Mean-Variance Hedging and Forward-Backward Stochastic Differential Filtering Equations

Abstract

This paper is concerned with a mean-variance hedging problem with partial information, where the initial endowment of an agent may be a decision and the contingent claim is a random variable. This problem is explicitly solved by studying a linear-quadratic optimal control problem with non-Markov control systems and partial information. Then, we use the result as well as filtering to solve some examples in stochastic control and finance. Also, we establish backward and forward-backward stochastic differential filtering equations which are different from the classical filtering theory introduced by Liptser and Shiryayev (1977), Xiong (2008), and so forth.

1. Introduction and Problem Formulation

We begin with a finite time horizon [0,𝑇] for 𝑇>0, a complete filtered probability space (Ω,ℱ,(ℱ_𝑡),ℙ) on which an 𝐑^𝑚-valued standard Brownian motion (𝑊(⋅)) is defined. Moreover, we let the natural filtration ℱ_𝑡=𝜎{𝑊(𝑠);0≤𝑠≤𝑡}, 0≤𝑡≤𝑇, and ℱ=ℱ_𝑇.

Suppose there is a financial market in which 𝑚+1 securities can be continuously traded. One of them is a bond whose price 𝐵(⋅) satisfies d𝐵(𝑡)=𝑟(𝑡)𝐵(𝑡)d𝑡,(1.1) where 𝑟(𝑡) is the interest rate of the bond at time 𝑡. The other 𝑚 assets are stocks whose dynamicses are subject to the following stochastic differential equations (SDEs): d𝑆_𝑖(𝑡)=𝜇_𝑖(𝑡)𝑆_𝑖(𝑡)d𝑡+𝜎_𝑖(𝑡)𝑆_𝑖(𝑡)d𝑊_𝑖(𝑡),(1.2) where 𝜇_𝑖(𝑡) and 𝜎_𝑖(𝑡) are called the appreciation rate of return and volatility coefficient of the 𝑖th stock.

Suppose there is an agent who invests in the bond and stocks, whose decision cannot influence the prices in the financial market. We assume that the trading of the agent is self-financed, that is, there is no infusion or withdrawal of funds over [0,𝑇]. We denote by 𝜋_𝑖(𝑡) the amount that the agent invests in the 𝑖th stock and by 𝑥^𝜋(𝑡) the wealth of the agent with an initial endowment 𝑥₀>0. Then the agent has 𝑥^𝜋(𝑡)−∑^𝑚𝑖=1𝜋_𝑖(𝑡) savings in a bank. Under the forgoing notations and interpretations, the wealth 𝑥^𝜋(⋅) is modeled by d𝑥^𝜋(𝑡)=𝑟(𝑡)𝑥^𝜋(𝑡)+^𝑚_𝑖=1𝜇_𝑖(𝑡)−𝑟(𝑡)𝜋_𝑖(𝑡)d𝑡+^𝑚_𝑖=1𝑥𝜎_𝑖(𝑡)𝜋_𝑖(𝑡)d𝑊_𝑖(𝑡),_𝜋(0)=𝑥₀.(1.3)

Generally speaking, it is impossible for the agent to know all the events occurred in the financial market. For instance, if the agent has not enough time or great vigor to observe all the prices of the 𝑚+1 assets, then the agent will only observe some data of all the prices. Without loss of generality, we denote by 𝒵_𝑡 the information available to the agent at time 𝑡, which is a subfiltration of ℱ_𝑡. Suppose a process only adapted to 𝒵_𝑡 is called observable. Therefore, the agent has to choose a portfolio strategy according to the observable filtration 𝒵_𝑡. A portfolio strategy 𝜋(⋅)=(𝜋₁(⋅),…,𝜋_𝑚(⋅)) is called admissible if 𝜋_𝑖(𝑡) is a 𝒵_𝑡-adapted, square-integrable process with values in 𝐑. The set of the admissible portfolio strategies is denoted by 𝒰_ad.

We give the following hypothesis.(H1)The coefficients 𝑟(⋅),𝜇_𝑖(⋅),𝜎_𝑖(⋅), and 𝜎_𝑖(⋅)⁻¹ are uniformly bounded and deterministic functions with values in 𝐑.

For any 𝜋(⋅)∈𝒰_ad, (1.3) admits a unique solution under Hypothesis (H1). If we define 𝑣_𝑖(⋅)=𝜎_𝑖(⋅)𝜋_𝑖(⋅), then (1.3) is rewritten as d𝑥^𝑣(𝑡)=𝑟(𝑡)𝑥^𝑣(𝑡)+^𝑚_𝑖=1𝜇_𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)𝑣^𝑖(𝑡)d𝑡+^𝑚_𝑖=1𝑥𝑣_𝑖(𝑡)d𝑊_𝑖(𝑡),_𝑣(0)=𝑥₀.(1.4)

Let 𝜉>0 be a given contingent claim, which is a 𝒵_𝑇-measurable, square-integrable random variable. Furthermore, we suppose 𝜉 is larger than or equal to 𝑥₀e^{∫𝑇0𝑟(𝑡)d𝑡}, where the value 𝑥₀e^{∫𝑇0𝑟(𝑡)d𝑡} coincides with the amount that the agent would earn when the initial wealth 𝑥₀ was invested in the bond at the interest rate 𝑟(⋅) for the entire investment period.

Define a cost functional 𝐽𝑣(⋅);𝑥₀=12𝔼||𝑥^𝑣(𝑇)−𝜉||².(1.5) Note that the above 𝜉 can contain 𝔼[𝑥^𝑣(𝑇)∣𝒵_𝑇] as a special case. For a priori given initial wealth 𝑥₀, (1.5) measures the risk that the contingent claim 𝜉 cannot be reached. The agent’s objective is subjecmtitno𝑣𝐽(⋅)𝑣∈(⋅)𝒰;𝑥_0ad(𝑥_𝑣,(⋅);𝑣(⋅))satisfies(1.3)or(1.4).(PIMV)

The above problem formulates a mean-variance hedging problem with partial information. For simplicity, hereinafter we denote it by the notation “Problem (PIMV)”, short for the “partial information mean-variance hedging problem”. In particular, if we let ℱ_𝑡=𝒵_𝑡, 0≤𝑡≤𝑇, then Problem (PIMV) reduces to the case with complete information. See, for example, Kohlmann and Zhou [1] for more details.

Because the contingent claim 𝜉 in (1.5) is random and the initial endowment 𝑥₀ in (1.3) may be a decision, our Problem (PIMV) is distinguished from the existing literature. See, for example, Pham [2], Xiong and Zhou [3], Hu and Øksendal [4], and so forth. Motivated by Problem (PIMV), we study a general linear-quadratic (LQ) optimal control problem with partial information in Section 2. By a combination of the martingale representation theorem, the technique of “completing the square”, and conditional expectation, we derive a corresponding optimal control which is denoted by a related optimal state equation, a Riccati differential equation and a backward stochastic differential equation (BSDE). To demonstrate the applications of our results, we work out some partial information LQ examples and obtain some explicitly observable optimal controls by filtering for BSDEs. Also, we establish some backward and forward-backward stochastic differential filtering equations which are different from the classical ones.

In Section 3, we use the result established in Section 2 to derive an optimal portfolio strategy of Problem (PIMV), which is denoted by the sum of a replicating portfolio strategy for the contingent claim 𝜉 and a Merton’s portfolio strategy. To explicitly illustrate Problem (PIMV), we provide a special but nontrivial example in this section. In terms of filtering theory, we derive the corresponding risk measure. Furthermore, we use some numerical simulations and three figures to illustrate the risk measure and the optimal portfolio strategy.

