2. Problem formulation

Let $( {\Omega,{\mathcal {F}},{\{ {\mathcal {F}} \} _{t \ge 0}},\mathbb {P}})$ be a complete probability space, where ${{\mathcal {F}}_t}$ represents the information available in the financial market up to time $\it t$ and $\mathbb {P}$ is the real-world probability measure. It is assumed that a DC pension plan member contributes to the pension account continuously until the retirement time $T$. Suppose that the assets in the financial market can be traded continuously without constraints on the amount of transactions, and there is no additional cost such as transaction fee or tax.

2.1. Financial market

Assume that the financial market consists of four assets: a risk-free asset, a zero-coupon bond, an inflation-indexed bond and a stock. The first asset in the financial market is the risk-free asset whose price process $A(t)$ is given by

(1) \begin{equation} \frac{{{{\rm d}}A(t)}}{{A(t)}} = R(t){{\rm d}}t,{{\rm }}\quad A( 0 ) = {A_0}, \end{equation}

where $R(t)$ is the nominal interest rate whose evolution satisfies the CIR model:

(2) \begin{equation} {{\rm d}}R(t) = ( {a - bR(t)} )\,{{\rm d}}t - {\sigma _R}\sqrt {R(t)} \,{{\rm d}}{W_1}(t),\quad R( 0 ) = {R_0}. \end{equation}

Here $a,b > 0$, $b$ denotes the speed of the mean reversion, ${\sigma _R}$ is the volatility and ${W_1}(t)$ is a standard Brownian motion on $( {\Omega,{\mathcal {F}},{\{{\mathcal {F}}\}_{t \ge 0}},\mathbb {P}} )$. If $2a \ge \sigma _R^2$, then for all $t > 0$ , the nominal interest rate is strictly positive almost surely.

The second asset is the zero-coupon bond closely correlated with the stochastic interest rate. Let $B( {t,T} )$ denote the price of this zero-coupon bond at time $t$ with a payoff of 1 at maturity $T$. With the market price of risk corresponding to ${W_1}(t)$ being given by $\lambda _{R}\sqrt {R(t)}$ and by the no-arbitrage pricing theory, $B( {t,T} )$ can be expressed as the following form (see [Reference Guan and Liang17,Reference Han and Hung19]):

$$B( {t,T} ) = \exp [ {{h_0}( {T-t} ) - {h_1}( {T-t} )R(t)}],$$

where

\begin{align*} {h_1}( {T - t} ) & = \frac{{2( {{{{\rm e}}^{2d( {T - t} )}} - 1} )}}{{( {c + 2d} )( {{{{\rm e}}^{2d( {T - t} )}} - 1} ) + 4d}},\\ {h_0}( {T-t} ) & ={-} \frac{a}{{\sigma _R^2}}\left[ {2\log \frac{{( {c + 2d} )( {{e^{2d( {T - t} )}} - 1} ) + 4d}}{{4d}} - ( {c + 2d} )( {T - t} )} \right],\\ c & = b - {\lambda _R}{\sigma _R},\\ d& = \frac{1}{2}\sqrt {{{( {b - {\lambda _R}{\sigma _R}} )}^2} + 2\sigma _R^2} . \end{align*}

The dynamics of $B( {t,T} )$ follows

(3) \begin{equation} \begin{aligned} \dfrac{{{{\rm d}}B( {t,T} )}}{{B( {t,T} )}} & = ( {R(t) + {\lambda _R}\sqrt {R(t)} {\sigma _B}( {T - t} )} )\,{{\rm d}}t + {\sigma _B}( {T - t} )\,{{\rm d}}{W_1}(t),\\ B( {T,T}) & = 1, \end{aligned} \end{equation}

with ${\sigma _B}( {T - t} ) = {\sigma _R}{h_1}( {T - t} )\sqrt {R(t)}$ being the volatility.

As argued by Boulier et al. [Reference Boulier, Huang and Taillard3], there are not always zero-coupon bonds corresponding to the specified maturing date in the market. Thus, a rolling bond with a fixed maturity ${\tau _1}$ will be incorporated in our analysis, which can be used to hedge the interest rate risk. The price dynamics of the rolling bond is as follows:

(4) \begin{equation} \frac{{{{\rm d}}{B_{{\tau _1}}}(t)}}{{{B_{{\tau _1}}}(t)}} = ( {R(t) + {\lambda _R}\sqrt {R(t)} {\sigma _B}( {{\tau _1}} )} )\,{{\rm d}}t + {\sigma _B}( {{\tau _1}} )\,{{\rm d}}{W_1}(t), \end{equation}

where ${\sigma _B}( {{\tau _1}} ) = {\sigma _R}{h_1}( {{\tau _1}} )\sqrt {R(t)}$. In addition, the relationship between $B( {t,T} )$ and ${B_{{\tau _1}}}(t)$ is

$$\frac{{{{\rm d}}B( {t,T} )}}{{B( {t,T} )}} = \frac{{{h_1}( {T - t} )}}{{{h_1}( {{\tau _1}} )}}\frac{{{{\rm d}}{B_{{\tau _1}}}(t)}}{{{B_{{\tau _1}}}(t)}} + \left( {1 - \frac{{{h_1}( {T - t} )}}{{{h_1}( {{\tau _1}} )}}} \right)\frac{{{{\rm d}}A(t)}}{{A(t)}}.$$

In the sequel, the rolling bond with a fixed maturity ${\tau _1}$ is referred to as the zero-coupon bond.

Since the investment management of the pension is a decades-long project, the inflation inevitably erodes the purchasing power of plan members’ wealth. Therefore, it is necessary to take into account the inflation in the asset allocation for a DC plan. The consumer price index (CPI) is usually used as an index to represent the level of inflation. We suppose the price index $P(t)$ satisfies the following stochastic differential equation:

(5) \begin{equation} \begin{aligned} \frac{{{{\rm d}}P(t)}}{{P(t)}} & = {\mu _P}\,{{\rm d}}t + {\sigma _{P1}}\sqrt {R(t)} \,{{\rm d}}{W_1}(t) + {\sigma _{P2}}\,{{\rm d}}{W_2}(t),\\ P( 0 ) & = {P_0}, \end{aligned} \end{equation}

where ${\mu _P} > 0$ represents the expected inflation rate, ${W_2}(t)$ is a standard Brownian motion independent of ${W_1}(t)$ and ${\sigma _{P1}}>0$ and ${\sigma _{P2}} > 0$ are the volatility parameters. The inflation-indexed bond is issued in the financial market to help plan members manage the inflation risk. Based on the no-arbitrage pricing theory and similar to the procedure for deriving the explicit expression of $B(t,T)$, we can know that the price of the inflation-indexed bond at time $t$ with the maturity $T$ and a payoff of $P ( T )$, denoted by $I( {t,T} )$, has the explicit form:

$$I( {t,T} ) = P(t)\exp [ {{q_0}( {T-t} ) - {q_1}( {T-t} )R(t)}],$$

where

\begin{align*} {q_1}( {T - t} ) & = \frac{{2( {1 + {\sigma _{P1}}{\lambda _R}} )( {{{{\rm e}}^{2f( {T - t} )}} - 1} )}}{{( {g + 2f} )( {{{{\rm e}}^{2f( {T - t} )}} - 1} ) + 4f}},\\ {q_0}( {T-t} ) & ={-} \dfrac{a}{{\sigma _R^2}}\left[ {2\log \frac{{( {g + 2f} )( {{e^{2f( {T - t} )}} - 1} ) + 4f}}{{4f}} - ( {g + 2f} )( {T - t} )} \right]\\ & \quad +{\mu _P}( {T - t} ) - {\sigma _{P2}}{\lambda _P}( {T - t} ),\\ g & = b - {\sigma _R}{\lambda _R} + {\sigma _R}{\sigma _{P1}}, \\ f & = \frac{1}{2}\sqrt {{{( {b - {\sigma _R}{\lambda _R} + {\sigma _R}{\sigma _{P1}}} )}^2} + 2\sigma _R^2( {1 + {\lambda _R}{\sigma _{P1}}} )} , \end{align*}

and ${\lambda _P} > 0$ denotes the market price of risk with respect to ${W_2}(t)$. By the Itô formula, we have

