Optimal Dynamic Asset Allocation: A Stochastic Invariance Approach

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1 Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 26 ThA7.4 Optimal Dynamic Asset Allocation: A Stochastic Invariance Approach Gianni Pola & Giordano Pola Abstract Optimal Asset Allocation deals with how to divide the investor s wealth across some asset classes in order to maximize the investor s gain. We consider the Optimal Asset Allocation in a multi period investment settings: Optimal Dynamic Asset Allocation provides the (optimal) re balancing policy to accomplish some investment s criteria. Given a sequence of target sets, which represent the portfolio specifications at each re balancing time, an optimal portfolio allocation is synthesized for maximizing the joint probability for the portfolio to fulfill the target sets requirements. The approach pursued is based on Dynamic Programming. The optimal solution is shown to conditionally depend on the portfolio realization, thus providing a practical scheme for the dynamic portfolio re balancing. Finally some case studies are given to show the proposed methodology. Keywords: Optimal Dynamic Asset Allocation, Portfolio Re balancing, Multi Period Investment, Optimal Control, Dynamic Programming, Stochastic Invariance Problem. I. INTRODUCTION In finance industry, portfolio allocations are usually achieved by an optimization process. An asset class is a specific category of investments, such as stocks, bonds, cash, and commodities. Asset allocation deals with how to divide the investor s wealth across some asset classes, selected from a given asset classes menu, in order to maximize the investor s gain. Standard approach for Optimal Asset Allocation is based on Markowitz Mean Variance model [1]. This approach is static by its nature. It permits the investor to make a one shot allocation to a given time horizon: portfolio re balancing during the investment life time is not faced in the model. The main assumption of the Markowitz approach is that asset classes performances are distributed according to a multivariate gaussian distribution. Recently a new focus has been placed on a dynamic formulation of asset allocation. The main reason is that dynamic allocation gives way to a tighter controlled portfolio evolution during the investment life time. Multi period investment strategies can be split into three main categories [5]. Tactical Asset Allocation (TAA) refers to an investment policy in a short time horizon. Strategic Asset Allocation (SAA) differs essentially from TAA in being it tuned for longer investment time horizons. Dynamic Asset This work has been partially supported by the HYCON Network of Excellence, contract number FP6-IST and by Ministero dell Istruzione, dell Universita e della Ricerca under Project MACSI (PRIN5). Gianni Pola is with CAAM SGR SpA, Quantitative Research, Piazzale Cadorna 3, 2123 Milan, Italy, gianni.pola@caamsgr.it Giordano Pola is with the Department of Electrical Engineering and Computer Science, Center of Excellence DEWS, University of L Aquila, Poggio di Roio, 674 L Aquila, Italy, pola@ing.univaq.it Allocation (DAA) lies in between TAA and SAA: whereas SAA looks for the best asset mix in the long run on average, DAA takes into account asset classes dynamics during the investment life time. Optimal Dynamic Asset Allocation (ODAA) deals with how to optimally allocate a multi period investment. First studies in ODAA (see [9] and [12]) faced the problem of how to divide the investment among the stock and the money markets. More recently, some works appeared in literature concerning optimal strategies for long lived investors under stochastic investment opportunities (see [3] and the references therein): in particular [2] studies the portfolio re balancing in the presence of stochastic variation in the interest rate, [3] considers the effects of inflation in a portfolio of stock or nominal bonds, [1] and [13] take into account the uncertainty in the asset returns prediction. In this paper we consider the ODAA from a Control System Theory perspective. We will show that the ODAA can be reformulated as a suitable optimal control problem. Given a sequence of target sets, which represent the portfolio specifications, an optimal portfolio allocation is synthesized for maximizing the joint probability for the portfolio to fulfill the target sets requirements. The proposed optimal control problem has been solved by using a Dynamic Programming [4] approach; in particular the ODAA is shown to be cast into the framework of the Stochastic Invariance Problem, as studied in [11]. The optimal solution is shown to conditionally depend on the portfolio realization, thus providing a practical scheme for dynamic portfolio re balancing. ODAA is contrasted to Markowitz formalism [1]; in particular it is shown that