In Section 4, we compare our results with the existing ones.

Finally, for the convenience of the reader, we state a classical filtering equation for SDEs which is used in Section 3 of this paper.

2. An LQ Optimal Control Problem with Partial Information

In this section, we study a partial information LQ optimal control problem, which is a generalization of Problem (PIMV).

Let us now begin to formulate the LQ problem. Consider a stochastic control system d𝑥^𝑣(𝑡)=𝐴(𝑡)𝑥^𝑣(𝑡)+^𝑚_𝑖=1𝐵_𝑖(𝑡)𝑣_𝑖(𝑡)+𝑔(𝑡)d𝑡+^𝑚_𝑖=1𝑥𝑣_𝑖(𝑡)d𝑊_𝑖(𝑡),_𝑣(0)=𝑥₀.(2.1) Here 𝑥^𝑣(t), 𝑥₀, 𝑣_𝑖(𝑡), 𝑔(𝑡)∈𝐑^𝑛, 𝐴(𝑡) and 𝐵_𝑖(𝑡)∈𝐑^𝑛×𝑛; 𝑣(⋅)=(𝑣₁(⋅),…,𝑣_𝑚(⋅)) is a control (process) with values in 𝐑^𝑛×𝑚. We suppose 𝑣(𝑡) is 𝒵_𝑡-adapted, where 𝒵_𝑡 is a given subfiltration of ℱ_𝑡 representing the information available to a policymaker at time 𝑡. We say that the control 𝑣(⋅) is admissible and write 𝑣(⋅)∈𝒰_ad if 𝑣(⋅)∈𝐿^2𝒵(0,𝑇;𝐑^𝑛×𝑚), that is, 𝑣(𝑡) is a 𝒵_𝑡-adapted process with values in 𝐑^𝑛×𝑚 and satisfies 𝔼^𝑇0||𝑣(𝑡)||²d𝑡<+∞.(2.2)

The following basic hypothesis will be in force throughout this section.(H2)𝐴(⋅), 𝐵_𝑖(⋅) are uniformly bounded and deterministic functions, 𝑥₀ is ℱ₀-adapted, and 𝑔(⋅)∈𝐿^2ℱ(0,𝑇;𝐑^𝑛).

For any 𝑣(⋅)∈𝒰_ad, control system (2.1) admits a unique solution under Hypothesis (H2). The associated cost functional is 𝐽𝑣(⋅);𝑥₀=12𝔼||𝑥^𝑣(𝑇)−𝜉||²,(2.3) where 𝜉 is a given ℱ_𝑇-measurable, square-integrable random variable. The LQ optimal control problem with partial information is subjecmtitno𝑣𝐽(⋅)𝑣∈(⋅)𝒰;𝑥_0ad(𝑥_𝑣,(⋅);𝑣(⋅))satisfies(2.1).(PILQ) An admissible control 𝑢(⋅) is called optimal if it satisfies 𝐽𝑢(⋅);𝑥₀=min_{𝑣(⋅)∈𝒰ad}𝐽𝑣(⋅);𝑥₀.(2.4) The solution 𝑥(⋅) and cost functional (2.3) along with 𝑢(⋅) are called the optimal state and the value function, respectively.

Problem (PILQ) is related to the recent work by Hu and Øksendal [4], where an LQ control for jump diffusions with partial information is investigated. Due to some characteristic setup, our Problem (PILQ) is not covered by [4]. See, for example, Section 4 in this paper for some detailed comments. Since the nonhomogeneous term in the drift of (2.1) is random and the observable filtration 𝒵_𝑡 is very general, it is not easy to solve Problem (PILQ). To overcome the resulting difficulty, we shall adopt a combination method of the martingale representation theorem, the technique of “completing the square”, and conditional expectation. This method is inspired by Kohlmann and Zhou [1], where an LQ control problem with complete information is studied.

To simplify the cost functional (2.3), we define 𝑦^𝑣(𝑡)=𝑥^𝑣(𝑡)−𝔼𝜉∣ℱ_𝑡.(2.5) Since 𝔼[𝜉∣ℱ_𝑡] is an ℱ_𝑡-martingale, by the martingale representation theorem (see e.g., Liptser and Shiryasyev [5]), there is a unique 𝑧_𝑖(⋅)∈𝐿^2ℱ(0,𝑇;R^𝑛) such that 𝔼𝜉∣ℱ_𝑡=𝔼𝜉+^𝑚_𝑖=1^𝑡0𝑧_𝑖(𝑠)d𝑊_𝑖(𝑠).(2.6) Applying Itô’s formula to (2.1) and (2.5)-(2.6), we have d𝑦^𝑣(𝑡)=𝐴(𝑡)𝑦^𝑣(𝑡)+^𝑚_𝑖=1𝐵_𝑖(𝑡)𝑣_𝑖(𝑡)+ℎ(𝑡)d𝑡+^𝑚_𝑖=1𝑣_𝑖𝑦(𝑡)−𝑧_𝑖(𝑡)d𝑊_𝑖(𝑡),_𝑣(0)=𝑦₀=𝑥₀−𝔼𝜉(2.7) with ℎ(𝑡)=𝑔(𝑡)+𝐴(𝑡)𝔼𝜉∣ℱ_𝑡,(2.8) and cost functional (2.3) reduces to 𝐽𝑣(⋅);𝑦₀=12𝔼||𝑦^𝑣(𝑇)||².(2.9)

Then Problem (PILQ) is equivalent to minimize (2.9) subject to (2.6)-(2.7) and 𝒰_ad. To solve the resulting problem, we first introduce a Riccati differential equation on 𝐑^𝑛×𝑛𝑃̇(𝑡)+𝑃(𝑡)𝐴(𝑡)+𝐴(𝑡)^𝜏𝑃(𝑡)−^𝑚_𝑖=1𝑃(𝑡)>𝑃0(𝑇,)0=𝑃≤𝐼(,𝑡𝑡)≤𝐵_𝑖𝑇(𝑡.)𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝑃(𝑡)=0,(2.10) Note that (2.7) contains a nonhomogeneous term ℎ(⋅). For this, we also introduce a BSDE on 𝐑^𝑛, −d𝛼(𝑡)=𝐴(𝑡)^𝜏−∑^𝑚_𝑖=1𝑃(𝑡)𝐵_𝑖(𝑡)𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝛼(𝑡)−^𝑚+𝑃(𝑡)ℎ(𝑡)+_𝑚∑_𝑖=1𝑃(𝑡)𝐵_𝑖(𝑡)𝑃(𝑡)⁻¹𝛽_𝑖(𝑡)_𝑖∑₌₁𝐵_𝑖(𝑡)𝛼𝑧_𝑖((𝑇𝑡))=0d.𝑡−∑^𝑚_𝑖=1𝛽_𝑖(𝑡)d𝑊_𝑖(𝑡),(2.11) Assume that the following hypothesis holds.(H3)For any 0≤𝑡≤𝑇, 𝐴(𝑡)+𝐴(𝑡)^𝜏≥^𝑚_𝑖=1𝐵_𝑖(𝑡)𝐵_𝑖(𝑡)^𝜏.(2.12)

Under Hypotheses (H2) and (H3), according to [1, Theorem 4.2], it is easy to see that (2.10) admits a unique solution, and then (2.11) admits a unique ℱ_𝑡-adapted solution (𝛼(⋅),𝛽₁(⋅),…,𝛽_𝑚(⋅)).