(6) \begin{equation} \begin{aligned} \dfrac{{{{\rm d}}I( {t,T} )}}{{I( {t,T} )}} & = ( {R(t) + {\lambda _R}\sqrt {R(t)} {\sigma _{I1}}( {T - t} ) + {\lambda _P}{\sigma _{I2}}} )\,{{\rm d}}t\\ & \quad + {\sigma_{I1}}( {T - t} )\,{{\rm d}}{W_1}(t) + {\sigma _{I2}}\,{{\rm d}}{W_2}(t),\\ I( {T,T} ) & = P( T ), \end{aligned} \end{equation}

where ${\sigma _{I1}}( {T - t} )$ and ${\sigma _{I2}}$ are the volatility parameters.

Analogous to the rolling zero-coupon bond, we introduce the rolling inflation-indexed bond with constant maturity ${\tau _2}$, which can serve as an instrument to hedge the inflation risk. The price process of the rolling inflation-indexed bond evolves as

(7) \begin{align} \dfrac{{{{\rm d}}{I_{{\tau _2}}}(t)}}{{{I_{{\tau _2}}}(t)}} & = ( {R(t) + {\lambda _R}\sqrt {R(t)} {\sigma _{I1}}( {{\tau _2}} ) + {\lambda _P}{\sigma _{I2}}} )\,{{\rm d}}t\nonumber\\ & \quad+ {\sigma _{I1}}( {{\tau _2}} )\,{{\rm d}}{W_1}(t) + {\sigma _{I2}}\,{{\rm d}}{W_2}(t), \end{align}

where ${\sigma _{I1}}( {{\tau _2}} ) = ( {{\sigma _{P1}} + {\sigma _R}{q_1}( {{\tau _2}} )} )\sqrt {R(t)}$. The relation between $I( {t,T} )$ and ${I_{{\tau _2}}}(t)$ is described by

$$\frac{{{{\rm d}}I( {t,T} )}}{{I( {t,T} )}} = \frac{{{{\rm d}}{I_{{\tau _2}}}(t)}}{{{I_{{\tau _2}}}(t)}} + \frac{{{q_1}( {T - t} ) - {q_1}( {{\tau _2}} )}}{{{h_1}( {{\tau _1}} )}}\left( {\frac{{{{\rm d}}{B_{{\tau _1}}}(t)}}{{{B_{{\tau _1}}}(t)}} - \frac{{{\rm d}M(t)}}{{M(t)}}} \right).$$

The rolling bond with a constant maturity ${\tau _2}$ is referred to as the inflation-indexed bond in the rest of this article.

The fourth asset in the financial market is the stock whose return rate depends on a Markov chain. Before introducing the stock price process, some notations related to this Markov chain are given. Let ${\mathbf {X}}: = {\{ {{\mathbf {X}}(t)} \}_{t \ge 0}}$ be a continuous-time Markov chain with $N$ states on $( {\Omega,{\mathcal {F}},{\{{\mathcal {F}}\}_{t \ge 0}},\mathbb {P}})$. The Markov chain can be applied to describe the uncertainty of the economy. The state space of ${\mathbf {X}}$ is represented by a set of standard unit vectors ${\mathcal {L}}: = \{ {{{\mathbf {e}}_1},{{\mathbf {e}}_2}, \ldots,{{\mathbf {e}}_N}} \} \subset {\mathbb {R}^N}$. Here ${\mathbf {Q}}: = {\{ {{\mathbf {Q}}(t)} \}_{t \ge 0}}$ is the transition rate matrix of ${\mathbf {X}}$ and ${\mathbf {Q}}(t) = {[ {{q_{ij}}(t)} ]_{i,j = 1,2, \ldots,N}}$ with ${q_{ij}}(t)$ representing the instantaneous transition rate of the Markov chain from the $i$th state to the $j$th state. The natural filtration generated by the Markov chain ${\mathbf {X}}$ is denoted as ${{\mathbb {F}}^{\mathbf {X}}}: = {\{ {{\mathcal {F}}_t^{\mathbf {X}}} \}_{t \ge 0}}$, where ${\mathcal {F}}_t^{\mathbf {X}} = \sigma \{ {{\mathbf {X}}( u ):u \in [ {0,t} ]} \}$. The price process of the stock is specified by

(8) \begin{equation} \frac{{{{\rm d}}S(t)}}{{S(t)}} = \mu ( {t,{\mathbf{X}}} )\,{{\rm d}}t + {\sigma _S}\,{{\rm d}}{W_3}(t),\quad S( 0 ) = {S_0}, \end{equation}

where $\mu ( {t,{\mathbf {X}}} ): = \langle {{\mathbf {\mu }},{\mathbf {X}}(t)} \rangle$ ($\langle { \cdot, \cdot } \rangle$ represents the inner product) is the expected return rate of the stock at time $t$, ${\mathbf {\mu }} = ( {{\mu _1},{\mu _2}, \ldots,{\mu _N}} )' \in {\mathbb {R}^N}$, ${\sigma _S} > 0$ is the volatility and ${W_3}(t)$ is a standard Brownian motion independent of ${W_1}(t)$, ${W_2}(t)$ and ${\mathbf {X}}(t)$. In investment practice, only the stock price process can be observed but not the specific economic states, that is, ${\mathbf {X}}(t)$ is not observable, and it is called a hidden Markov chain. From (8), we know that ${W_3}(t)$ has the corresponding market price of the risk ${\lambda _S}(t): = ({1}/{{{\sigma _S}}})( {\mu ( {t,{\mathbf {X}}} ) - R(t)} )$. The filtration generated by the stock price process is denoted as ${{\mathbb {F}}^S}: = {\{ {{\mathcal {F}}_t^S} \}_{t \ge 0}}$, where ${\mathcal {F}}_t^S = \sigma \{ {S( u ):u \in [ {0,t} ]} \}$.

2.2. Wealth process and optimization problem

Denote the filtrations generated by ${W_1}: = {\{ {{W_1}(t)} \}_{t \ge 0}}$, ${W_2}: = {\{ {{W_2}(t)} \}_{t \ge 0}}$ and ${W_3}: = {\{ {{W_3}(t)} \}_{t \ge 0}}$ by ${\mathbb {F}^{{W_1}}}: = {\{ {{\mathcal {F}}_t^{{W_1}}} \}_{t \ge 0}}$, ${\mathbb {F}^{{W_2}}}: = {\{ {{\mathcal {F}}_t^{{W_2}}} \}_{t \ge 0}}$ and ${\mathbb {F}^{{W_3}}}: = {\{ {{\mathcal {F}}_t^{{W_3}}} \}_{t \ge 0}}$, respectively. The observable information for the member is $\mathbb {G}: = {\{ {{{\mathcal {G}}_t}} \}_{t \ge 0}}$, where ${{\mathcal {G}}_t}: = {\mathcal {F}}_t^{{W_1}} \vee {\mathcal {F}}_t^{{W_2}} \vee {\mathcal {F}}_t^S$.

In a DC pension plan, the members continuously contribute to their accounts until the retirement date $T$. The instantaneous contribution rate is denoted as $C(t)$, which is usually a preset proportion of the salary of the member. However, the member's salary is usually influenced by many stochastic factors. Therefore, we assume that $C(t)$ is a $\mathbb {G}$-adapted process.