in the single period limit and gaussian hypothesis, optimal solution of the ODAA problem is in the boundary of the feasible region, while Markowitz optimal solution lies in the so called Efficient Frontier. The proposed approach seems particularly appealing since: (i) it provides an optimal re balancing policy which conditionally depends on the portfolio realization; (ii) it does not assume any particular distribution on the stochastic random variables involved; (iii) it does not assume stationarity in the stochastic dynamics for asset-classes. Features (i) and (ii) are shown to overcome Markowitz limits and feature (iii) is very useful for Scenario Analysis [6]. The benefit of the approach developed in this paper is illustrated by means of two practical case studies. The first one shows the benefit of ODAA over Markowitz method in treating strongly asymmetric and leptokurtic asset classes. The second case study includes Cash, Bond and Equity global indices. We show that whilst the re balancing period /6/$2. 26 IEEE. 2589

2 45th IEEE CDC, San Diego, USA, Dec , 26 ThA7.4 decreases, the probability to meet the investors specifications increases. This paper is organized as follows. In Section II preliminary definitions are introduced. Section III briefly recalls main results of Markowitz approach. Section IV presents our approach while Section V illustrates two case studies. Finally Section VI offers some conclusive remarks. II. PRELIMINARIES AND BASIC DEFINITIONS An asset class is a specific category of investments. Assets within the same class generally exhibit similar risk characteristics. Portfolio construction deals with how to divide the investors wealth across some asset classes for maximizing the investors gain. Consider an investment universe made of m asset classes. Given k N, define the vector: w k = [ w k (1) w k (2) w k (m) ] T R m, where the entries are the performances, or equivalently the returns, which are associated with the asset classes at time k. The performance w k (i), linked to the i th asset class at time k, is defined as the percentage variation of the asset class price in the time interval [k 1,k), i.e. w k (i) = z k(i) z k 1 (i), (1) z k 1 (i) where z k (i) and z k 1 (i) are the prices of the i th asset class, respectively, at time k and k 1. From equation (1), it follows that the asset classes performance is downside bounded, i.e. w k (i) [ 1, + ), i =1, 2,..., m, k N. The asset classes performances w k (i) are modelled as discrete time stochastic processes and hence w k is a random vector describing the asset classes performance at time k. Let p wk be the probability density function associated with w k. We assume that w k is a correlated random vector. In general, correlation can be split into two categories: synchronous correlation, deals with correlation among asset classes at the same time interval (i.e. correlation among w k (i) and w k (j), for each i, j =1, 2,...,m); time lagged correlation, refers to correlation among asset classes at lagged time interval (i.e. correlation among w k (i) and w k (j), with k k and for each i, j =1, 2,...,m). Synchronous correlation is much more relevant in financial time series [8] and therefore we assume that w k is characterized only by synchronous correlation. The Expected Return (ER) for the i th asset class at time k is defined by: µ k (i) =E[w k (i)], where E[ ] is the usual expectation operator. For notational simplicity we denote by µ k R m the collection of the asset classes ERs at time k, i.e. µ k = [ µ k (1) µ k (2) µ k (m) ] R m. Covariance matrix (CM) at time k is defined by: S k (i, j) =E[(w k (i) E[w k (i)])(w k (j) E[w k (j)])]. Asset classes standard deviation is known in the finance industry as volatility: it is a risk measure expressing the variability of the asset class performance around the ER. A portfolio allocation is a vector u R m, where the i th entry expresses the amount of the investment in the i th asset class. For example: if m =3, u =[.3.5.2] indicates that we are investing 3% of our wealth in the first asset class, 5% in the second, and 2% in the latter. Let u k be the portfolio allocation at time k N. Usually in the investment process some constraints are imposed on u k. The most commons are: Budget constraint: i u k(i) =1. This equality means that the investor s wealth is fully invested in the portfolio; Long only constraint: u k (i), i =1,...,m. This inequality implies that short selling is not allowed; Risk budget constraint: (u T k 1 S ku k 1 ) 1/2 σ max. This relation indicates that portfolio risk is up ward bounded by σ max. Sometimes, some further constraints are imposed on u k and for this reason we prefer to consider the most general case by setting U k R m as the collection of all the portfolio allocations allowed in the investment process at time k, i.e. any admissible portfolio