For any admissible pair (𝑣(⋅),𝑦^𝑣(⋅)), using Itô’s formula to (1/2)𝑦^𝑣(⋅)^𝜏𝑃(⋅)𝑦^𝑣(⋅)+𝛼(⋅)^𝜏𝑦^𝑣(⋅), integrating from 0 to 𝑇, taking the expectations and trying to complete a square, then we have 𝐽𝑣(⋅);𝑦₀=𝐽_ℱ𝑦₀+12𝔼^𝑇0𝑚_𝑖=1𝑣_𝑖=𝐽_ℱ(𝑡)+𝐿^𝑣𝑖(𝑡)^𝜏𝑃(𝑡)𝑣_𝑖(𝑡)+𝐿^𝑣𝑖(𝑡)d𝑡𝑦₀+12𝔼^𝑇0𝑚_𝑖=1𝔼𝑣_𝑖(𝑡)+𝐿^𝑣𝑖(𝑡)^𝜏𝑃(𝑡)𝑣_𝑖(𝑡)+𝐿^𝑣𝑖(𝑡)∣𝒵_𝑡d𝑡,(2.13) where 𝐽_ℱ𝑦₀=12𝑦^𝜏0+1𝑃(0)𝑦₀+𝑦^𝜏0𝛼(0)2𝔼^𝑇02𝛼(𝑡)^𝜏ℎ(𝑡)−2^𝑚_𝑖=1𝛽_𝑖(𝑡)𝑧_𝑖(𝑡)+^𝑚_𝑖=1−𝑧_𝑖(𝑡)^𝜏𝑃(𝑡)𝑧_𝑖(𝑡)_𝑚_𝑖=1𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝛼(𝑡)+𝑃(𝑡)⁻¹𝛽_𝑖(𝑡)−𝑧_𝑖(𝑡)^𝜏×𝑃(𝑡)𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝛼(𝑡)+𝑃(𝑡)⁻¹𝛽_𝑖(𝑡)−𝑧_𝑖(𝑡)d𝑡,(2.14)𝐿^𝑣𝑖(𝑡)=𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝑃(𝑡)𝑦^𝑣(𝑡)+𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝛼(𝑡)+𝑃(𝑡)⁻¹𝛽_𝑖(𝑡)−𝑧_𝑖(𝑡).(2.15) Since 𝐽_ℱ(𝑦₀) is independent of 𝑣_𝑖(⋅), the integrand in (2.13) is quadratic with respect to 𝑣_𝑖(⋅) and 𝑃(⋅)>0, then it follows from the property of conditional expectation that the minimum of 12𝔼^𝑇0𝑚_𝑖=1𝔼𝑣_𝑖(𝑡)+𝐿^𝑣𝑖(𝑡)^𝜏𝑃(𝑡)𝑣_𝑖(𝑡)+𝐿^𝑣𝑖(𝑡)∣𝒵_𝑡d𝑡(2.16) over all 𝒵_𝑡-adapted 𝑣_𝑖(𝑡) is attained at 𝑢_𝑖(𝑡)==−−𝑃𝔼(𝑡𝐿)_𝑖−1(𝑡)∣𝒵_𝑡−𝑃(𝑡)₋₁𝐵_𝑖(𝑡)^𝜏𝑃(𝑡)𝔼𝑦(𝑡)∣𝒵_𝑡+𝔼𝛼(𝑡)∣𝒵_𝑡𝔼𝛽_𝑖(𝑡)∣𝒵_𝑡+𝔼𝑧_𝑖(𝑡)∣𝒵_𝑡(2.17) where 𝐿_𝑖(𝑡)=𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝑃(𝑡)𝑦(𝑡)+𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝛼(𝑡)+𝑃(𝑡)⁻¹𝛽_𝑖(𝑡)−𝑧_𝑖(𝑡)(2.18) and 𝑦(⋅) is the solution of the SDE with 𝑢_𝑖(⋅) replaced by (2.17) d𝑦(𝑡)=𝐴(𝑡)𝑦(𝑡)+^𝑚_𝑖=1𝐵_𝑖(𝑡)𝑢_𝑖(𝑡)+ℎ(𝑡)d𝑡+^𝑚_𝑖=1𝑢_𝑖𝑦(0)=𝑦₀(𝑡)−𝑧_𝑖(𝑡)d𝑊_𝑖(𝑡),=𝑥₀−𝔼𝜉.(2.19)

Now, we are in the position to derive an optimal feedback control in terms of the original optimal state variable 𝑥(⋅). Substituting (2.5) into (2.17), we get 𝑢_𝑖(𝑡)==−−𝔼𝑃(𝑡𝐿)_𝑖−1(𝑡)∣𝒵_𝑡−𝑃(𝑡)₋₁𝐵_𝑖(𝑡)^𝜏𝑃(𝑡)𝔼𝑥(𝑡)∣𝒵_𝑡−𝑃(𝑡)𝔼𝜉∣𝒵_𝑡+𝔼𝛼(𝑡)∣𝒵_𝑡𝔼𝛽_𝑖(𝑡)∣𝒵_𝑡+𝔼𝑧_𝑖(𝑡)∣𝒵_𝑡,(2.20) where 𝑥(⋅) satisfies the SDE with 𝑢_𝑖(⋅) replaced by (2.20) d𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡)+^𝑚_𝑖=1𝐵_𝑖(𝑡)𝑢_𝑖(𝑡)+𝑔(𝑡)d𝑡+^𝑚_𝑖=1𝑥(0)=𝑥₀𝑢_𝑖(𝑡)d𝑊_𝑖(𝑡),.(2.21)

Furthermore, we define for any 0≤𝑡≤𝑇𝑞𝑝(𝑡)=𝔼𝜉∣ℱ_𝑡−𝑃(𝑡)⁻¹𝛼(𝑡),_𝑖(𝑡)=𝑧_𝑖(𝑡)−𝑃(𝑡)⁻¹𝛽_𝑖(𝑡).(2.22) Applying Itô’s formula to (2.6) and (2.10)-(2.11), we can check that (𝑝(⋅),𝑞₁(⋅),…,𝑞_𝑚(⋅)) is the unique solution of the BSDE d𝑝(𝑡)=𝐴(𝑡)𝑝(𝑡)+^𝑚_𝑖=1𝐵_𝑖(𝑡)𝑞_𝑖(𝑡)+𝑔(𝑡)d𝑡+^𝑚_𝑖=1𝑝(𝑇)=𝜉.𝑞_𝑖(𝑡)d𝑊_𝑖(𝑡),(2.23)

Finally, substituting (2.17) into (2.13), we get the value function 𝐽_𝒵𝑦₀=𝐽_ℱ𝑦₀+12𝔼^𝑇0𝑚_𝑖=1𝔼𝐿_𝑖(𝑡)^𝜏𝑃(𝑡)𝐿_𝑖(𝑡)∣𝒵_𝑡−𝔼𝐿_𝑖(𝑡)^𝜏∣𝒵_𝑡𝑃(𝑡)𝔼𝐿_𝑖(𝑡)∣𝒵_𝑡𝑑𝑡,(2.24) where 𝐽_ℱ(𝑦₀) and 𝐿_𝑖(⋅) are defined by (2.14) and (2.18), respectively.