Suppose ${V_0}$ is the initial wealth of the member's pension account. Let ${x_B}(t)$, ${x_I}(t)$ and ${x_S}(t)$ represent the proportions of wealth invested in the zero-coupon bond, the inflation-indexed bond and the stock at time $t$, respectively. Then the wealth proportion invested in the risk-free asset is $1 - {x_B}(t) - {x_I}(t) - {x_S}(t)$. If the investment strategy is denoted as $\mathbf {x} := \{ \mathbf {x} ( t ) \} _{t\in [ 0,T ] } = \{ ( x_{B} ( t ) , x_{I} ( t ) , x_{S} ( t ) )^{\mathrm {T} } \}_{t\in [ 0,T ] }$, the nominal wealth process ${V^{\mathbf {x}}}$ corresponding to the investment strategy ${\mathbf {x}}$ evolves according to

(9) \begin{equation} \begin{aligned} {{\rm d}}{V^{\mathbf{x}}}(t) & = {V^{\mathbf{x}}}(t)( {R(t) + {{\mathbf{x}}^{{\rm T}}}(t){\bf \Sigma} {\bf \Lambda} (t)})\,{{\rm d}}t + C(t)\,{{\rm d}}t + {V^{\mathbf{x}}}(t){{\mathbf{x}}^{{\rm T}}}(t){\bf \Sigma} \,{{\rm d}}{\mathbf{W}}(t),\\ {V^{\mathbf{x}}}( 0 ) & = {V_0}, \end{aligned} \end{equation}

where

\begin{align*} {\bf{\Sigma }}& : = \left( {\begin{array}{ccc} {{\sigma _B}({{\tau _1}} )} & 0 & 0\\ {{\sigma _{I1}}( {{\tau _1}} )} & {{\sigma _{I2}}} & 0\\ 0 & 0 & {{\sigma _S}} \end{array}} \right),\\ {\bf \Lambda} (t)& : = {( {{\lambda _R}\sqrt {R(t)} ,{\lambda _P},{\lambda _S}(t)} )^{{\rm T}}},\\ {\mathbf{W}}(t)& : = {( {{W_1}(t),{W_2}(t),{W_3}(t)} )^{{\rm T}}}. \end{align*}

Let ${\mathcal {A}}$ be the set of all admissible strategies that satisfy the following properties:

1. (1) ${\mathbf {x}}: = {\{ {{\mathbf {x}}(t)} \}_{t \in [ {0,T} ]}}$ is $\mathbb {G}$-progressively measurable;

2. (2) $\int _0^T {{{\| {{\mathbf {x}}(t)} \|}^2}\,{{\rm d}}t} < \infty$, $\mathbb {P}$ - a.s.;

3. (3) the stochastic differential equation (9) related to the nominal wealth process ${V^{\mathbf {x}}}$ has a unique strong solution.

In order to protect the benefit of the plan member more effectively, a minimum guarantee constraint should be imposed on the DC pension fund. Assume that the value of the minimum guarantee at retirement time $T$ is $G( T )$, which is ${{\mathcal {G}}_T}$-adapted. It is required that the nominal wealth value at time $T$ does not fall below $G( T )$. The goal of the plan member is to find an optimal investment strategy in ${\mathcal {A}}$ that minimizes the risk measure $\rho$ of the terminal wealth ${V^{\mathbf {x}}}( T )$ over the level of minimum guarantee, that is

(10) \begin{equation} \begin{aligned} & \inf_{{\mathbf{x}} \in {\mathcal{A}}} \rho ( {{V^{\mathbf{x}}}( T ) - G(T )} ),\\ & {\rm s.t.}\quad {V^{\mathbf{x}}}( T ) \ge G( T ). \end{aligned} \end{equation}

The convex risk measure, which is a generalization of the coherent risk measure, is selected to measure the risk. It can reflect the nonlinear increase of portfolio risk along with the size of assets caused by the liquidity risk. The definition of the convex risk measure is as follows.

Definition 1. Let ${\mathcal {S}}$ be a set of random variables with lower bounds on $( {\Omega,{\mathcal {F}},{\mathbb {P}}} )$, then the convex risk measure $\rho$ is a functional $\rho :{\mathcal {S}} \to \mathbb {R}$ that satisfies the following properties:

1. (1) Additivity: For any $L \in {\mathcal {S}}$ and $K \in \mathbb {R}$, $\rho ( {L + K} ) = \rho ( L ) - K$.

2. (2) Monotonicity: For any ${L_1},{L_2} \in {\mathcal {S}}$, if ${L_1}( \omega ) \le {L_2}( \omega )$ for any $\omega \in \Omega$, then $\rho ( {{L_1}} ) \ge \rho ( {{L_2}} )$.

3. (3) Convexity: For any ${L_1},{L_2} \in {\mathcal {S}}$, $\rho ( {\alpha {L_1} + ( {1 - \alpha }){L_2}}) \le \alpha \rho ( {{L_1}} ) + ( {1 - \alpha } )\rho ( {{L_2}})$, $\alpha \in ( {0,1} )$.

Here, the random variable $L$ describes the financial position and can be regarded as a loss variable.

The lemma below gives a representation of the convex risk measure.

Lemma 1. Denote by ${{\mathcal {M}}_a}$ a family of measures absolutely continuous with respect to the probability measure $\mathbb {P}$. Define a function $\eta :{{\mathcal {M}}_a} \to \mathbb {R}$ such that $\eta ( \mathbb {Q} ) < \infty$ for any $\mathbb {Q}\in {{\mathcal {M}}_a}$. Then the convex risk measure $\rho ( L )$ of $L \in {\mathcal {S}}$ is continuous and can be expressed as

(11) \begin{equation} \rho ( L ) = \sup_{\mathbb{Q} \in {{\mathcal{M}}_a}} \{ {{{\mathbb{E}}^\mathbb{Q}}[ { - L} ] - \eta (\mathbb{Q} )} \}, \end{equation}

where ${\mathbb {E}}^\mathbb {Q}$ represents the expectation under the measure $\mathbb {Q}$.

Proof. See Frittelli and Rosazza Gianin [Reference Frittelli and Rosazza Gianin15] or Föllmer and Schied [Reference Föllmer and Schied14].

From Lemma 1, three components must be determined to define a convex risk measure: the family of probability measures ${{\mathcal {M}}_a}$, the penalty function $\eta$ and the loss variable $L$. The family ${{\mathcal {M}}_a}$ is interpreted as a set of generalized scenarios representing contingencies of future market or economic situations. The penalty function $\eta$ penalizes the improper selection of a probability model. In order to obtain the analytical expression of the optimal strategy, we suppose that the probability measure family ${{\mathcal {M}}_a}$ is generated by the Girsanov's transformation and the penalty function $\eta$ has a special form. Thus, problem (10) will be converted into a stochastic differential game problem.

3. Problem reduction

Problem (10) is a portfolio optimization problem with partial observation and state constraints. We first transform it into a portfolio optimization problem with complete information by using the filtering method. Then, to deal with the state constraints, we construct the auxiliary processes.

3.1. Filtering estimate

The filtering method has been widely applied to study the stochastic control problem with incomplete information. For the intensive exposition of this theory, see Kallianpur [Reference Kallianpur20], Elliott et al. [Reference Elliott, Moore and Aggoun12] and Liptser and Shiryaev [Reference Liptser and Shiryaev23]. Next, we extract the unobservable information about market states from the observable stock price using the filtering technique.