allocation u k at time k must satisfy u k U k. As defined by equation (1) for a single asset class, we define the portfolio performance, or equivalently the portfolio return, as: r k = x k x k 1, (2) x k 1 where x k and x k 1 are the portfolio value, respectively, at times k and k 1. It can be shown [7] that the portfolio performance r k at time k is given by: The portfolio ER at time k is given by: r k = u T k 1w k. (3) µ k = E[r k ]=u T k 1E[w k ]. Analogously the portfolio volatility at time k is given by σ k =(E[(r k µ k ) 2 ]) 1/2 =(u T k 1S k u k 1 ) 1/2. Optimal Asset Allocation deals with how to select u k so that a given optimality criterion is fulfilled. Traditional approaches in the finance industry make use of the Markowitz model. We briefly review the classical results of the Markowitz method in the next section to then introducing our approach in Section IV. III. MARKOWITZ APPROACH Markowitz method [1] provides an optimal solution to the asset allocation problem: in particular the optimality criterion is the risk minimization. Main assumptions in the Markowitz s world are: All investors have the same time horizon: they make a one shot allocation, and they are not allowed to re balance their portfolio during the investment life time; 259

3 45th IEEE CDC, San Diego, USA, Dec , 26 ThA7.4 Fig. 1. Feasible region C k and the Efficient Frontier. All investors agree on the same stochastic model of the asset classes performance. Moreover the asset classes performance is assumed to be multivariate gaussian distributed. All investors agree on the asset classes ERs and CM, as well. The last assumption makes the portfolio performance gaussian distributed, and hence fully described by the distribution mean (portfolio ER) and standard deviation (portfolio volatility). 1 For any given k N, the portfolio allocation set U k is a collection of m dimensional vectors. However a low dimensional representation for U k can be provided by a bi dimensional plane, called the Mean Variance plane. A Mean Variance plot reports the portfolio volatility on the X axis and the portfolio ER on the Y axis. For any given k N, U k spans a bi dimensional region C k in the Mean Variance plane, defined as follows: C k = {( µ, σ) : u k U k s.t. µ = u T k µ k, σ =(u T k S k u k ) 1 2 }, where µ k and S k are respectively the asset classes ERs and CM at time k. C k is called feasible region for the portfolio allocations. The boundary of C k is denoted by C k. The feasible region is in general unbounded; however some constraints on u k make it a bound set (e.g. budget and long only constraints.) Figure 1 shows the feasible region of a three asset classes menu (budget and long only constraints imposed; details in Section V-B.) Asset classes are represented in the Mean Variance plane (Figure 1; black spots.) Portfolio allocation with maximum ER is given by u k =[1]. Let us consider an asset classes menu, characterized by ERs and CM given respectively by µ k and S k, and a target return r. Investors wish to minimize portfolio risk among the portfolios ensemble, while guaranteeing the target return r. The optimization problem is then formalized as follows: { minuk U k u T k S ku k, u T k µ (4) k = r. Feasible target returns belong to the interval [r min, r max ], where: r min = ( arg inf u T ) T k S k u k µk, u k U k r max = sup u k U k u T k µ k. 1 It should be stressed that a gaussian modelling is not appropriate to represent the portfolio return r k that takes value in [ 1, ). By varying target return r in [r min, r max ], the optimization problem (4) gives back the collection of all optimal portfolios, known in the literature as the Efficient Frontier (Figure 1; black line). It is worth pointing out that the Efficient Frontier is a proper subset of C k. For more details see [7]. Only multivariate gaussian asset classes are fully described by the Markowitz framework. Strongly asymmetric and leptokurtic asset classes [8] present distributional features that are very important in the asset allocation, although completely ignored in this approach. (See Section V-A for a practical example.) IV. OPTIMAL DYNAMIC ASSET ALLOCATION The aim of this section is to approach the portfolio allocation from a different perspective. In particular, we provide an optimal asset allocation solution which goes beyond the gaussian hypothesis and single period approach of the Markowitz formulation. A non linear stochastic discrete time dynamical control system, modelling the portfolio value dynamics, can be obtained by combining equations (2) and (3), as follows: x k+1 = x k (1 + u T k w k+1 ), k N, (5) where: x k R is the portfolio value at time k; w k R m is a random vector describing the asset classes returns, with probability density function p wk ; u k U k is the control at time k, representing the portfolio allocation. Portfolio value at time k =is assumed to be known; for simplicity x =1. Let (Ω, F,P) be the probability space associated with system (5). In a dynamic asset allocation settings, a good mathematical modelling of the control u k is the class of time varying feedback control strategies, that is, u k = g(k, x k ) U k, k N, (6) for some g(k,.) :R U k, k N. For any time k N, denote by U k the class of time varying feedback functions of the form (6). Let us consider a finite time horizon N. Our approach deals with how to choose the N controls u k U k,k=,...,n 1 in order to fulfill some specifications on the portfolio value x k at time k =1,...,N. The specifications are defined by means of a sequence of sets, {Σ k } k=1,...,n. The investor wishes to have a portfolio value x k at time k that is in Σ k, i.e. x k Σ k, and therefore the controller synthesis problem deals with how to select controls u k, k =, 1,..., N 1 in order to maximize the joint probability quantity: P ({ω Ω:x Σ,x 1 Σ 1,...,x N Σ N }). (7) More formally, Problem 1: (Optimal Dynamic Asset Allocation (ODAA)) Given a finite time horizon N N and a sequence of target sets {Σ k } k=,1,...,n, where Σ k are Borel subsets of R, for 2591

4 45th IEEE CDC, San Diego, USA, Dec , 26 ThA7.4 any k =, 1,..., N, find optimal portfolio allocations u k U k,k=, 1,..., N 1 that maximizes the joint probability (7), i.e. sup P ({ω Ω:x Σ,...,x N Σ N }). u k U k,k=,...,n 1 (8) The problem above can be cast into the framework of the Stochastic Invariance Problem, as introduced and studied in [11]. In the following, we specialize the results of [11] to our framework so that the optimal solution of the ODAA Problem can be synthesized. Given N N, and π = {u,u 1,..., u N 1 } with u i U i for any i =, 1,..., N 1, set: π k = {u k,u k+1,..., u N 1 }, k =, 1,...,N 1; then π = π. The vector π is said to be a control policy. Let p f(x,uk,w k+1 ) be the probability density function associated with the random variable: f(x, u k,w k+1 )=x(1 + u T k w k+1 ). Inspired by [4], we now define the cost function associated with the probability quantity (7) and hence to the ODAA Problem. Given a sequence of sets Σ k, k =, 1,..., N, we introduce the following cost function V which associates a real number V (k, x, π k ) [, 1] to a triple (k, x, π k ) by: I ΣN (x), if k = N; V (k, x, π k )= Σ k+1 V (k +1,z,π k+1 )p f(x,uk,w k+1 )(z)dz if k = N 1,N 2,...,, where I A (x) is the indicator function of a Borel subset A of Σ N, i.e. { 1, x A I A (x) =, otherwise. Notice that cost function V is defined backwards, as required in standard Dynamic Programming algorithms [4]. The following result gives way for rewriting the probability quantity (7) in terms of the cost function V : Proposition 1: Given N N, a control policy π and a sequence of sets {Σ k } k=,1,...,n, P ({ω Ω:x Σ,...,x N Σ N })=V (,x,π). (9) From Proposition 1, it is possible to reformulate the ODAA Problem, as follows. Problem 1: (Optimal Dynamic Asset Allocation (ODAA)) Given a finite time horizon N N and a sequence of target sets {Σ k } k=,1,...,n, where Σ k are Borel subsets of R, for any k =, 1,..., N, compute: π = arg sup V (,x,π). (1) π We now give the main result of this section that provides an algorithm solving the ODAA Problem, or equivalently equation (1). Theorem 1: The optimal value of the ODAA Problem is equal to p = J (x ), where J (x) is given by the last step of the following algorithm, J N (x) =I ΣN (x), (11) J k (x) = sup J k+1 (z)p f(x,uk,w k+1)(z)dz (12) u k U k Σ k+1 k = N 1,N 2,...,. Furthermore, if û k (x) =u k maximizes the right hand side of equation (12) for each x Σ k and k =, 1,..., N 1, then the class of policies ˆπ = {û, û 1,..., û N 1 } is optimal. Remark 1: The motivation for using supremum instead of maximum in optimization problem (12) is that no regularity assumptions have been imposed on the portfolio allocations set U k. However, if for example budget and long only constraints are considered, U k is compact and hence the continuity of the function to be optimized in (12) ensures the existence of the maximum; thus in this case it is possible to replace in (12) supremum by maximum. We conclude by giving a formal comparison between Markowitz method and our approach in the single period limit. The following result states that under appropriate assumptions, the optimal portfolio allocation of the ODAA Problem is in the boundary of the feasible region. Proposition 2: Suppose that the feasible region C k is compact and that: (single period limit) N =1; (gaussian hypothesis) w k is distributed according to a multivariate gaussian distribution. Then, the optimal portfolio allocation of the ODAA Problem is in C k. V. CASE STUDIES In this section ODAA is applied for finding the optimal allocation to two different data sets. A. Synthetic data set In this case study we consider an asset class ensemble composed by three un correlated synthetic asset classes which exhibit significant asymmetry and leptokurtic behavior. Figure 2 (upper left panel) depicts the marginal probability density functions of w k for the asset classes: asset classes 1 and 2 are distributed according to a gamma distribution, while asset class 3 is gaussian like distributed. (Distributions are assumed to be stationary in k.) More precisely the marginal probability density functions are given below: w k (1) = γ, w k (2) = 2ρ γ, w k (3) = η, where ρ R, γ is Γ(α, β) distributed with α =1and β = ρ, and η is a gaussian random variable with mean and standard deviation equal to ρ. In this case study we set ρ =.3. The asset classes considered are set to have the same mean and 2592