Theorem 2.1. Let Hypotheses (H2) and (H3) hold. Then the optimal control of Problem (PILQ) is 𝑢_𝑖(t)=−𝑃(𝑡)⁻¹𝐵_𝑖(𝑡)^𝜏𝑃(𝑡)𝔼𝑥(𝑡)∣𝒵_𝑡−𝔼𝑝(𝑡)∣𝒵_𝑡+𝔼𝑞_𝑖(𝑡)∣𝒵_𝑡,(2.25) where 𝑥(⋅) and (𝑝(⋅),𝑞₁(⋅),…,𝑞_𝑚(⋅)) are the solutions of (2.21) and (2.23), respectively; the corresponding value function is given by (2.24).

Remark 2.2. Note that the dynamics of BSDE (2.23) is similar to control system (2.1) except for the state constraint, which shows a perfect relationship between stochastic control and BSDE. This interesting phenomenon is first found by [1], to our best knowledge. Also, [1] finds that the solution (𝑝(⋅),𝑞₁(⋅),…,𝑞_𝑚(⋅)) of (2.23) can be regarded as the optimal state-control pair (𝑥(⋅),𝑢(⋅)) of an LQ control problem with complete information, in which the initial state 𝑥₀ is an additional decision. That is, 𝑝(⋅)=𝑥(⋅) and 𝑞_𝑖(⋅)=𝑢_𝑖(⋅) with 𝑢(⋅)=(𝑢₁(⋅),…,𝑢_𝑚(⋅)). However, this conclusion is not true in our partial information case. For clarity, we shall illustrate it by the following example.

Example 2.3. Without loss of generality, we let Hypothesis (H2) hold and 𝑛=1 in Problem (PILQ).

Since 𝑃(⋅) defined by (2.10) is a scalar, it is natural that (2.10) admits a unique solution. Consequently, (2.11) also admits a unique solution. Note that Hypothesis (H3) is not used in this setup. Define Δ(⋅)=𝑥(⋅)−𝑝(⋅).(2.26) From Itô’s formula, (2.21)-(2.23) and (2.25), we get dΔ(𝑡)=𝐴(𝑡)Δ(𝑡)−^𝑚_𝑖=1𝐵_𝑖(𝑡)²𝔼Δ(𝑡)∣𝒵_𝑡+^𝑚_𝑖=1+𝐵_𝑖(𝑡)𝔼𝑞_𝑖(𝑡)∣𝒵_𝑡−𝑞_𝑖(𝑡)d𝑡_𝑚_𝑖=1𝐵_𝑖Δ(0)=𝑥₀(𝑡)𝔼𝑞_𝑖(𝑡)∣𝒵_𝑡−𝑞_𝑖(𝑡)−𝐵_𝑖(𝑡)𝔼Δ(𝑡)∣𝒵_𝑡d𝑊_𝑖(𝑡),−𝑝(0).(2.27) Hereinafter, we set 𝑌(𝑡)=𝔼𝑌(𝑡)∣𝒵_𝑡,𝑌(⋅)=𝑥(⋅),𝑝(⋅),𝑞₁(⋅),𝑞₂(⋅),𝑔(⋅),𝑋(⋅)orΔ(⋅),0≤𝑡≤𝑇,(2.28) where the signal 𝑌(𝑡) is an ℱ_𝑡-adapted and square-integrable stochastic progress, while the observation is the component of the 𝑚-dimensional Brownian motion (𝑊(⋅)). Without loss of generality, we let the observable filtration 𝒵_𝑡 be 𝒵_𝑡=𝜎𝑊₁(𝑠),…,𝑊_𝑙(𝑠);0≤𝑠≤𝑡,0≤𝑡≤𝑇,1≤𝑙≤𝑚−1.(2.29) In this setting, we call (2.28) the optimal filtering of the signal 𝑌(𝑡) with respect to the observable filtration 𝒵_𝑡 in the sense of square error. See, for example, [5, 6] for more details.

Note that (𝑊₁(⋅),…,𝑊_𝑙(⋅)) is independent of (𝑊_𝑙+1(⋅),…,𝑊_𝑚(⋅)), 𝑥₀ and 𝑝(0) are deterministic. Taking the conditional expectations on both sides of (2.27), we get the optimal filtering equation of Δ(𝑡) with respect to 𝒵_𝑡dΔ(𝑡)=𝐴(𝑡)−^𝑚_𝑖=1𝐵_𝑖(𝑡)²Δ(𝑡)d𝑡−^𝑙_𝑖=1Δ(0)=𝑥₀𝐵_𝑖(𝑡)Δ(𝑡)d𝑊_𝑖(𝑡),−𝑝(0).(2.30) Note that Δ(⋅) satisfies a homogeneous linear SDE and hence must be identically zero if 𝑥₀=𝑝(0).

Thereby, if the decision 𝑥₀ takes the value 𝑝(0) in Example 2.3, then the next corollary follows from Theorem 2.1.

Corollary 2.4. The optimal control of Example 2.3 is 𝑢_𝑖(𝑡)=𝔼𝑞_𝑖(𝑡)∣𝒵_𝑡,0≤𝑡≤𝑇.(2.31) In particular, if 𝒵_𝑡=ℱ_𝑡, 0≤𝑡≤𝑇, then it reduces to the case of [1], that is, 𝑢_𝑖(⋅)=𝑞_𝑖(⋅).

From Theorem 2.1 and Corollary 2.4, we notice that the optimal control strongly depends on the conditional expectation of (𝑝(𝑡),𝑞₁(𝑡),…,𝑞_𝑚(𝑡)) with respect to 𝒵_𝑡, 0≤𝑡≤𝑇, where (𝑝(⋅),𝑞₁(⋅),…,𝑞_𝑚(⋅)) is the solution of BSDE (2.23). Since 𝒵_𝑡 is very general, the conditional expectation is, in general, infinite dimensional. Then it is very hard to find an explicitly observable optimal control by some usual methods. However, it is well known that such an optimal control plays an important role in theory and reality. For this, we desire to seek some new technique to further research the problem in the rest of this section. Recently, Wang and Wu [7] investigate the filtering of BSDEs and use a backward separation technique to explicitly solve an LQ optimal control problem with partial information. Please refer to Wang and Wu [8] and Huang et al. [9] for more details about BSDEs with partial information. Inspired by [7, 9], we shall apply the filtering of BSDEs to study the conditional expectation mentioned above. Note that there is no general filtering result for BSDEs in the published literature. In the rest of this section, we shall present two examples of such filtering problems. Combining Theorem 2.1 with a property of conditional expectation, we get some explicitly observable optimal controls. As a byproduct, we establish two new kinds of filtering equations, which are called as backward and forward-backward stochastic differential filtering equations. The result enriches and develops the classical filtering-control theory (see e.g., Liptser and Shiryayev [5], Bensoussan [10], Xiong [6], and so on).

Example 2.5. Let Hypothesis (H2) hold, 𝑛=1, and 𝑚=2 in Problem (PILQ). Suppose the observable filtration is denoted by 𝒵_𝑡=𝜎𝑊₁(𝑠);0≤𝑠≤𝑡,0≤𝑡≤𝑇.(2.32)

From Theorem 2.1, the optimal control is 𝑢_𝑖(𝑡)==−−𝐵𝐵_𝑖𝑖(𝑡)𝔼𝑥(𝑡)∣𝒵_𝑡−𝔼𝑝(𝑡)∣𝒵_𝑡+𝔼𝑞_𝑖(𝑡)∣𝒵_𝑡(𝑡)[𝑥̂(𝑡)−𝑝̂(𝑡)]+𝑞̂_𝑖(𝑡),(2.33) where (𝑝(⋅),𝑞₁(⋅),𝑞₂(⋅)) is the unique solution of d𝑝(𝑡)=𝐴(𝑡)𝑝(𝑡)+²_𝑖=1𝐵_𝑖(𝑡)𝑞_𝑖(𝑡)+𝑔(𝑡)d𝑡+²_𝑖=1𝑝(𝑇)=𝜉𝑞_𝑖(𝑡)d𝑊_𝑖(𝑡),(2.34) and 𝑥(⋅) satisfies (2.21) with 𝑚=2.