The Wonham filter of the hidden Markov chain is denoted as

(12) \begin{equation} {p_k}(t): = \mathbb{P}( {{{\mathbf{X}}(t) = {{\mathbf{e}}_k}} \,|\,{{\mathcal{G}}_t}} ),\quad k = 1,2, \ldots ,N. \end{equation}

Let ${\mathbf {p}}(t) = {( {{p_1}(t),{p_2}(t), \ldots,{p_N}(t)})^{{\rm T}}}$. Define the innovation process:

$$\widehat {{W_3}}(t): = {W_3}(t) + \frac{1}{{{\sigma _S}}}\int_0^t {( {\mu ( {u,{\mathbf{X}}} ) - \widehat \mu ( {u,{\mathbf{X}}} )} )}\, {{\rm d}}u,$$

where $\widehat \mu ( {t,{\mathbf {X}}} ) = \mathbb {E}[ {{\mu ( {t,{\mathbf {X}}} )} \,|\,{{\mathcal {G}}_t}} ] = \langle {{\mathbf {\mu }},{\mathbf {p}}(t)} \rangle$ is the filtered estimate of $\mu ( {t,{\mathbf {X}}} )$.

Lemma 2. We have the following results:

1. (1) $\widehat {{W_3}}: = {\{ {\widehat {{W_3}}(t)} \}_{t \ge 0}}$ is a $\mathbb {G}$-adapted Brownian motion.

2. (2) The filter processes ${p_k}(t)$ specified by (12) satisfy that for $t \ge 0$,

   \begin{align*} {{\rm d}}{p_k}(t) & = \sum_{j = 1}^N {{q_{jk}}(t)} {p_j}(t)\,{{\rm d}}t + \frac{1}{{{\sigma _S}}}( {{\mu _k} - \widehat \mu ( {t,{\mathbf{X}}} )} ){p_k}(t)\,{{\rm d}}\widehat {{W_3}}(t),\\ {p_k}( 0 ) & = P( {{\mathbf{X}}( 0 ) = {{\mathbf{e}}_k}} ),\quad k = 1, \ldots ,N. \end{align*}
   We can rewrite it in matrix form as follows:
   $${{\rm d}}{\mathbf{p}}(t) = {\mathbf{Q}}'{\mathbf{p}}(t)\,{{\rm d}}t + \frac{1}{{{\sigma _S}}}{\mathbf{N}}(t){\mathbf{p}}(t)\,{{\rm d}}\widehat {{W_3}}(t),$$
   where ${\mathbf {N}}(t){{\rm =\ diag(}}{\mathbf {\mu }} - \widehat \mu ( {t,{\mathbf {X}}} ){{\mathbf {1}}_N}{{\rm )}}$.

3. (3) $\sigma ( {S( u ):u \in [ {0,t} ]}) = \sigma ( {\widehat {{W_3}}( u ):u \in [ {0,t} ]} ) = \sigma ( {{\mathbf {p}}( u ):u \in [ {0,t} ]} )$.

Proof. See Elliott et al. [Reference Elliott, Moore and Aggoun12] for the detailed proof, which is omitted here.

Let $\widehat {{\lambda _S}}(t): = ({1}/{{{\sigma _S}}})( {\widehat \mu ( {t,{\mathbf {X}}} ) - R(t)} )$, which is the filtered estimate of the market price of risk ${\lambda _S}(t)$. By using the innovation process and the filtered estimate, the dynamics of the nominal wealth process (9) can be rewritten as

(13) \begin{equation} \begin{aligned} {{\rm d}} {{V^{\mathbf{x}}}}(t) & = {{V^{\mathbf{x}}}}(t)( {R(t) + {{\mathbf{x}}^{{\rm T}}}(t){\bf \Sigma} {\widehat {\bf{\Lambda}}} (t)} )\,{{\rm d}}t + C(t)\,{{\rm d}}t + {{V^{\mathbf{x}}}}(t){{\mathbf{x}}^{{\rm T}}}(t){\bf {\Sigma}} \,{{\rm d}}\widetilde {\mathbf{W}}(t),\\ {{V^{\mathbf{x}}}}( 0 ) & = {V_0}, \end{aligned} \end{equation}

where ${\widehat {\bf {\Lambda }}} (t) = {( {{\lambda _R}\sqrt {R(t)} ,{\lambda _P},\widehat {{\lambda _S}}(t)} )^{{\rm T}}}$ and $\widetilde {\mathbf {W}}(t): = {( {{W_1}(t),{W_2}(t),\widehat {{W_3}}(t)} )^{{\rm T}}}$.

To simplify the notation, ${V^{\mathbf {x}}}$ will be abbreviated as $V$ below.

3.2. Auxiliary problem

In Section 3.1, we have converted problem (10) into an investment optimization problem with complete information, but it is still the one with state constraints which is difficult to solve directly. We will transform it into an unconstrained stochastic control problem by constructing auxiliary processes. This approach has been adopted by Deelstra et al. [Reference Deelstra, Grasselli and Koehl7], Han and Hung [Reference Han and Hung19] and Guan and Liang [Reference Guan and Liang17].

Define the auxiliary processes:

\begin{align*} F(t)& : = \frac{1}{{H(t)}}\mathbb{E}\left[ {\left. {\int_t^T {H( u )C( u )\,{{\rm d}}u} } \,\right|{{\mathcal{G}}_t}} \right],\\ G(t)& : = \frac{1}{{H(t)}}\mathbb{E}[ {{G( T )H( T )} |{{\mathcal{G}}_t}}], \end{align*}

where

$$H(t): = \exp \left[ { - \int_0^t ( {R( u ) + \dfrac{1}{2}{{\| {{\mathbf{\Lambda }}( u )} \|}^2} ) \,{{\rm d}}u - \int_0^t {{{\mathbf{\Lambda }}^{{\rm T}}}(t)\,{{\rm d}}\widetilde {\mathbf{W}}( u )} } } \right]$$

is a $\mathbb {G}$-adapted pricing kernel. $F(t)$ represents the present value of the future contribution from $t$ to $T$, and $G(t)$ denotes the present value of the level of minimum guarantee $G( T )$ at time $t$.

By the property (3) of Lemma 2 and the martingale representation theorem, there exist unique, $\mathbb {G}$-adapted square integrable processes ${\mathbf {u}}(t): = {( {{u_1}(t),{u_2}(t),{u_3}(t)})^{{\rm T}}}$ and ${\mathbf {v}}(t): = {( {{v_1}(t),{v_2}(t),{v_3}(t)})^{{\rm T}}}$ such that

(14) \begin{align} {{\rm d}}( {F(t)H(t)} ) & ={-} H(t)C(t)\,{{\rm d}}t + {{\mathbf{u}}^{{\rm T}}}(t)\,{{\rm d}}\widetilde {\mathbf{W}}(t), \end{align}

(15) \begin{align} {{\rm d}}( {G(t)H(t)} ) & = {{\mathbf{v}}^{{\rm T}}}(t)\,{{\rm d}}\widetilde {\mathbf{W}}(t). \end{align}

Define a surplus process $Z(t) = V(t) + F(t) - G(t)$, which equals the value of assets in the pension account plus the present value of the future contribution, and then minus the discounted value of the minimum guarantee level. The following results show that $Z(t)$ is self-financing and solving the optimization problem (10) is equivalent to solving an auxiliary problem related to $Z(t)$.

