Consumption-portfolio optimization and filtering in a hidden Markov-modulated asset price model

We study a consumption-portfolio optimization problem in a hidden Markov-modulated asset price model with multiple risky assets, where model uncertainty is present. Under this modeling framework, the appreciation rates of risky shares are modulated by a continuous-time, finite-state hidden Markov chain whose states represent different modes of the model. We consider the situation where an economic agent only has access to information about the price processes of risky shares and aims to maximize the expected, discounted utility from intermediate consumption and terminal wealth within a finite horizon. The standard innovations approach in filtering theory is then used to transform the partially observed consumption-portfolio optimization problem to the one with complete observations. Robust filters of the chain and estimates of some other parameters are presented. Using the stochastic maximum principle, we derive a closed-form solution of an optimal consumption-portfolio strategy in the case of a power utility.


Introduction. Portfolio optimization is an important research topic in finance.
Markowitz [16] pioneered the use of quantitative methods for an optimal portfolio selection problem and developed the celebrated mean-variance approach for portfolio optimization. He considered a single-period model and provided an elegant mathematical formulation of the portfolio selection problem, where the problem was reduced to a mean-variance optimization problem under the normality assumption for the return rates of individual securities. Merton [17,18] considered the portfolio optimization problem in a continuous-time framework. Under the assumption that the dynamics of a risky asset follow a Geometric Brownian Motion (GBM) and some specific forms of utility functions, Merton derived a closed-form solution of an optimal portfolio strategy in a continuous-time setting. Although Merton's approach to portfolio optimization problem produces beautiful theoretical results, there are some shortcomings from the practical perspective. For example, the GBM assumption is not consistent with many important 'stylized' features of assets' returns, such as the asymmetry and heavy-tailedness of the distribution of returns, time-varying conditional volatility, regime switches and others. However, this seems to be far from the reality, especially if one wishes to consider investment problems under economic uncertainties over a long time period, where structural changes in macroeconomic conditions may occur several times and cause fundamental changes in investment opportunity sets. Consequently, it may pay us dividends to consider a portfolio optimization problem in a more flexible model which can incorporate empirical features of the returns of risky assets and describe the stochastic evolution of investment opportunity sets.
Markovian regime-switching models have received a considerable attention from both academic researchers and market practitioners in economics and finance. Under regime-switching models, some model parameters or coefficients can change over time according to different states of a market or an economy which are described by the values of a state process, say, an underlying Markov chain. In econometrics and statistics, some early contributions on regime-switching models may be attributed to the works of Quandt [21], Goldfeld and Quandt [9], Tong [31,32], and Hamilton [11]. There is quite a large literature on portfolio optimization in regime-switching models. Some of them include Zhou and Yin [37], Yin and Zhou [33], Sotomayor and Cadenillas [30], Zhang et al. [35], Yiu et al. [34], Shen and Siu [25] and Zhang et al. [36], to name a few.
It is worth pointing out that the works aforementioned on portfolio optimization in regime-switching models mostly assumed that the underlying Markov chain is observable. However, in reality, the underlying state of the economy, or the Markov chain, is not directly observable. The model, where the underlying Markov chain is unobservable, is usually called the hidden Markov model (HMM). Some works on portfolio optimization problem under the HMM include Rieder and Bäuerle [22], Sass and Haussmann [23], Putschögl and Sass [20], Elliott et al. [6] and Elliott and Siu [7], Korn et al. [15], Siu [26,27,28,29], among others. Indeed, the unobservability of the states introduces model uncertainty, which is an important issue that needs to be addressed in a portfolio optimization problem. The importance of model uncertainty was highlighted in many fields including economics, finance, insurance, statistics and engineering. There are two major approaches to model uncertainty, namely, the Bayesian approach and the robust approach. The Bayesian approach supposes that the "true" state of the underlying model is represented by a set of random quantities. The robust approach assumes that the modeler surrounds an approximating model by introducing a family of "neighborhood" models via perturbations. Schied et al. [24] gave a good survey on the theory of robust preferences and robust portfolio selection. Particularly, Schied et al. [24] considered a dynamic version of model uncertainty, where the uncertain drift of the risky asset price process was assumed to be stochastic and time-dependent. For a systematic account of robustness and model uncertainty in economic modeling, one may also refer to Hansen and Sargent [12].