Similar to Example 2.3, the optimal filtering equation of 𝑥(⋅) is d𝑥̂(𝑡)=𝐴(𝑡)𝑥̂(𝑡)+²_𝑖=1𝐵𝑥_𝑖̂((0𝑡))𝑢=_𝑖(𝑥𝑡)₀+𝑔̂(𝑡)d𝑡+𝑢₁(𝑡)d𝑊₁(𝑡),.(2.35) We proceed to calculate the optimal filtering of (𝑝(⋅),𝑞₁(⋅),𝑞₂(⋅)). Recalling BSDE (2.34) and noting that the observable filtration is 𝒵_𝑡, it follows that d𝑝̂(𝑡)=𝐴(𝑡)𝑝̂(𝑡)+²_𝑖=1𝑝̂(𝑇𝐵)_𝑖=(𝑡)𝔼𝑞̂_𝑖𝜉(𝑡∣)𝒵+_𝑇𝑔̂(𝑡)d𝑡+𝑞̂₁(𝑡)d𝑊₁(𝑡),.(2.36) As (2.36) is a (non-Markov) BSDE, we call it a backward stochastic differential filtering equation which is different from the classical filtering equation for SDEs. Since 𝑞̂₂(⋅) is absent from the diffusion term in (2.36), then we are uncertain that if (2.36) admits a unique solution (𝑝(̂⋅),𝑞̂₁(⋅),𝑞̂₂(⋅)). But, we are sure that it is true in some special cases. See the following example, in which we establish a forward-backward stochastic differential filtering equation and obtain a unique solution of this equation.

Example 2.6. Let all the assumptions hold and 𝑔(⋅)≡0 in Example 2.5. For simplicity, we set the random variable 𝜉=𝑋(𝑇), where 𝑋(⋅) is the solution of d𝑋(𝑡)=𝐾(𝑡)𝑋(𝑡)d𝑡𝑋+𝑀(0)₁=(𝑡)𝑋d𝑊₁₀(𝑡)+𝑀₂(𝑡)d𝑊₂(𝑡),.(2.37) Assume that 𝐾(⋅), 𝑀₁(⋅), and 𝑀₂(⋅) are bounded and deterministic functions with values in 𝐑; 𝑋₀ is a constant.

Similar to Example 2.5, the optimal control is 𝑢_𝑖(𝑡)=−𝐵_𝑖(𝑡)[𝑥̂(𝑡)−𝑝̂(𝑡)]+𝑞̂_𝑖(𝑡),(2.38) where (𝑥̂(⋅),𝑝(̂⋅),𝑞̂₁(⋅),𝑞̂₂(⋅),𝑋(⋅)) is the solution of d𝑥̂(𝑡)=𝐴(𝑡)𝑥̂(𝑡)+²_𝑖=1𝑥̂(0)=𝑥₀𝐵_𝑖(𝑡)𝑢_𝑖(𝑡)d𝑡+𝑢₁(𝑡)d𝑊₁(𝑡),d𝑝̂(𝑡)=𝐴,(𝑡)𝑝̂(𝑡)+₂(2.39)_𝑖=1𝑝̂(𝑇)=𝑋(𝑇),𝐵_𝑖(𝑡)𝑞̂_𝑖(𝑡)d𝑡+𝑞̂₁(𝑡)d𝑊₁(𝑡),(2.40)

d𝑋(𝑡)=𝐾(𝑡)𝑋𝑋((0𝑡))d=𝑡𝑋+₀𝑀₁(𝑡)d𝑊₁(𝑡),.(2.41) It is remarkable that (2.35) together with (2.40)-(2.41) is a forward-backward stochastic differential filtering equation. To our best knowledge, this is also a new kind of filtering equation.

We now desire to give a more explicitly observable representation of 𝑢_𝑖(⋅). Due to the terminal condition of (2.40), we get by Itô’s formula and the method of undetermined coefficients, 𝑝̂(⋅𝑞)̂_𝑖=Φ(⋅)𝑋(⋅)+Ψ(⋅),(⋅)=Φ(⋅)𝑀_𝑖(⋅).(2.42) Here 𝑋(⋅) is the solution of (2.41), and Ψ(𝑡)=−^𝑇𝑡2_𝑖=1Φ(𝑡)e=e^{−∫𝑠𝑡𝐴(𝑟)d𝑟}Φ(𝑠)𝐵_{𝑖∫𝑇𝑡(𝐾(𝑠)−𝐴(𝑠))𝑑𝑠}(𝑠)𝑀_𝑖(𝑠)d𝑠,.(2.43) Thus, the optimal control is 𝑢_𝑖(𝑡)=−𝐵_𝑖(𝑡)𝑥̂(𝑡)−Φ(𝑡)𝑋(𝑡)−Ψ(𝑡)+Φ(𝑡)𝑀_𝑖(𝑡),(2.44) where 𝑋(⋅) satisfies (2.41) and 𝑥̂(⋅) is the solution of d𝑥̂(𝑡)=𝐴(𝑡)−²_𝑖=1𝐵_𝑖(𝑡)²𝑥̂(𝑡)+²_𝑖=1+𝐵𝐵_𝑖(𝑡)Φ(𝑡)𝑀_𝑖(𝑡)+𝐵_𝑖(𝑡)(Φ(𝑡)𝑥̂(𝑡)+Ψ(𝑡))d𝑡₁𝑥̂(0)=𝑥₀(𝑡)Φ(𝑡)𝑋(𝑡)−𝑥̂(𝑡)+Ψ(𝑡)+Φ(𝑡)𝑀₁(𝑡)d𝑊₁(𝑡),.(2.45) Since 𝑋(⋅) is the solution of (2.41), it is easy to see that the above equation admits a unique solution 𝑥̂(⋅). Now 𝑢_𝑖(𝑡), 0≤𝑡≤𝑇, defined by (2.44) is an explicitly observable optimal control.

Remark 2.7. BSDE theory plays an important role in many different fields. Then we usually treat some backward stochastic systems with partial information. For instance, to get an explicitly observable optimal control in Theorem 2.1, it is necessary to estimate (𝑝(𝑡),𝑞₁(𝑡),…,𝑞_𝑚(𝑡)) depending on the observable filtration 𝒵_𝑡. However, there are short of some effective methods to deal with these estimates. In this situation, although the filtering of BSDEs is very restricted, it can be regarded as an alternative technique (just as we see in Examples 2.2-2.3). By the way, the study of Problem (PILQ) motivates us to establish some general filtering theory of BSDEs in future work. To our best knowledge, this is a new and unexplored research field.