5 45th IEEE CDC, San Diego, USA, Dec , 26 ThA7.4 pdf weight w u u 1 portfolio value x 1 Σ 1 Σ 2 Σ 3 1 Σ time (years) u x 2 Fig. 2. Synthetic data set. Marginal probability density functions (upper left panel.) Target sets Σs (upper right panel.) Optimal Dynamic Asset Allocation (lower panels.) standard deviation, although they are very different in terms of skewness and kurtosis, as shown in Table below: Asset Mean St Dev Skewness Kurtosis asset asset asset Since mean and standard deviation of the asset classes considered are the same, Markowitz optimizer is not able to distinguish among them and in fact, the Markowitz optimal solution is given by the equally weighted portfolio, i.e. u =[ 1/3 1/3 1/3 ]. The ODAA Problem that we want to solve considers a 3 years investment with a 1 year re balancing period (N =3). The target sets Σs selected are: Σ = {1}, Σ 1 = [(1 + θ) 1, + ), Σ 2 = [(1 + θ) 2, + ), Σ 3 = [(1 + θ) 3, + ), (13) where θ is a parameter set to.5: this corresponds to ask the optimizer to find an (optimal) allocation able to beat a fixed year target return equal to 5%. In order to reduce computational complexity of Algorithm (11) (12) the portfolio allocations are discretized by a grid of 1% step or, in other words, portfolio allocations u k (i) are set to be multiple of 1%. This choice is also motivated by a practical implementation reason: since re balancing period considered is 1 year, a portfolio manager wants to make a marked portfolio re balance. Thus, given three asset classes and the constraints imposed (budget and long only constraints), the number of admissible portfolio allocations contained in U k is (This procedure has been taken for each k =, 1, 2.) Σ includes just one point, and hence the optimal allocation at time k =trivially does not depend on the portfolio realization. Given the target sets selected, the optimal allocation at time k =is given by 1% in the gaussian asset (see Figure 2; lower left panel). Much more interesting are the optimal allocations at time k =1and k =2which highlight the dependence of the optimal solution on the portfolio realization (see Figure 2; lower central and right panels). For example when x 2 crosses a value around 1.12 the optimizer prefers to move portfolio weights from asset 2 to assets 1 and 3. The maximal probability obtained in this case study is p =.9. (The probability is quite small, due to the target sets specification as in (13).) B. Financial data set This section considers a more realistic asset classes menu. The investment s universe consists of m =3asset classes: Cash Euro, Global Bond and Global Equity Indices. Details on the indices used in the analysis are reported below. Label Asset Index C Cash Euro JP Morgan EMU 3 months cash B Global Bond JP Morgan GBI+ E Global Equity Morgan Stanley ACWF Indices are priced in Euro, and time series starts from December 1997 for Cash Euro and the Global Bond, and December 22 for the Global Equity Index; time series end in December 25. Time series are monthly data. Time horizon is 3 years and re balancing period is 1 year (N =3). In order to make a straight comparison with the Markowitz formalism we modelled asset classes dynamics with a multivariate gaussian distribution. (This assumption is supported by the Jarque-Bera test at 95% CL.) Here below the estimated performances and volatilities (annualized values) are reported: Label ER Volatility C 3.38%.31% B 5.% 5.4% E 13.3% 1.41% The estimated correlation matrix is: C B E C B E We assume stationarity in the dynamics for w k. Figure 3 (upper left panel) reports the marginal probability density functions. The target sets Σs have been designed as in (13) with θ =.8 (see Figure 3; upper right panel): the investor wishes to beat an (annual) performance equal to 8%. Admissible portfolio allocations have been discretized as in the previous case study. Optimal Dynamic Asset Allocation are illustrated in Figure 3 (lower panels). The allocation at the beginning of the investment is 1% in the Equity Market. The portion of the investment in this asset class gets lower as the portfolio value increases: the optimizer prefers to switch to low volatile asset-classes as long as the target return is easier to be reached. The maximal probability p for the portfolio to stay in the target sets is.57. This case study allows us to make a straight comparison with 2593