In this paper, we investigate a consumption-portfolio optimization problem of an economic agent in a hidden Markov-modulated asset price model in presence of model uncertainty, where the "true" state or mode of the model evolves over time according to a continuous-time, finite-state, hidden Markov chain. This is called the HMM approach in Elliott and Siu [7], which may be thought of as a Bayesian approach to model uncertainty. We consider a financial market consisting of one risk-free bond and n risky shares. The price dynamics of the risky shares are modeled as a Markovian regime-switching, multi-dimensional, GBM. More specifically, the appreciation rates of the risky shares are modulated by a hidden Markov chain and the random shocks of the shares are modeled by an n-dimensional standard Brownian motion. Consequently, the modeling framework considered here allows the flexibility that the price processes of the risky shares may be correlated. Besides investing in the bond and the shares, the agent can consume part of his/her wealth over time. We consider the situation where the agent aims to maximize the expected, discounted utility from intermediate consumption and terminal wealth only based on observations of the prices of the risky shares. As in the literature using the standard innovations approach in filtering theory, the problem with partial observations is transformed to the one with complete observations first. Then robust filters of the chain and estimates of some other parameters of the hidden Markov model with multiple correlated observations and Gaussian noises are presented based on the standard reference probability approach in filtering. Finally the stochastic maximum principle is applied to derive a closed-form solution to the problem in the case of a power utility.
The rest of this paper is organized as follows. The next section presents the model dynamics. Section 3 formulates the consumption-portfolio optimization problem under the hidden Markov model and discusses the standard innovations approach. In Section 4, robust filters and estimates are presented. The closed-form solution of the optimal consumption-portfolio strategy is derived in Section 5. The final section summarizes the paper. Some standard proofs are relegated in an Appendix.

The model dynamics.
First of all, we present the basic notation to be used throughout the paper.
: the set of real numbers; C : the transpose of any vector or matrix C; tr(C): the trace of a square matrix C; C, D : the inner product of C and D, that is C, D := tr(C D); ||C||: the Euclidean norm of C, that is ||C|| 2 = C, C ; diag(C): the diagonal matrix with the elements of a vector C on the diagonal; 1 n : the n-dimensional vector, where all entries are equal to 1, i.e., 1 n := (1, 1, · · · , 1) ∈ n . To avoid confusion, only the vectors or matrices associated with different states of the hidden Markov chain will be denoted in boldfaced letters unless otherwise stated.
We follow Elliott and Siu [7] to introduce the HMM approach for model uncertainty in a continuous-time setting. Consider a complete probability space (Ω, F , P), where P is a real-world probability measure. Let T := [0, T ] be a finite-time horizon, where T < ∞. To describe the transitions of the "true" state, or mode, of the model over time, we consider a continuous-time, finite-state, hidden Markov chain X := {X(t)|t ∈ T } on (Ω, F , P) taking values in a finite-state space S := {s 1 , s 2 , · · · , s N }. The states of the chain X represent different hidden states of the model. In practice, the "true" state of the model in force at a particular time is not observable and will have to be estimated by some observable market and economic information. In our case, the observable information is the price processes of the risky shares.
Without loss of generality, as in Elliott et al. [5], we identify the states of the chain with the set of standard unit vectors E := {e 1 , e 2 , · · · , e N } ⊂ N , where the j th component of e i is the Kronecker delta δ ij for each i, j = 1, 2, · · · , N . This is called the canonical state space representation of the chain. Let A := [a ij ] i,j=1,2,··· ,N be the rate matrix of the chain X under P, where a ij is a constant transition intensity of the chain X from state 'e j ' to state 'e i '. Note that for each i, j = 1, 2, · · · , N with i = j, a ij ≥ 0 and N i=1 a ij = 0, so a ii ≤ 0. We further assume a ij > 0, so a ii < 0. Then under the canonical representation of the state space E , Elliott et al. [5] obtained the following semimartingale dynamics for the chain: where {M(t)|t ∈ T } is an N -valued, (F X , P)-martingale. Here F X := {F X (t)|t ∈ T } is the right-continuous, P-complete, natural filtration generated by the chain X. The representation (1) describes how the hidden "true" state of the model evolves over time.