Proposition 1. The surplus process $Z(t)$ is self-financing, and there exists a $\mathbb {G}$-progressively measurable process ${\bf {\pi }}(t): = {( {{\pi _B}(t),{\pi _I}(t),{\pi _S}(t)} )^{{\rm T}}}$ such that

(16) \begin{equation} \begin{aligned} \dfrac{{{{\rm d}}Z(t)}}{{Z(t)}} & = (R(t) + {{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma }}{\widehat {\bf{\Lambda}}}(t) )\,{{\rm d}}t + {{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma}} \,{{\rm d}}\widetilde {\mathbf{W}}(t),\\ Z( 0 ) & > 0. \end{aligned} \end{equation}

The constraint ${V^{\mathbf {x}}}( T ) \ge G( T )$ in problem (10) is equivalent to $Z( 0 ) \ge 0$, and problem (10) is equivalent to the following auxiliary problem:

(17) \begin{equation} \begin{aligned} \inf_{{\bf{\pi }} \in \widetilde {\mathcal{A}}} \rho ( {{Z^{\bf{\pi }}}( T )} ),\\ {\rm s.t.}\quad Z( 0 ) \ge 0. \end{aligned} \end{equation}

Moreover, ${\mathbf {x}}(t)$ and ${\bf {\pi }}(t)$ have the following one-to-one correspondence:

(18) \begin{align} x_{B} ( t ) & = \frac{1}{V ( t ) \sigma _{B} ( {\tau_1} ) } \left[ Z(t)\left (\pi _{B} ( t ) \sigma _{B} ( {\tau_1} ) - \left ( \lambda _{R} \sqrt{R ( t ) } - \frac{\lambda _{P} \sigma _{I1} ( \tau_2 ) }{\sigma _{I2} } \right ) \right )\right.\nonumber\\ & \quad \left.-\left ( \frac{u_{1} ( t ) }{H ( t ) } - \frac{v_{1} ( t ) }{H ( t ) } \right ) + \left ( \frac{u_{2} ( t ) \sigma _{I1} ( \tau_2 ) }{H ( t ) \sigma _{I2} } - \frac{v_{2} ( t ) \sigma _{I1} ( \tau_2 ) }{H ( t ) \sigma _{I2} } \right ) \right] , \end{align}

(19) \begin{align} {x_I}(t)& = \frac{1}{V ( t ) {\sigma _{I2}}} {\left[ {Z(t)( {{\pi _I}(t){\sigma _{I2}} - {\lambda _P}} ) - \frac{{{u_2}(t)}}{{H(t)}} + \frac{{{v_2}(t)}}{{H(t)}}} \right]} , \end{align}

(20) \begin{align} {x_S}(t) & = \frac{1}{V ( t ) {\sigma _{S}}}{\left[ {Z(t)( {{\pi _S}(t){\sigma _S} - \widehat {{\lambda _S}}(t)} ) - \frac{{{u_3}(t)}}{{H(t)}} + \frac{{{v_3}(t)}}{{H(t)}}} \right]}, \end{align}

where the set of admissible strategies $\widetilde {\mathcal {A}}$ satisfies: ${\mathbf {\pi }} \in \widetilde {\mathcal {A}}$ if and only if the original strategy ${\mathbf {x}}$ associated with ${\bf {\pi }}$ according to (18)–(20) is admissable, i.e., ${\mathbf {x}} \in {\mathcal {A}}$.

Proof. See Appendix A.

4. Solution to the problem

In this section, we first derive the explicit optimal investment strategy of the auxiliary problem (17). Then, following the relationship between ${\mathbf {x}}$ and ${\bf {\pi }}$ in (18)–(20), we derive the optimal strategy for the original optimization problem (10).

Before solving problem (17), three components of the convex risk measure in Lemma 1 should be specified: the family of probability measures ${{\mathcal {M}}_a}$, the penalty function $\eta$ and the loss variable $L$.

The family of probability measures ${{\mathcal {M}}_a}$ is generated via the Girsanov's transformation. Let ${\bf {\theta }} = {\{ {{\bf {\theta }}(t)} \}_{t \ge 0}}: = {\{ {{{( {{\theta _1}(t),{\theta _2}(t),{\theta _3}(t)} )}^{{\rm T}}}} \}_{t \ge 0}} \in {\mathbb {R}^3}$ be a $\mathbb {G}$-progressively measurable process. When ${\bf {\theta }}$ satisfies

$${\mathbb{P}}\left( {\int_0^T {{{\| {{\bf\theta }(t)} \|}^2}\,{{\rm d}}t < \infty } } \right) = 1,$$

the $\mathbb {G}$-adapted process ${\Lambda ^{\bf {\theta }}}: = {\{ {{\Lambda ^{\bf {\theta }}}(t)} \}_{t \ge 0}}$ defined by

$${\Lambda ^{\bf{\theta }}}(t): = 1 + \int_0^t {{\Lambda ^{\bf{\theta }}}( u )} {{\bf{\theta }}^{{\rm T}}}( u )\,{{\rm d}}\widetilde {\mathbf{W}}( u )$$

is a local martingale. Furthermore, if ${\mathbb {E}}[ {{\Lambda ^{\bf {\theta }}}( T )} ] = 1$, then ${\Lambda ^{\bf {\theta }}}$ is a martingale. Denote the set of all $\mathbb {G}$-progressively measurable processes ${\bf {\theta }}$ satisfying ${\mathbb {E}}[ {{\Lambda ^{\mathbf {\theta }}}( T )}] = 1$ by ${\mathbf {\Theta }}$. For each ${\bf {\theta }} \in {\mathbf {\Theta }}$, define a $\mathbb {G}$-adapted probability measure ${{\mathbb {P}}^{\bf {\theta }}}$ which is absolutely continuous with respect to $\mathbb {P}$:

$${\left. {\frac{{{{\rm d}}{{\mathbb{P}}^{\bf{\theta }}}}}{{{{\rm d}}{\mathbb{P}}}}} \right|_{{{\mathcal{G}}_T}}}: = {\Lambda ^{\bf{\theta }}}( T ).$$

Then ${{\mathcal {M}}_a}$ is specified as ${{\mathcal {M}}_a} = {{\mathcal {M}}_a}( {\mathbf {\Theta }} ): = {\{ {{{\mathbb {P}}^{\bf {\theta }}}} \}_{{\bf {\theta }} \in {\mathbf {\Theta }}}}$.

For each $( {{\bf {\pi }},{\bf {\theta }}} ) \in \widetilde {\mathcal {A}} \times {\mathbf {\Theta }}$, let ${{\mathbf {Y}}^{{\bf {\pi }},{\bf {\theta }}}}: = {\{ {{{\mathbf {Y}}^{{\bf {\pi }},{\bf {\theta }}}}(t)} \}_{t \ge 0}}$ be the controlled state process, where ${{\mathbf {Y}}^{{\bf {\pi }},{\bf {\theta }}}}(t) = {( {Y_1^{\bf {\theta }}(t),Y_2^{\bf {\pi }}(t)} )^{{\rm T}}}: = {( {{\Lambda ^{\bf {\theta }}}(t),{Z^{\bf {\pi }}}(t)} )^{{\rm T}}} \in {{\mathbb {R}}^2}$ satisfying

(21) \begin{equation} \begin{aligned} {{\rm d}}Y_1^{\bf{\theta }}(t) & = Y_1^{\bf{\theta }}(t){{\bf{\theta }}^{{\rm T}}}(t)\,{{\rm d}}\widetilde {\mathbf{W}}(t),\quad Y_1^{\bf{\theta }}( 0 ) = {y_1} = 1,\\ {{\rm d}}Y_2^{\bf{\pi }}(t) & = Y_2^{\bf{\pi }}(t)( {R(t) + {{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma }}{\bf{\widehat \Lambda}} (t)} )\,{{\rm d}}t + {{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma}} \,{\rm d}\widetilde {\mathbf{W}}(t),\quad Y_2^{\bf{\pi }}( 0 ) = {y_2} = Z( 0 ). \end{aligned} \end{equation}

To simplify the notation, we use ${\mathbf {Y}}(t)$, ${Y_1}(t)$ and ${Y_2}(t)$ to represent ${{\mathbf {Y}}^{{\bf {\pi }},{\bf {\theta }}}}(t)$, $Y_1^{\bf {\theta }}(t)$ and $Y_2^{\bf {\pi }} (t)$, respectively.