3. Solution to the Problem (PIMV)

We now regard Problem (PIMV) as a special case of Problem (PILQ). Consequently, we can apply the result there to solve the Problem (PIMV). From Theorem 2.1, we get the optimal portfolio strategy 𝜋^∗𝑖(𝑡)=𝜋^∗𝑖1(𝑡)+𝜋^∗𝑖2(𝑡)(3.1) with 𝜋^∗𝑖1(𝑡)=−𝜇^𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)²𝜋𝔼𝑥^∗(𝑡)∣𝒵_𝑡−𝔼𝑝(𝑡)∣𝒵_𝑡_∗𝑖2,(𝑡)=1𝜎_𝑖(𝑡)𝔼𝑞^𝑖(𝑡)∣𝒵_𝑡.(3.2) Here (𝑝(⋅),𝑞₁(⋅),…,𝑞_𝑚(⋅)) and 𝑥^∗(⋅) are the solutions of d𝑝(𝑡)=𝑟(𝑡)𝑝(𝑡)+^𝑚_𝑖=1𝜇_𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)𝑞^𝑖(𝑡)𝑑𝑡+^𝑚_𝑖=1d𝑥_∗𝑝(𝑇)=𝜉,𝑞_𝑖(𝑡)d𝑊_𝑖(𝑡),(3.3)(𝑡)=𝑟(𝑡)𝑥^∗(𝑡)+^𝑚_𝑖=1𝜇_𝑖(𝑡)−𝑟(𝑡)𝜋^∗𝑖(𝑡)𝑑𝑡+^𝑚_𝑖=1𝑥𝜎_𝑖(𝑡)𝜋^∗𝑖(𝑡)d𝑊_𝑖(𝑡),_∗(0)=𝑥₀.(3.4) So we have the following theorem.

Theorem 3.1. If Hypothesis (H1) holds, then the optimal portfolio strategy of Problem (PIMV) is given by (3.1).

We now give a straightforward economic interpretation of (3.1). Introduce an adjoint equation −d𝜃(𝑠)=𝑟(𝑠)𝜃(𝑠)𝑑𝑠+^𝑚_𝑖=1𝜇_𝑖(𝑠)−𝑟(𝑠)𝜎_𝑖(𝑠)𝜃^𝑖(𝑠)d𝑊_𝑖(𝑠),𝜃(𝑡)=1,0≤𝑡≤𝑠≤𝑇.(3.5) Applying Itô’s formula to 𝑝(⋅)𝜃(⋅), 𝑝(𝑡)=e^{−∫𝑇𝑡𝑟(𝑠)d𝑠}𝔼𝜉e^{−∫𝑇𝑡∑𝑚𝑖=1(𝜇𝑖(𝑠)−𝑟(𝑠))/(𝜎𝑖(𝑠))d𝑊𝑖∫(𝑠)−(1/2)𝑇𝑡∑𝑚𝑖=1((𝜇𝑖(𝑠)−𝑟(𝑠))/𝜎𝑖(𝑠))2𝑑𝑠}∣ℱ_𝑡.(3.6) Note that 𝒵_𝑡 is a subfiltration of ℱ_𝑡, 0≤𝑡≤𝑇. Then we have 𝔼𝑝(𝑡)∣𝒵_𝑡=e^{−∫𝑇𝑡𝑟(𝑠)𝑑𝑠}𝔼𝜉e^{−∫𝑇𝑡∑𝑚𝑖=1(𝜇𝑖(𝑠)−𝑟(𝑠))/(𝜎𝑖(𝑠))d𝑊𝑖∫(𝑠)−(1/2)𝑇𝑡∑𝑚𝑖=1((𝜇𝑖(𝑠)−𝑟(𝑠))/(𝜎𝑖(𝑠)))2d𝑠}∣𝒵_𝑡,(3.7) which is the partial information option price for the contingent claim 𝜉. According to Corollary 2.4, 𝜋^∗𝑖2(⋅) is the partial information replicating portfolio strategy for the contingent claim 𝜉 when the initial endowment 𝑥₀ is the initial option price 𝑝(0). Then 𝜋^∗𝑖1(⋅) defined by (3.2) is exactly the partial information Merton’s portfolio strategy for the terminal utility function 𝑈(𝑥)=𝑥² (see e.g., Merton [11]). That is, the optimal portfolio strategy (3.1) is the sum of the partial information replicating portfolio strategy for the contingent claim 𝜉 and the partial information Merton’s portfolio strategy. Consequently, if the initial endowment 𝑥₀ is different from the initial option price 𝑝(0) necessary to hedge the contingent claim 𝜉, then 𝑥₀−𝑝(0) should be invested according to Merton’s portfolio strategy.

In particular, suppose the contingent claim 𝜉 is a constant. In this case, it is easy to see that the solution (𝑝(⋅),𝑞₁(⋅),…,𝑞_𝑚(⋅)) of (3.3) is𝑝(𝑡)=e^{−∫𝑇𝑡𝑟(𝑠)d𝑠}𝜉,𝑞_𝑖(𝑡)=0,0≤𝑡≤𝑇.(3.8) So we have the following corollary.

Corollary 3.2. Let Hypothesis (H1) hold and 𝜉 be a constant. Then the optimal portfolio strategy of Problem (PIMV) is 𝜋^∗𝑖(𝑡)=−𝜇^𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)²𝔼𝑥^∗(𝑡)∣𝒵_𝑡−e^{−∫𝑇𝑡𝑟(𝑠)d𝑠}𝜉,0≤𝑡≤𝑇.(3.9)

Remark 3.3. The solution (𝑝(⋅),𝑞₁(⋅),…,𝑞_𝑚(⋅)) defined by (3.8) has a straightforward interpretation in financial terms. That is, to achieve a deterministic wealth level 𝜉 at the terminal time 𝑇, the agent should only invest a risk-free asset (a bond) and cannot invest any risky assets (stocks). Therefore, the optimal portfolio strategy obtained in Corollary 3.2 is only the partial information Merton’s portfolio strategy.

The left part of this section will focus on a special mean-variance hedging problem with partial information. By virtue of filtering theory, we get an explicitly observable optimal portfolio strategy as well as a risk measure. We also plot three figures and give numerical simulations to illustrate the theoretical result.

Example 3.4. Let 𝑚=2 and all the conditions in Corollary 3.2 hold. Suppose the observable filtration of an agent is 𝒵_𝑡=𝜎𝑆₁(𝑠)∶0≤𝑠≤𝑡,0≤𝑡≤𝑇.(3.10) It implies that the agents can observe all the past prices of 𝑆₁(⋅), but due to some limit factors (e.g., bad behavior of the stock 𝑆₂(⋅) or time and energy of the investor) they cannot (do not want to) observe 𝑆₂(⋅).

Set 𝒮₁(⋅)=log𝑆₁(⋅), where log𝑥 (𝑥>0) denotes a logarithm function. It follows from Itô’s formula that d𝒮₁(𝑡)=𝜇₁(𝑡)−12𝜎¹(𝑡)²d𝑡+𝜎¹(𝑡)d𝑊₁(𝑡).(3.11) Since 𝜇₁(⋅) and 𝜎₁(⋅) are deterministic functions (see Hypothesis (H1)), the above filtration 𝒵_𝑡 is equivalently rewritten as 𝒵_𝑡=𝜎𝒮₁(𝑠)∶0≤𝑠≤𝑡=𝜎𝑊₁(𝑠)∶0≤𝑠≤𝑡.(3.12)

Similar to Example 2.5, we get from Corollary 3.2𝜋^∗𝑖(𝑡)=−𝜇^𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)²𝑥̂^∗(𝑡)−e^{−∫𝑇𝑡𝑟(𝑠)d𝑠}𝜉,(3.13) where 𝑥̂^∗(⋅) is the solution of d𝑥̂^∗(𝑡)=𝑟(𝑡)−²_𝑖=1𝜇^𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)²𝑥̂^∗(𝑡)+²_𝑖=1𝜇^𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)²−𝜇e^{−∫𝑇𝑡𝑟(𝑠)d𝑠}𝜉𝑑𝑡₁(𝑡)−𝑟(𝑡)𝜎₁(𝑡)𝑥̂^∗𝑥̂_∗(𝑡)−e^{−∫𝑇𝑡𝑟(𝑠)d𝑠}𝜉d𝑊₁(𝑡),(0)=𝑥₀.(3.14) Now 𝜋^∗𝑖(𝑡), 0≤𝑡≤𝑇, defined by (3.13), is an observable optimal portfolio strategy.