The financial market consists of one risk-free bond B := {B(t)|t ∈ T } and n risky shares S k := {S k (t)|t ∈ T }, k = 1, 2, · · · , n. Let r be a constant, continuously compounded risk-free interest rate such that r > 0. Then the evolution of the price process of the bond B over time is given by Let µ k := {µ k (t)|t ∈ T } be the process describing the evolution of the appreciation rate of the share S k over time, for each k = 1, 2, · · · , n. Suppose that the chain X determines µ k (t) as Here µ k := (µ 1 k , µ 2 k , · · · , µ N k ) ∈ N with µ i k > r, for each i = 1, 2, · · · , N and k = 1, 2, · · · , n; µ i k is the appreciation rate of the share S k corresponding to the i-th state of the model. The inner product ·, · selects which component of the vector of the appreciation rate µ k is in force at a particular time t based on the hidden state of the model at this time. That is, 'µ k (t) = µ i k ' if and only if 'X(t) = e i ', for each k = 1, 2, · · · , n and i = 1, 2, · · · , N .
Let W := {W (t)|t ∈ T } = {(W 1 (t), W 2 (t), · · · , W n (t)) |t ∈ T } be an ndimensional standard Brownian motion on (Ω, F , P) with respect to its rightcontinuous, P-complete, natural filtration. To simplify our discussion, we assume that the Brownian motion W and the Markov chain X are stochastically independent under P throughout this paper.
Suppose that β kd is the volatility rate of the share S k corresponding to the random shock described by the Brownian motion W d , for each k = 1, 2, · · · , n and d = 1, 2, · · · , n. We assume that β kd is a positive constant instead of assuming that it is modulated by the chain. There are some technical difficulties in filtering when β kd is modulated by the chain. Furthermore, it is rather uneasy to interpret the information structure when a hidden-Markov modulated volatility is considered. For details, one may refer to, for example, Guo [10]. Then we assume that the price processes of the risky shares are governed by a hidden Markov-modulated multi-dimensional geometric Brownian motion as follows: or equivalently, where β k := (β k1 , β k2 , · · · , β kn ) ∈ n , for each k = 1, 2, · · · , n. Write S(t) := (S 1 (t), S 2 (t), · · · , S n (t)) ∈ n , for each t ∈ T . We can rewrite Eq. (5) in the vector form: where µ(t) := (µ 1 (t), µ 2 (t), · · · , µ n (t)) ∈ n , β := [β kd ] k,d=1,2,··· ,n ∈ n×n and s := (s 1 , s 2 , · · · , s n ) ∈ n . Since this paper focuses on the consumption-portfolio optimization in a hidden Markov model, we only consider the complete market case. So we assume that the number of risky shares and the number of randomness are the same; otherwise, the market is incomplete. This will highlight how to use filtering techniques to deal with model uncertainty rather than how to tackle the incomplete market.
Throughout this paper, we assume that the following non-degeneracy condition is satisfied, i.e., where δ is some positive constant and I n×n is the (n × n)-identity matrix.
) for each t ∈ T and k = 1, 2, · · · , n. By Itô's differentiation rule, it is easy to see that where β k := ||β k || and || · || denotes the Euclidean norm in n . Write g k (t) := µ k (t) − 1 2 β 2 k , for each t ∈ T and k = 1, 2, · · · , n. Then the return process Y k of the k-th share can be rewritten as: The vector of the return processes Y : Y n (t)) |t ∈ T } is governed by the following stochastic differential equation: where g(t) := (g 1 (t), g 2 (t), · · · , g n (t)) ∈ n and 0 n := (0, 0, · · · , 0) ∈ n . Note that both the appreciation rate µ k and the Brownian motion W d are unobservable to the agent, for each k = 1, 2, · · · , n and d = 1, 2, · · · , n. The agent can only observe the price process S k or the return process Y k of the risky share. As pointed out by Merton [19], estimates of appreciation rates based on realized returns data are noisy. Therefore, incorporating the model uncertainty of the appreciation rate is an important issue.