Next, we specify the penalty function $\eta$. As is proved in Delbaen et al. [Reference Delbaen, Peng and Rosazza Gianin8,Reference Delbaen, Peng and Rosazza Gianin9], the penalty term of a time-consistent dynamic convex risk measure has an integral representation. Let ${\mathcal {C}}( {[ {0,T} ] ; {{\mathbb {R}}^2}} )$ be the space of ${\mathbb {R}}^2$-valued continuous functions on $[ {0,T} ]$. Suppose that $\lambda ( { \cdot, \cdot, \cdot, \cdot } ):[ {0,T} ] \times {\mathcal {C}}( {[ {0,T} ] ; {{\mathbb {R}}^2}} ) \times {\mathbb {R}^3} \times {\mathbb {R}^3} \to {\mathbb {R}}$ and $h( \cdot ):{{\mathbb {R}}^2} \to {\mathbb {R}}$ are two measurable convex functions. From now on, a strategy $( {{\bf {\pi }},{\bf {\theta }}} )$ is admissible if

1. (1) ${\mathbb {E}}[ {\int _0^T {{{| {\lambda ( {t,{\mathbf {Y}}( \cdot ),{\bf {\pi }}(t),{\bf {\theta }}(t)} )} |}\,^2}{{\rm d}}t + {{| {h( {{\mathbf {Y}}( T )} )} |}^2}} } ] < \infty$;

2. (2) ${\mathbb {E}}[ {\int _0^T {Y_1^2(t)Y_2^2(t){{( {R(t) + {{\bf {\pi }}^{{\rm T}}}(t){\bf {\Sigma }}\widehat {\bf {\Lambda }}(t) + {{\bf {\pi }}^{{\rm T}}}(t){\bf {\Sigma \theta }}(t)} )}^2}\,{{\rm d}}t} } ] < \infty$.

Let $\overline {\mathcal {A}} \times \overline {\mathbf {\Theta }}$ denote the set of all admissible strategies. The penalty function $\eta$ is specified as the following form:

(22) \begin{equation} {\eta} ( {{\bf{\theta }},{\bf{\pi }}} ): = {\mathbb{E}}\left[ {\int_0^T {\lambda ( {t,{\mathbf{Y}}({\cdot} ),{\bf{\pi }}(t),{\bf{\theta }}(t)} )\,{{\rm d}}t + h( {{\mathbf{Y}}( T )} )} } \right],\quad( {{\bf{\pi }},{\bf{\theta }}} ) \in \overline {\mathcal{A}} \times \overline {\mathbf{\Theta }} . \end{equation}

This expression for ${\eta } ( {{\bf {\theta }},{\bf {\pi }}} )$ is a generalization of that derived in Delbaen et al. [Reference Delbaen, Peng and Rosazza Gianin8,Reference Delbaen, Peng and Rosazza Gianin9] for a time-consistent dynamic convex risk measure. It consists of two parts: one part is an integral related to the paths of the state process ${\bf Y}$ and the control processes $\bf {\pi }$ and $\bf {\theta }$, the other part depends on the terminal value of the state process ${\bf Y}$.

Assume that the loss variable $L$ for a pension plan member under the strategy ${\bf {\pi }}$ is $- {Z^{\bf {\pi }}}( T )$. Then the convex risk measure in the auxiliary problem (17) is

$$\rho ( {{Z^{\bf{\pi }}}( T )} ): = \sup_{{\bf{\theta }} \in \overline {\mathbf{\Theta }} } \{ {{{\mathbb{E}}^{{{\mathbb{P}}^{\bf{\theta }}}}}[ { - {Z^{\bf{\pi }}}( T )} ] - \eta ( {{\bf{\theta }},{\bf{\pi }}} )} \}.$$

The optimization goal of the member is to select an investment strategy ${\bf {\pi }} \in \overline {\mathcal {A}}$ to minimize the risk $\rho ( {{Z^{\bf {\pi }}}( T )})$, that is, we need to solve the following optimization problem:

(23) \begin{equation} \Phi ( {Z( 0 )} ): = \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } \rho ( {{Z^{\bf{\pi }}}( T )} ) = \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } \left\{ {\sup_{{\bf{\theta }} \in \overline {\mathbf{\Theta }} } \{ {{{\mathbb{E}}^{{{\mathbb{P}}^{\bf{\theta }}}}}[ { - {Z^{\bf{\pi }}}( T )} ] - \eta ( {{\bf{\theta }},{\bf{\pi }}} )} \}}\right\}. \end{equation}

Substituting (21) and (22) into (23) and by the Bayes’ rule, we have

(24) \begin{equation} \Phi ( {Z( 0 )} ) = \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } \sup_{{\bf{\theta }} \in \overline {\mathbf{\Theta }} } {\mathbb{E}}\left[ { - {Y_1}( T ){Y_2}( T ) - \int_0^T {\lambda ( {t,{\mathbf{Y}}({\cdot} ),{\bf{\pi }}(t),{\bf{\theta }}(t)} )\,{{\rm d}}t - h( {{\mathbf{Y}}( T )} )} } \right] = \Phi ( {\mathbf{y}} ), \end{equation}

where $Y( 0 ) = {\mathbf {y}} = ( {{y_1},{y_2}} ) = ( {1,Z( 0 )})$. For $( {{\bf {\pi }},{\bf {\theta }}} ) \in \overline {\mathcal {A}} \times \overline {\mathbf {\Theta }}$, write the value function of problem (17) as

(25) \begin{equation} {J^{{\bf{\pi }},{\bf{\theta }}}}( {\mathbf{y}} ): = {\mathbb{E}}\left[ { - {Y_1}( T ){Y_2}( T ) - \int_0^T {\lambda ( {t,{\mathbf{Y}}({\cdot} ),{\bf{\pi }}(t),{\bf{\theta }}(t)})\,{{\rm d}}t - h( {{\mathbf{Y}}( T )} )} } \right]. \end{equation}

Then problem (17) is a zero-sum stochastic differential game between the plan member and the market:

(26) \begin{equation} \Phi ( {Z( 0 )} ) = \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } \sup_{{\bf{\theta }} \in \overline {\mathbf{\Theta }} } {J^{{\bf{\pi }},{\bf{\theta }}}}( {\mathbf{y}} ) = {J^{{{\bf{\pi }}^*},{{\bf{\theta }}^*}}}( {\mathbf{y}} ). \end{equation}

Here, $( {{{\bf \pi }^*},{{\bf {\theta }}^*}} )$ is called the saddle point of the game problem. In this two-party game between the plan member and the market, the member chooses a trading strategy ${\bf {\pi } }$ to minimize the risk, while the market selects a probability measure indexed by ${\bf {\theta }}$ corresponding to the worst-case scenario in which the risk is maximized to counter the member's behavior.

The BSDE approach is applied to solve the game problem (26) with complete information. Before solving the problem, we define the following notations:

1. (1) ${{\mathcal {L}}^2}( {{{\mathcal {G}}_T}} )$: The space of square integrable and ${{\mathcal {G}}_T}$-measurable random variables;

2. (2) ${\mathcal {S}}_n^2( {[0,T]} )$: the space of $\mathbb {G}$-progressively measurable processes $\phi :\Omega \times [ {0,T} ] \to {{\mathbb {R}}^n}$ such that for each $\omega \in \Omega$, $\phi ( {\omega,t} )$ is continuous in $t$ and ${\mathbb {E}}[ {\sup _{t \in [ {0,T} ]} {{\| {\phi (t)} \|}^2}}] < \infty$;

3. (3) ${\mathcal {H}}_n^2( {[0,T]} )$: the space of $\mathbb {G}$-progressively measurable processes $\phi :\Omega \times [ {0,T} ] \to {{\mathbb {R}}^n}$ satisfying ${\mathbb {E}}[ {\int _0^T {{{\| {\phi (t)} \|}^2}\,{{\rm d}}t} } ] < \infty$.