We now calculate the risk measure (or the value function) of the agent’s goal RM²∶=2𝐽𝑢(⋅);𝑥₀=min_{𝑣(⋅)∈𝒰ad}2𝐽𝑣(⋅);𝑥₀.(3.15) From (3.14), we derive 𝔼𝑥̂^∗(𝑡)=𝑥₀e^{∫𝑡0∑[𝑟(𝑠)−2𝑖=1((𝜇𝑖(𝑠)−𝑟(𝑠))/(𝜎𝑖(𝑠)))2]𝑑𝑠}+𝜉^𝑡02_𝑖=1𝜇^𝑖(𝑠)−𝑟(𝑠)𝜎_𝑖(𝑠)²𝔼𝑥̂_∗e^{∫𝑡𝑠∑[𝑟(𝜈)−2𝑖=1((𝜇𝑖(𝜈)−𝑟(𝜈))/(𝜎𝑖(𝜈)))2]d∫𝜈−𝑇𝑠𝑟(𝜈)d𝜈}d𝑠,(𝑡)²=𝑥²⁰e^{∫𝑡0∑[2𝑟(𝑠)−2𝑖=1((𝜇𝑖(𝑠)−𝑟(𝑠))/(𝜎𝑖(𝑠)))2−((𝜇2(𝑠)−𝑟(𝑠))/(𝜎2(𝑠)))2]d𝑠}+^𝑡0e^{∫𝑡𝑠∑[2𝑟(𝜈)−2𝑖=1((𝜇𝑖(𝜈)−𝑟(𝜈))/(𝜎𝑖(𝜈)))2−((𝜇2(𝜈)−𝑟(𝜈))/(𝜎2(𝜈)))2]d𝜈}×e^{−∫𝑇𝑠𝑟(𝜈)d𝜈}𝜉2𝜇²(𝑠)−𝑟(𝑠)𝜎₂(𝑠)²+𝜇𝔼𝑥̂^∗(𝑠)₁(𝑡)−𝑟(𝑡)𝜎₁(𝑡)²e^{−∫𝑇𝑠𝑟(𝜈)d𝜈}𝜉d𝑠.(3.16) Combining (3.4), (3.14), Itô’s formula with Lemma A.1, d𝑥^∗(𝑡)²=2𝑟(𝑡)𝑥^∗(𝑡)²+e^{∫−2𝑇𝑡𝑟(𝑠)d𝑠}𝜉²−𝑥̂^∗(𝑡)²²_𝑖=1𝜇^𝑖(𝑡)−𝑟(𝑡)+2𝜇₁𝜎_𝑖(𝑡)²d𝑡(𝑡)−𝑟(𝑡)𝜎₁(𝑡)e^{∫−2𝑇𝑡𝑟(𝑠)d𝑠}𝑥𝜉−𝑥̂^∗(𝑡)𝑥̂^∗(𝑡)d𝑊₁(𝑡),_∗(0)²=𝑥²⁰.(3.17) Solving the above equation, 𝔼𝑥^∗(𝑇)²=𝑥²⁰e^{2∫𝑇0𝑟(𝑠)d𝑠}+^𝑇02_𝑖=1𝜇^𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)²𝜉²−e^{2∫𝑇𝑡𝑟(𝑠)d𝑠}𝔼𝑥̂^∗(𝑡)²d𝑡.(3.18) Applying a property of conditional expectation and throughout integration by parts, we get RM²=𝔼𝑥^∗(𝑇)²−2𝜉𝔼𝑥̂^∗(𝑇)+𝜉²=𝜉−𝑥₀e^{∫𝑇0𝑟(𝑡)d𝑡}²1−^𝑇0𝜌(𝑡)d𝑡(3.19) with 𝜌(𝑡)=²_𝑖=1𝜇^𝑖(𝑡)−𝑟(𝑡)𝜎_𝑖(𝑡)²e^{−∫𝑡0[∑2𝑖=1((𝜇𝑖(𝑠)−𝑟(𝑠))/(𝜎𝑖(𝑠)))2+((𝜇2(𝑠)−𝑟(𝑠))/(𝜎2(𝑠)))2]d𝑠}.(3.20) So we have the following proposition.

Proposition 3.5. The optimal portfolio strategy and the risk measure are given by (3.13) and (3.19), respectively.

To further illustrate the optimal portfolio strategy (3.13) and the risk measure (3.19), we plot three figures and give some numerical results here. Suppose 𝑟=0.06, 𝜇₁=0.12, 𝜇₂=0.18, 𝜎₁=0.12, and 𝜎₂=0.24. Taking 𝑇=1 year, we get from (3.19) 𝜉=𝑥₀e^0.06+1131+2e^−(3/4)RM.(3.21)

In Figure 1, we let $0≤𝑥₀≤$2 million and $0≤RM≤$ 1 million. The plane explicitly describes the relationship among 𝜉, 𝑥₀, and RM. In detail, the bigger 𝑥₀ and RM, the bigger 𝜉.

Figure 1 

The relationship among 𝜉, 𝑥0 and RM.

In Figure 2, we let the investment goal of the agent 𝜉=$ 1.2 million. The beeline shows that RM is a deceasing function of the initial endowment 𝑥₀. In particular, when 𝑥₀=$ 1.13 million, we have RM=$ 0. This means that to achieve the investment goal $ 1.2 million at the end of one year, the agent only needs to invest $ 1.13 million in the bond at the interest rate 6%; moreover, there is no risk for the investment strategy.

Figure 2 

The relationship between RM and 𝑥0.

In Figure 3, we let 𝑥₀=$ 1 million. The beeline implies that RM is an increasing function of the investment goal 𝜉. Consider now the agent who has an initial endowment 𝑥₀=$ 1 million and wishes to obtain an expected return rate 20% in one year. Taking 𝑥₀=$ 1 million and 𝜉=$ 1.2 million, we get RM=$ 0.1112 million, meaning that the risk of the investment goal is as high as 11.12%.

Figure 3 

The relationship between RM and 𝜉.

Furthermore, we calculate the Merton’s portfolio strategy. Let 𝑟=0.06, 𝑇=1, 𝑥₀=$ 1 million, and 𝜉=$ 1.2 million.(1)Set 𝜇₁=0.12, 𝜇₂=0.18, 𝜎₁=0.12, and 𝜎₂=0.24. By (3.13), the amount of money the agent should invest in the 𝑖th stock is 𝜋^∗𝑖(𝑡)=𝜇^𝑖−0.06𝜎^2𝑖1.2e^{−0.06(1−𝑡)}−𝑥̂^∗(𝑡).(3.22)

In particular, at the initial time 𝑡=0, 𝜋^∗1(0)=$ 0.5422 million and 𝜋^∗2(0)=$ 0.2711 million, which implies that the agent needs to invest $ 0.5422 million and $ 0.2711 million in the stocks 𝑆₁(⋅) and 𝑆₂(⋅), respectively, and invest in the bond for an amount of 1−(0.5422+0.2711)=$0.1867million.(3.23)(2)Set 𝜇₁=0.12, 𝜇₂=0.18, 𝜎₁=0.12, and 𝜎₂=0.17. Similarly, we have 𝜋^∗1(0)=$ 0.5422 million, 𝜋^∗2(0)=$ 0.5403 million, and (0.5422+0.5403)−1=$0.0825million,(3.24)

which implies that the agent needs to borrow $ 0.0825 million from a bank and invest the amount $ 1.0825 million in the two stocks 𝑆₁(⋅) and 𝑆₂(⋅). This is indeed an aggressive policy.