We now specify the information structure of our model. Let be the right-continuous, P-complete, natural filtrations generated by the processes S k and Y k , respectively, for each k = 1, 2, · · · , n. Since F S k and F Y k are equivalent, we choose to use F Y k as the observed information structure.
Consequently, F Y , F X and G describe the flows of observable information, hidden information and full information, respectively.

YANG SHEN AND TAK KUEN SIU
3. The consumption-portfolio optimization problem. In this section, we formulate the portfolio optimization problem of the economic agent. Denote by π k (t), k = 1, 2, · · · , n, the amount of the agent's wealth invested in the share S k at time t, and c(t) the instantaneous amount of the agent's wealth consumed per unit of time at time t. We call π := {π(t)|t ∈ T } = {(π 1 (t), π 2 (t), · · · , π n (t)) |t ∈ T } and c := {c(t)|t ∈ T } a portfolio process and a consumption process of the agent. Specifically, the following definition is standard in the existing literature: Denote by V (t) := V π,c (t) the total wealth of the agent at time t corresponding to the pair of the portfolio and consumption processes (π, c). The portfolio process π is said to be c-financed if the amount of the agent's wealth invested in the bond is V (t) − n k=1 π k (t) at time t, for each t ∈ T . Suppose that (1) the stocks can be traded continuously over time; (2) there are no transaction costs, taxes, and short-selling constraints in trading; (3) the portfolio process π is c-financed. Then the wealth process V := {V (t)|t ∈ T } is governed by We consider the consumption-portfolio optimization problem where the economic agent aims to maximize the expected, discounted utility from intermediate consumption and terminal wealth over the set of admissible portfolio and consumption processes with partial observations. The definitions of the admissible set and the utility function are given below.
Definition 3.2. A pair of portfolio and consumption processes (π, c) is said to be admissible for the initial wealth v > 0 if the stochastic differential equation (11) admits a unique strong solution V such that Write A(v) for the space of admissible pairs (π, c) associated with the initial wealth v > 0. Note that V (t) ≥ 0 may be interpreted as a solvency constraint. Let U 1 and U 2 be the utility functions of intermediate consumption and terminal wealth of the agent, respectively. Suppose that ρ is a positive constant, which represents the discount rate of the agent. The objective of the agent is to maximize the expected, discounted utility from intermediate consumption and terminal wealth: subject to the wealth process (11) over the class Since the investor only has access to the information about the share prices, the consumption-portfolio optimization problem under the hidden Markov model is a stochastic control problem with partial observations. In this paper, we employ the standard innovations approach in filtering theory to transform the problem with partial information to the one with complete information. Particularly, using the separation principle, the two problems, namely, the filtering problem and the consumption-portfolio optimization problem are separated, so that these two problems can be solved independently.
For any integrable, G-adapted, process ξ : That is, ξ(t) is a version of the conditional expectation of ξ(t) with respect to P given F Y (t). It is known that the F Y -optional projection takes care of the measurability in (t, ω) ∈ T × Ω.
It is then not difficult to see that under P, the return process and the wealth process can be written in terms of the filtered estimate g(t) of g(t) and the innovations process W (t) as follows: and Therefore, the consumption-portfolio optimization problem with partial observations is transformed to the one with complete observations: subject to (π, c) ∈ A 1 (v) and (V (t), π, c) satisfy (15) .
4. Filtering. To discuss the consumption-portfolio optimization problem (16), robust filters of the hidden Markov chain and estimates of some other model parameters should first be derived. Then some techniques in stochastic optimal control theory can be employed to find an optimal pair of portfolio and consumption processes. We present robust filters of the hidden Markov chain and estimates of other parameters in Subsections 4.1 and 4.2, respectively. The filtering results to be presented in this section are standard and are derived using standard reference probability approach to filtering, see, for example, Elliott et al. [5]. Since the proofs of results in this section are standard, they are all placed in an Appendix. Notice that only robust filters of the hidden Markov chain will be used in expressions for optimal portfolio and consumption processes (see Theorem 5.1), specifically in the optional projection of µ, i.e., µ, while estimates of other parameters will not be needed there. The readers may skip Subsection 4.2 and jump to Section 5 if they wish.