For the controlled state process ${\bf Y}$, by the Itô formula,

(27) \begin{align} {Y_1}( T ){Y_2}( T ) & = {y_1}{y_2} + \int_0^T {{Y_1}(t){Y_2}(t)} ( {R(t) + {{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma }}\widehat {\bf{\Lambda }}(t) + {{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma \theta }}(t)} )\,{{\rm d}}t\nonumber\\ & \quad + \int_0^T {{Y_1}(t)} {Y_2}(t)( {{{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma }} + {{\bf{\theta }}^{{\rm T}}}(t)} )\,{{\rm d}}\widetilde {\mathbf{W}}(t). \end{align}

For each $( {t,Y( \cdot ),{\bf {\pi }}(t),{\bf {\theta }}(t)} ) \in [ {0,T} ] \times {\mathcal {C}}( {[ {0,T} ];{{\mathbb {R}}^2}} ) \times {{\mathbb {R}}^3} \times {{\mathbb {R}}^3}$, let

(28) \begin{align} \widetilde \lambda ( {t,{\mathbf{Y}}({\cdot} ),{\bf{\pi }}(t),{\bf{\theta }}(t)} )& : = {Y_1}(t){Y_2}(t)( {R(t) + {{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma }}\widehat {\bf{\Lambda }}(t) + {{\bf{\pi }}^{{\rm T}}}(t){\bf{\Sigma \theta }}(t)} )\nonumber\\ & \quad + \lambda ( {t,{\mathbf{Y}}({\cdot} ),{\bf{\pi }}(t),{\bf{\theta }}(t)}). \end{align}

Due to $( {{\bf {\pi }},{\bf {\theta }}} ) \in \overline {\mathcal {A}} \times \overline {\bf {\Theta }}$, we have ${\mathbb {E}}[ {\int _0^T {{{| {\widetilde \lambda ( {t,{\mathbf {Y}}( \cdot ),{\bf {\pi }}(t),{\bf {\theta }}(t)} )}|}^2}\,{{\rm d}}t} } ] < \infty$. Then value function (25) can be rewritten as

$${J^{{\bf{\pi ,\theta }}}}( {\mathbf{y}} ): ={-} {y_1}{y_2} + {\mathbb{E}}\left[ { - \int_0^T {\widetilde \lambda ( {t,{\mathbf{Y}}({\cdot} ),{\bf{\pi }}(t),{\bf{\theta }}(t)} )\,{{\rm d}}t - h( {{\mathbf{Y}}( T )} )} } \right].$$

Denote

(29) \begin{equation} {\widetilde J^{{\bf{\pi ,\theta }}}}( {\mathbf{y}} ): = {\mathbb{E}}\left[ { - \int_0^T {\widetilde \lambda ( {t,{\mathbf{Y}}({\cdot} ),{\mathbf{\pi }}(t),{\bf{\theta }}(t)} )\,{{\rm d}}t - h( {{\mathbf{Y}}( T )} )} } \right]. \end{equation}

Then game problem (26) is equivalent to the following problem:

(30) \begin{equation} \widetilde \Phi ( {\mathbf{y}} ): = {\widetilde J^{{{\bf{\pi }}^{\mathbf{*}}}{\mathbf{,}}{{\bf{\theta }}^*}}}( {\mathbf{y}} ) = \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } \sup_{{\bf{\theta }} \in \overline {\bf{\Theta }} } {\widetilde J^{{\bf{\pi ,\theta }}}}( {\mathbf{y}} ). \end{equation}

The Hamiltonian $H:[0,T] \times {\mathcal {C}}([0,T];{{\mathbb {R}}^2}) \times {{\mathbb {R}}^3} \times {\mathbb {R}^3} \times {\mathbb {R}^3} \to {\mathbb {R}}$ of problem (30) is defined as

(31) \begin{equation} H( {t,{\mathbf{Y}}({\cdot} ),{{\mathbf{U}}_2}(t),{\bf{\pi }}(t),{\bf{\theta }}(t)} ): ={-} \widetilde \lambda ( {t,{\mathbf{Y}}({\cdot} ),{\bf{\pi }}(t),{\bf{\theta }}(t)} ). \end{equation}

To ensure the existence of the value function in the stochastic differential game, we suppose that the Hamiltonian satisfies the Isaacs’ condition, i.e.

$$\inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } \sup_{{\bf{\theta }} \in \overline {\bf{\Theta }} } H( {t,{\mathbf{Y}}({\cdot} ),{{\mathbf{U}}_2}(t),{\bf{\pi }}(t),{\bf{\theta }}(t)} ) = \sup_{{\bf{\theta }} \in \overline {\mathbf{\Theta }} } \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } H( {t,{\mathbf{Y}}({\cdot} ),{{\mathbf{U}}_2}(t),{\bf{\pi }}(t),{\bf{\theta }}(t)}).$$

The Isaacs’ condition is satisfied if $\widetilde \lambda$ is concave in ${\bf {\pi }}$ and convex in ${\bf {\theta }}$. Define

$$H( {t,{\mathbf{Y}}({\cdot} ),{{\mathbf{U}}_2}(t),{{\bf{\pi }}^*}(t),{{\bf{\theta }}^*}(t)} ): = \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } \sup_{{\bf{\theta }} \in \overline {\mathbf{\Theta }} } H( {t,{\mathbf{Y}}({\cdot} ),{{\mathbf{U}}_2}(t),{\bf{\pi }}(t),{\bf{\theta }}(t)}),$$

which will be regarded as the driver of the BSDE corresponding to the value function of game problem (30). The following theorem establishes the relation between the value function of the differential game problem (30) and the solution of this BSDE.

Theorem 1. When the Isaacs’ condition holds, the following BSDE

(32) \begin{equation} \begin{aligned} - {{\rm d}}{U_1}(t) & = H( {t,{\mathbf{Y}}({\cdot} ),{{\mathbf{U}}_2}(t),{{\bf{\pi }}^*}(t),{{\bf{\theta }}^*}(t)} )\,{{\rm d}}t - {\mathbf{U}}_2^{{\rm T}}(t)\,{{\rm d}}\widetilde {\mathbf{W}}(t),\\ {U_1}( T ) & ={-} h( {{\mathbf{Y}}( T )} ), \end{aligned} \end{equation}

admits a unique solution ${\{ {( {{U_1}(t),{{\mathbf {U}}_2}(t)} )} \}_{t \in [ {0,T} ]}} \in {\mathcal {S}}_1^2( {[0,T]} ) \otimes {\mathcal {H}}_3^2( {[0,T]} )$. Furthermore, ${\{ {( {{{\bf {\pi }}^{\mathbf {*}}}(t),{{\bf {\theta }}^{\mathbf {*}}}(t)} )} \}_{t \in [ {0,T} ]}}$ is the saddle point of game problem (30) and the associated value function is

$${U_1}( 0 ) = {\widetilde J^{{{\bf{\pi }}^*}{\mathbf{,}}{{\bf{\theta }}^*}}}( {{{\mathbf{Y}}_0}} ) = \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } \sup_{{\bf{\theta }} \in \overline {\mathbf{\Theta }} } {\widetilde J^{{\bf{\pi ,\theta }}}}( {{{\mathbf{Y}}_0}} ) = \sup_{{\bf{\theta }} \in \overline {\mathbf{\Theta }} } \inf_{{\bf{\pi }} \in \overline {\mathcal{A}} } {\widetilde J^{{\bf{\pi ,\theta }}}}( {{{\mathbf{Y}}_0}} ).$$