4. Comparison with Existing Results

The subject of stochastic control with partial information has been discussed by many researchers, such as Bensoussan [10] and Xiong and Zhou [3]. Usually, they made one of the following two assumptions: (i) the filtration 𝒵 is a sigma algebra generated by some observable process; (ii) the control systems are Markovian. From this viewpoint, our work cannot be covered by their results.

Note that our work is related to the recent paper by Hu and Øksendal [4]. In what follows, we shall give some detailed comparisons between them.(1)Comparing with [4], the distinctive characteristics of our Problem (PIMV) are the following four points. First, since 𝜉 in our cost functional (1.5) is a random variable, then it partly generalizes that of [4]; meanwhile, our cost functional (1.5) can measure the risk that the contingent claim cannot be reached. Second, we give a possible formulation of partial information in the setting of finance and interpret the economic meaning of the optimal portfolio with partial information. Third, in terms of filtering theory, we explicitly compute the observable optimal portfolio strategy and the risk measure of an agent in Example 3.4. Last but not least, we present some numerical results and figures to illustrate the optimal portfolio strategy and the risk measure. Although the example is special, it is nontrivial and contains some filtering techniques. These results show the practical sense of our paper.(2)Since the initial state 𝑥₀ in control system (2.1) may be a decision and 𝜉 in cost functional (2.3) is a random variable, our Problem (PILQ) is different from that of [4]. In particular, if the foregoing 𝑥₀ is also a decision, then the partial information optimal control can be denoted by a conditional expectation of the solution of the corresponding adjoint equation (recall Corollary 2.4). Obviously, this is different from [1, 4].(3)Filtering theory plays an important role in optimal control with partial information. To get an explicitly observable optimal control, it is necessary to compute the conditional expectation of the solution of BSDEs. However, it is not doable in most cases. In this paper, we try to solve some stochastic control problems by using filtering theory. As a byproduct, we establish backward and forward-backward stochastic differential filtering equations which are different from the existing literature about filtering theory. Although the technique of solving stochastic control is restricted and the filtering equations are linear, we can regard them as a contribution to filtering-control theory. By the way, the study about Problem (PILQ) motivates us to establish some general filtering theory of BSDEs in future work.

In [4], Hu and Øksendal obtain some optimal controls and value functions with complete and partial information. Note that they only denote the optimal controls by conditional expectation. We also notice there is not any filtering result in [4]. This is different from ours.

Appendix

A Classical Filtering Equation for SDEs

We present here a classical filtering result for the readers’ convenience which was employed in this paper. For a detailed discussion of filtering, we refer to the books [5, 6].

Consider the following 1-dimensional state and observation equations: 𝜉(𝑡)=𝜉𝜃₀(𝑡)=𝜃₀+^𝑡0ℎ(𝑠)d𝑠+𝑥(𝑡),+^𝑡0𝑎(𝑠,𝜔)d𝑠+^𝑡0𝑏(𝑠,𝜉)d𝑊(𝑠).(A.1) Here (𝑊(⋅)) is a 1-dimensional standard Brownian motion defined on the complete filtered probability space (Ω,ℱ,(ℱ_𝑡),𝑃) equipped with a natural filtration ℱ_𝑡=𝜎{𝑊(𝑠);0≤𝑠≤𝑡}, ℱ=ℱ_𝑇, 0≤𝑡≤𝑇; 𝑥(𝑡) is an ℱ_𝑡-martingale; ℎ(𝑡) is an ℱ_𝑡-adapted process with ∫^𝑇0|ℎ(𝑠)|d𝑠<+∞; the functional 𝑏(𝑡,𝑦), 𝑦∈𝐶_𝑇, 0≤𝑡≤𝑇, is ℬ_𝑡-measurable.

We need the following hypothesis.(H)For any 𝑦,𝑦∈𝐶_𝑇, 0≤𝑡≤𝑇, the functional 𝑏(𝑡,⋅) satisfies ||𝑏(𝑡,𝑦)−𝑏𝑡,𝑦||²≤𝐿₁^𝑡0𝑦(𝑠)−𝑏(𝑡,𝑦)₂𝑦(𝑠)²d𝐾(𝑠)+𝐿₂𝑦(𝑡)−𝑦(𝑡)²,≤𝐿₁^𝑡01+𝑦(𝑠)²d𝐾(𝑠)+𝐿₂1+𝑦(𝑡)²,(A.2)

where 𝐿₁ and 𝐿₂ are two nonnegative constants and 0≤𝐾(⋅)≤1 is a nondecreasing right continuous function. Moreover, sup_{0≤𝑡≤𝑇}𝔼𝜃(𝑡)²<+∞, ∫^𝑇0𝔼ℎ(𝑡)²d𝑡<+∞, ∫^𝑇0𝔼𝑎(𝑡,𝜔)²d𝑡<+∞, 𝑏(𝑡,𝑦)²≥𝐶>0.

The following result is due to of [5, Theorem 8.1].

Lemma A.1. Define 𝜃̂(t)=𝔼[𝜃(t)∣ℱ^𝜉_t]. Here 𝜃(t) can take 𝜃(t), h(t), D(t), a(t,𝜔), and 𝜃(t)a(t,𝜔); ℱ^𝜉_t=𝜎{𝜉(s)∶0≤𝑠≤t}, 0≤t≤T. If Hypothesis (H) holds, then the optimal nonlinear filtering equation is 𝜃̂(𝑡)=𝜃̂₀+^𝑡0ℎ(𝑠)d𝑠+^𝑡0𝐷(𝑠)+𝜃(𝑠)𝑎(𝑠,𝜔)−𝜃̂(𝑠)𝑎̂(𝑠,𝜔)𝑏(𝑠,𝜉)⁻¹d𝑊(𝑠),(A.3) where 𝑊(𝑡)=^𝑡0d𝜉(𝑠)−𝑎̂(𝑠,𝜔)d𝑠𝑏(𝑠,𝜉)(A.4) is a standard Brownian motion (with respect to ℱ^𝜉𝑡), and 𝐷(𝑡) is an ℱ_𝑡-adapted process with 𝐷(𝑡)=d⟨𝑥(𝑡),𝑊(𝑡)⟩d𝑡.(A.5)

Acknowledgments

The authors would like to thank the anonymous referee and the Editor for their insightful comments for improving the quality of this work. G. Wang acknowledges the financial support from the National Nature Science Foundation of China (11001156), the Natural Science Foundation of Shandong Province, China (ZR2009AQ017), and the Independent Innovation Foundation of Shandong University, China (2010GN063). Z. Wu acknowledges the financial support from the National Natural Science Foundation of China (10921101), the National Basic Research Program of China (973 Program, no. 2007CB814904), the Natural Science Foundation of Shandong Province, China (JQ200801 and 2008BS01024), and the Science Foundation for Distinguished Young Scholars of Shandong University, China (2009JQ004). The material in this paper was partially presented at the 27th Chinese Control Conference, July 16–18, 2008, Kunming, Yunnan, China.