From (17), we have where g k := ( g 1 k , g 2 k , · · · , g N k ) ∈ N and g i k := β −1 k g i k , for each k = 1, 2, · · · , n and i = 1, 2, · · · , N . Rewriting (20) in the vector form gives where g := g i k k=1,2,··· ,n;i=1,2,··· ,N ∈ n×N . Consider the following G-adapted process K := {K(t)|t ∈ T } defined by Since β is a constant matrix and the chain X has a finite number of states, the Novikov condition for the stochastic exponential in (22) is satisfied. Then, K is a (G, P)-martingale, and hence E[K(T )] = 1. Consequently a reference probability measure P equivalent to P is defined by setting dP dP G (T ) := K(T ) .
By Girsanov's theorem for the correlated Brownian motion (see Jeanblanc et al. [13]), the process { Y (t)|t ∈ T } is an n-dimensional, (G, P)-Brownian motion with the covariance matrix Σ. Since { W (t)|t ∈ T } and {X(t)|t ∈ T } are independent under P, the probability law of the chain remains unchanged under changing the measures from P to P. Define a transformed process Z := {Z(t)|t ∈ T } = {(Z 1 (t), Z 2 (t), · · · , Z n (t)) |t ∈ T } by putting It can be shown that Z is an n-dimensional, (G, P)-standard Brownian motion since Σ − 1 2 is the rotation matrix. It is obvious that the transformed process {Z(t)|t ∈ T } is an F Y -adapted process. Thus adopting Z as the new observation process will simplify the analysis.
For convenience of presentation, we shall use Eqs. (25) and (26) interchangeably in the sequel. The real-world probability measure P can then be re-constructed from the reference probability measure P on G (T ) as follows: It can be seen from Eqs. (21) and (23) that under P the observation process Z depends on the chain X.
Using a version of the Bayes' rule, for any G-adapted, integrable process ξ = {ξ(t)|t ∈ T }, where E[·] is the expectation taken under the reference measure P; σ(ξ(t)) and σ(1) are the optional projections of Λξ := {Λ(t)ξ(t)|t ∈ T } and Λ on the observed filtration F Y under P, respectively. Here σ(ξ(t)) is called an unnormalized, or Zakai form, of the filter of ξ(t).
The following proposition gives the Zakai stochastic differential equation governing the evolution of the unnormalized filter σ(X(t)) of the chain X. To simplify our notation, we write q(t) := σ(X(t)) , t ∈ T . Proposition 1. Let B k := diag( g k ), the diagonal matrix with entities given by the components of g k , for each k = 1, 2, · · · , n. Then, the unnormalized filter q(t) of X(t) is governed by the Zakai stochastic differential equation: Remark 1. With a slight abuse of notation, let B := diag( g), the three-dimensional diagonal array with non-zero entries given by the components in the matrix g. Note that each entry on the diagonal of B is a vector g i := ( g i 1 , g i 2 , · · · , g i n ) ∈ n , for each i = 1, 2, · · · , N . Then, Eq. (28) can be also written as where ⊗ is the tensor product of appropriate dimension. Although Eq. (29) looks apparently simpler than Eq. (28), it may not be convenient due to the multiplication with the three-dimensional array B. Here we adopt the presentation of Eq. (28) rather than Eq. (29) whenever the three-dimensional array may lead to confusion in the sequel.
The Zakai stochastic differential equation in Proposition 1 involves stochastic integrals, which may not be easy to implement in practice. So we now employ the gauge transformation technique of Clark [3] to simplify the stochastic differential equation in Proposition 1 and derive a path-wise linear ordinary differential equation for the transformed process of the Zakai filter. For each i = 1, 2, · · · , N , we consider a scalar-valued process γ i := {γ i (t)|t ∈ T } defined by The gauge transformation matrix Γ(t) is then defined as the following diagonal matrix: Since γ i (t) > 0, for each i = 1, 2, · · · N and t ∈ T , the inverse of Γ(t) exists. Write, for each t ∈ T , (Γ(t)) −1 for the inverse of Γ(t).
