, we will discuss another application of the study of the risk sensitive portfolio optimization problem
(2.1) to the consumption problem in (1.1). This application seems to not be expected from the results in the literature. We are motivated by the three recent papers, Fleming- Hernandez(2003, 2005), Fleming-Pang(2004). Fleming-Pang (2004) considers a model that the stock price is geometric Brownian motion and interest rate of the banking account is random and is an ergodic 1-d diffusion process. We also use an approach similar to Fleming-Pang (2004).
We show that a solution of HJB for (2.1) can be used to construct a supersolution for the HJB of (1.1). Then a nice solution of the HJB for (1.1) can be obtained. From this, (1.1) can be solved. We develop some useful ideas for general factor models with multiple stocks. An attempt to use the duality argument similar to that in Pliska(1986)and Cox-Huang(1989) is proposed in Castaneda-Hernandez (2005)for general factor models. A solution is given in the case of HARA utility. However, the result for general utilities is not satisfactory.
References Analytical Solution • J.A. Wachter(2002), Portfolio and consumption decisions under mean-reverting ruturns: an exact solution for complete markets, J. Financ. Quant. Anal. 37, 63-91. • Chacko, G. and Viceira, L.M.(2005), Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, Rev. Financ. Stud. 18, 1369-1402. • Jun Liu (2007), Portfolio selection in stochastic enviroments, Rev. Financ. Stud. 20, 1-39. Risk Sensitive Portfolio Optimization
• Fleming, W.H.(1995),Optimal investment models and risk-sensitive stochastic control. Mathematical Finance (Davis M, et al,ed), Spring-Verlag, Berlin. • W.H. Fleming and S. J. Sheu (1999), optimal long term growth rate of expected utility of wealth, Ann. Appl. Probab., Vol 9, 871-903 • W.H. Fleming and S.J. Sheu (1999), Risk sensitive control and an optimal investment model, Math. Finance 10, 197-213. • W.H. Fleming and S.J. Sheu (2002), Risk sensitive control and an optimal investment model II, Ann. Appl. Probab. 12, 730-767. • K. Kuroda and H. Nagai (2002), risk sensitive portfolio optimization
Recommend
More recommend