6 45th IEEE CDC, San Diego, USA, Dec , 26 ThA7.4 pdf weight w u C B E C B E portfolio value u x 1 Σ 1 Σ 2 1 Σ time (years) u x 2 Σ 3 Fig. 3. Financial data set. Marginal probability density functions (upper left panel). Σ sets (upper right panel). Optimal Dynamic Asset Allocation (lower panels). the Markowitz optimizer. In order to treat a multi period investment, a constant mix allocation is considered: in this approach the optimal allocation at each re balancing time is constant, and it is given by solving the optimization problem (4) with r =.8. (Figure 1 illustrates the feasible region and the Efficient Frontier for this case study.) The optimal solution is u = u 1 = u 2 = [ ]. The maximal probability p provided by the constant mix strategy is.3. Finally, we study the effect of the re balancing period size on the ODAA maximal probability: in particular we consider 6 months, 3 months and 1 month re balancing periods. We assume the same target sets as in (13) with θ =.8, at the end of any year, and no target sets in the intermediate periods. For example, in the 6 month re balancing case study, we set Σ = {1}, Σ 2 = [(1 + θ) 1, + ), Σ 4 = [(1 + θ) 2, + ), Σ 6 = [(1 + θ) 3, + ), Σ 1 =[, + ), Σ 3 =[, + ), Σ 5 =[, + ), Σ 7 =[, + ). (14) Notice that target sets in the first column of (14) coincide with those chosen for the 1 year re balancing period (see equations (13)). By applying Algorithm (11) (12) to these case studies, the following maximal probabilities are obtained: Re-balancing period p 1 year.57 6 months.62 3 months.65 1 month.69 Results above point out that the more often the portfolio allocation is revised, the bigger is the probability to achieve the investors goals. Indeed, this result is a direct consequence of the Dynamic Programming approach pursued in this paper. It is worth to stress that the constant mix approach does not produce similar results: in fact the probability is stuck to.3 for each re balancing time considered in the table above. The applications presented in this section should be considered for illustrating the methodology. The views expressed in this article are those of the authors and do not necessarily correspond to those of CAAM SGR SpA. VI. CONCLUSIONS In this paper we considered the Optimal Dynamic Asset Allocation. Given a specified finite time horizon and a sequence of target sets that the investors would like their portfolio to stay within, the optimal portfolio allocation is synthesized in order to maximize the joint probability for the portfolio value to verify the target sets requirements. The approach does not assume any specific distributions for the asset classes dynamics, thus being particularly appealing to treat non gaussian asset classes. The proposed optimal control problem has been solved by using a Dynamic Programming approach. A formal comparison with Markowitz portfolio optimization is addressed in the single period limit. Finally, some case studies are provided in order to illustrate the proposed methodology. REFERENCES [1] Barberis, N., Investing for the Long Run when Returns are Predictable, The Journal of Finance, 55: (2). [2] Brennan, M. J., Schwartz, E. S., Lagnado, R., Strategic Asset Allocation, Journal of Economic Dynamics and Control, 21: (1997). [3] Brennan, M. J., Xia, Y., Dynamic Asset Allocation under Inflation, The Journal of Finance, 57(3): (22). [4] Bertsekas, D. P., Dynamic Programming and Optimal Control, Athena Scientific, Belmont Massachusetts, Second Edition, Voll. 1,2. (21). [5] Hammer, A. D., Dynamic Asset Allocation: Strategies for the Stock, Bond, and Money Markets, John Wiley & Sons, Inc. (1991). [6] Hull, J. C., Options, Futures, & Other Derivatives, Prentice Hall International, Inc. (2). [7] Luenberger, D. G., Investment Science, Oxford University Press, New York, USA (1998). [8] Mantegna, R. N., Stanley, H. E., An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge UK (2). [9] Merton, R. C., Lifetime Portfolio Selection Under Uncertainty: the Continuous Time Case, Review of Economics and Statistics, 51: (1969). [1] Markowitz, H., Portfolio Selection, The Journal of Finance, 7(1): (1952). [11] Pola, G., Lygeros, J., Di Benedetto, M. D., Invariance in Stochastic Dynamical Systems, 17 th International Symposium on Mathematical Theory of Network and Systems (MTNS 26), Kyoto, Japan, July 24th 28th (26). [12] Samuelson, P. A., Lifetime Portfolio Selection by Dynamic Stochastic Programming, Review of Economics and Statistics, 51: (1969). [13] Xia, Y., Learning about Predictability: the effect of parameter uncertainty on dynamic asset allocation, The Journal of Finance, 56(1): (21). 2594