Applying Itô's differentiation rule gives Hence the gauge transformation matrix Γ := {Γ(t)|t ∈ T } follows the matrixvalued, stochastic differential equation: The following proposition gives the path-wise linear ordinary differential equation for the transformed process q.
Proposition 2. q satisfies the following path-wise linear ordinary differential equation: Remark 2. Note that the gauge transformation matrix Γ(t) is stochastic. Consequently the path-wise linear ordinary differential equation (31) is not deterministic.

Filtered estimates of other parameters.
In this subsection, we discuss a robust, filter-based, EM algorithm to estimate the unknown drift parameters µ i := (µ i 1 , µ i 2 , · · · , µ i n ) ∈ n , i = 1, 2, · · · , N , and the intensity parameters a ij of the chain X, i, j = 1, 2, · · · , N . To obtain these recursive estimates, the dynamics of some measure-valued quantities that are useful for the derivations of these estimates are first presented. For each t ∈ T and each i, j = 1, 2, · · · , N , let J ij (t) be the number of jumps of the chain X from e i to e j up to time t. That is, For each t ∈ T and each i = 1, 2, · · · , N , suppose that O i (t) is the occupation time of the chain X in state e i up to time t. That is, Furthermore, for each t ∈ T , k, l = 1, 2, · · · , n and i = 1, 2, · · · , N , write Clearly, from (18), we have For each t ∈ T , k, l = 1, 2, · · · , n and i = 1, 2, · · · , N , let G i kl (t) be the level integral with respect to β −1 k φ kl Z l (t) corresponding to the state e i up to time t. That is, The following theorem gives the stochastic differential equations governing the measure-valued quantities σ( ) and σ(X(t)G i kl (t)) are governed by the following stochastic differential equations: Aσ(X(s)H i kl (s))ds Similarly, the gauge transformation technique can be applied to simplify the stochastic differential equations (33)-(38) and to derive the corresponding pathwise linear ordinary differential equations for the transformed processes of the measure-valued quantities. These transformed, measure-valued, processes, denoted as σ(X(t)J ij (t)), σ(X(t)O i (t)), σ(X(t)H i kl (t)), σ(X(t)H i kl (t)), σ(X(t)H i kl (t)) and σ(X(t)G i kl (t)), are defined by The following proposition gives the dynamics of these transformed, measure-valued, processes.
Proposition 4. σ(X(t)J ij (t)), σ(X(t)O i (t)), σ(X(t)H i kl (t)), σ(X(t)H i kl (t)), σ(X (t)H i kl (t)) and σ(X(t)G i kl (t)) are governed by the following equations: σ(X(t)J ij (t)) = Aσ(X(s)H i kl (s))ds , (42) Aσ(X(s)G i kl (s))ds The unnormalized estimates of J ij (t), O i (t), H i kl (t), H i kl (t), H i kl (t) and G i kl (t) can then be determined by taking the inner products with 1 N . That is, ), 1 N , for each t ∈ T , k, l = 1, 2, · · · , n and i, j = 1, 2, · · · , N . From (32), it is clear that . For each t ∈ T and i = 1, 2, · · · , N , write k,l=1,2,··· ,n ∈ n×n , and Ξ i (t) : = n l=1 σ(G i 1l (t)) + Using the expectation-maximization (EM) algorithm, the revised, or updated, estimates of a ij and µ i = (µ i 1 , µ i 2 , · · · , µ i n ) ∈ n , for each i, j = 1, 2, · · · , N , are presented in the following theorem. The basic idea of the EM algorithm is to split the estimation of a model containing hidden quantities into two steps. The first step is the expectation step which evaluates the expectation of pseudo-likelihood function. The second step is the maximization step which aims to determine an optimal estimate of the unknown parameter by maximizing the expectation of the pseudo-likelihood function. The use of pseudo-likelihood function in the EM algorithm seems to be quite standard and has been discussed in, for example, Elliott et al. [5].
Theorem 4.1. Using the EM algorithm, the revised, or updated, estimates of a ij , for each t ∈ T , i, j = 1, 2, · · · , N and i = j, are given by Furthermore, if Π i (t) is nonsingular, for each t ∈ T and i = 1, 2, · · · , N , the estimate of µ i is given by Remark 3. Theorem 4.1 gives unique estimates of the appreciation rates µ i under the assumption that Π i (t) is nonsingular. In general, the system may not have a unique solution or there may be no solution at all. However, it is known that the Moore-Penrose inverse (pseudoinverse) for any matrix always exists and is unique. Let (Π i (t)) † be the Moore-Penrose inverse of Π i (t), for each t ∈ T and i = 1, 2, · · · , N . Using (Π i (t)) † , one may consider as a "best-fit" solution to the system of linear equations (47), for each t ∈ T and i = 1, 2, · · · , N . Recall that in regression analysis, a vector ξ i ∈ n is called a least squares solution to (47) if for each t ∈ T and i = 1, 2, · · · , N . Among all least squares solutions, a vector µ i is called a minimum-norm least squares solution to (47) if for all other least squares solutions ξ i . One may refer to Definition 2.1.1 in Campbell and Meyer [2] for more details about the minimum-norm least squares solution, which is called the minimal least squares solution therein. It follows from Theorem 2.1.1 in Campbell and Meyer [2] that (48) is the minimum-norm least squares solution to (47), i.e., It is well known that (Π i (t)) † = (Π i (t)) −1 if Π i (t) is nonsingular. Thus when Π i (t) is nonsingular, the minimum-norm least squares solution coincides with (46).
To compute the estimates a ij and µ i , for each i, j = 1, 2, · · · , N , one can implement a robust filter bank including the stochastic differential equations described by Propositions 2 and 4. Then these estimates can be computed by the standard procedures of the robust-filter-based EM algorithm in Elliott et al. [5]. 5. The maximum principle approach. In this section, we adopt the maximum principle approach to discuss the "filtered" consumption-portfolio optimization problem of the agent. With the robust filters and estimates presented in the previous section, we only need to solve a stochastic optimal control problem with complete observations. We derive a closed-form solution of the optimal consumptionportfolio strategy when the agent has a power utility.
We now define two auxiliary admissible sets A (v) and A 1 (v), where A (v) and A 1 (v) are the same as A(v) and A 1 (v), respectively, except that the wealth process V is allowed to be negative, i.e., V (t) ∈ , t ∈ T , P-a.s., in A (v) and A 1 (v).

(49)
Since no state constraint is imposed in Problem (49), we can apply the sufficient maximum principle directly. Indeed, we shall show that the original problem (16) and the auxiliary problem (49) have the same solution after Problem (49) is solved.
To pave the way for the maximum principle approach, we restate the problem in the language of the stochastic control. Recall that the controlled state process and the performance functional of the consumption-portfolio optimization problem (49) are given by (15) and (12), respectively. In this case, the Hamiltonian of the problem is The adjoint processes p := {p(t)|t ∈ T } and q := {q(t)|t ∈ T } form the solution pair of the following adjoint equation: Note that our consumption-portfolio problem is a special case of the stochastic control problem. Furthermore, from the condition that U 1 and U 2 are concave functions, we see that both the terminal cost and the Hamiltonian are concave. Therefore, the sufficient maximum principle can be applied in our problem. To make the problem solvable and explicit, we consider the power utility functions for the running and terminal costs. That is where 0 < γ < 1.
To find a solution (p, q) to (51), we now consider the following trial solution where f is a continuously differentiable function with terminal condition f (T ) = 1. It remains to determine the function f (t).
In other words, (π * , c * ) forms the optimal control pair of portfolio and consumption for both the auxiliary and original problems. Finally, we conclude this section with a theorem which presents the optimal portfolio and consumption processes and the associated wealth process of the original problem (16).
Theorem 5.1. The optimal portfolio and consumption processes and the associated wealth process of Problem (16) are given by  Conditioning both sides on F Y (t) gives Q(θ, θ) := E ln dP θ dP θ G (t) F Y (t) = N i,j=1,i =j (σ(J ij (t)) ln a ij − a ij σ(O i (t))) where R(θ) does not involve the parameter θ.