Inspired by the entropy penalty function proposed by Delbean et al. [Reference Delbaen, Peng and Rosazza Gianin8], we assume that

(33) \begin{equation} \lambda ( {t,{\mathbf{Y}}({\cdot} ),{\bf{\pi }}(t),{\bf{\theta }}(t)} ) = \frac{1}{{2( {1 - \gamma } )}}{Y_1}(t){Y_2}(t){\| {{\bf{\theta }}(t)}\|^2}. \end{equation}

Here, $1 - \gamma$ stands for the relative risk aversion of the member, and $\gamma < 1$. The corresponding $\eta ( {{\bf {\theta }},{\bf {\pi }}})$ is called the penalty function with quadratic form. From (21), for any $t \in [ {0,T} ]$, ${Y_1}(t) > 0$, ${Y_2}(t) > 0$, ${\mathbb {P}}$ - a.s. Then $\lambda ( {t,{\mathbf {Y}}( \cdot ),{\bf {\pi }}(t),{\bf {\theta }}(t)})$ is a convex function of ${\bf {\theta }}$. Moreover, $\widetilde \lambda$ defined by (28) is a concave function about $\bf {\pi }$ and a convex function about ${\bf {\theta }}$. Therefore, the Hamiltonian defined by (31) satisfies the Isaacs’ condition.

With this particular penalty function given by (33), we can derive the saddle point of game problem (30) and the optimal investment strategy explicitly.

Theorem 2. The saddle point $( {{{\bf \pi }^*}(t),{{\bf {\theta }}^*}}(t))$ of game problem (30) is given by

(34) \begin{align} {{\bf{\theta }}^*}(t) & = {( {\theta _1^*(t),\theta _2^*(t),\theta _3^*(t)})^{{\rm T}}}\nonumber\\ & = {( { - {\lambda _R}\sqrt {R(t)} , - {\lambda _P}, - \widehat {{\lambda_S}}(t)} )^{{\rm T}}}\nonumber\\ & ={-} \widehat {\mathbf{\Lambda }}{(t)^{{\rm T}}},\end{align}

(35) \begin{align}{{\bf{\pi }}^*}(t) & = {( {\pi _B^*(t),\pi _I^*(t),\pi _S^*(t)} )^{{\rm T}}}\nonumber\\ & = \left( \begin{array}{c} \dfrac{1}{{( {1 - \gamma } ){\sigma _R}{h_1}( {{\tau _1}} )}}\left( {{\lambda _R} - \dfrac{{{\lambda _P}( {{\sigma _{P1}} + {\sigma _R}{q_1}( {{\tau _2}} )} )}}{{{\sigma _{I2}}}}} \right)\\ \dfrac{{{\lambda _P}}}{{( {1 - \gamma } ){\sigma _{I2}}}}\\ \dfrac{{\widehat {{\lambda _S}}(t)}}{{( {1 - \gamma } ){\sigma _S}}} \end{array} \right). \end{align}

The optimal investment strategy of the pension member is

(36) \begin{align} x_B^*(t) & = \frac{1}{{V^* (t){\sigma _R}{h_1}( {{\tau _1}} )}}\left[ {Z(t)\left( {{\lambda _R} - \frac{{{\lambda _P}( {{\sigma _{P1}} + {\sigma _R}{q_1}( {{\tau _2}} )})}}{{{\sigma _{I2}}}}} \right)\frac{\gamma }{{1 - \gamma }}} \right]\nonumber\\ & \quad- \frac{1}{{V^* (t){\sigma _B}( {{\tau _1}} )}}\left[ {\frac{{{u_1}(t)}}{{H(t)}} - \frac{{{v_1}(t)}}{{H(t)}} - \frac{{{u_2}(t){\sigma _{I1}}( {{\tau _2}} )}}{{H(t){\sigma _{I2}}}} + \frac{{{v_2}(t){\sigma _{I1}}( {{\tau _2}} )}}{{H(t){\sigma _{I2}}}}} \right], \end{align}

(37) \begin{align} x_I^*(t) & = \frac{1}{{V^* (t){\sigma _{I2}}}}\left[ {Z(t){\lambda _P}\frac{\gamma }{{1 - \gamma }} - \frac{{{u_2}(t)}}{{H(t)}} + \frac{{{v_2}(t)}}{{H(t)}}} \right], \end{align}

(38) \begin{align} x_S^*(t)& = \frac{1}{{V^* (t){\sigma _S}}}\left[ {Z(t)\widehat {{\lambda _S}}(t)\frac{\gamma }{{1 - \gamma }} - \frac{{{u_3}(t)}}{{H(t)}} + \frac{{{v_3}(t)}}{{H(t)}}} \right], \end{align}

where $V^*(t)$ represents the optimal wealth level at time $t$.

In what follows, we specify the explicit forms of $C (t)$ and $G ( T )$ in order to derive concrete expressions for ${\bf u} (t)$ and ${\bf v} (t)$ in Proposition 1 and Theorem 2.

In practice, the contribution of a plan member is generally a preset proportion of the salary, which may be influenced by the interest rate and inflation to a certain extent. Therefore, it is assumed that the instantaneous contribution rate $C(t)$ follows the following stochastic differential equation:

(39) \begin{equation} \begin{aligned} \dfrac{{{{\rm d}}C(t)}}{{C(t)}} & = {\mu _C}\,{{\rm d}}t + {\sigma _{C1}}\sqrt {R(t)} \,{{\rm d}}{W_1}(t) + {\sigma _{C2}}\,{{\rm d}}{W_2}(t),\\ C( 0 ) & = {C_0}, \end{aligned} \end{equation}

where ${\mu _C} > 0$ is the expected growth rate of the instantaneous contribution rate, and ${\sigma _{C1}}$, ${\sigma _{C2}}$ are volatility parameters. We construct a fictitious derivative to replicate the continuous contribution. The payoff of the fictitious bond at maturity $s$ is $C( s )$, and its price at time $t$ is denoted as $D( {t,s} )$, $t \le s$. By the no-arbitrage pricing theory, we have

(40) \begin{align} D( {t,s} ) & = \frac{1}{{H(t)}}{\mathbb{E}}[ {{C( s )H( s )} |{{\mathcal{G}}_t}}]\nonumber\\ & = C(t)\exp {{f_0}( {t,s} ) - {f_1}( {t,s} )R(t)} ], \end{align}

where

\begin{align*} {f_1}( {t,s} ) & = \frac{{2( {1 + {\lambda _R}{\sigma _{C1}}} )( {{{{\rm e}}^{2j( {s - t} )}} - 1} )}}{{( {i + 2j} )( {{{{\rm e}}^{2j( {s - t} )}} - 1} ) + 4j}},\\ {f_0}( {t,s} ) & ={-} \frac{a}{{\sigma _R^2}}\left[ {2\log \left( {\frac{{( {i + 2j} )( {{{{\rm e}}^{2j( {s - t} )}} - 1} ) + 4j}}{{4j}}} \right) - ( {i + 2j} )( {s - t} )} \right]\\ & \quad + {\mu _C}( {s - t} ) - {\sigma_{C2}}{\lambda _P}( {s - t} ),\\ i& = b - {\sigma _R}{\lambda _R} + {\sigma _R}{\sigma _{C1}},\\ j & = \frac{1}{2}\sqrt {{{( {b - {\sigma _R}{\lambda _R} + {\sigma _R}{\sigma _{C1}}} )}^2} + 2\sigma _R^2( {1 + {\lambda _R}{\sigma _{C1}}})} . \end{align*}

$D( {t,s} )$ satisfies the following equation:

(41) \begin{equation} \begin{aligned} \frac{{{{\rm d}}D( {t,s} )}}{{D( {t,s} )}} & = ( {R(t) + {\lambda _R}\sqrt {R(t)} {\sigma _{D1}}( {t,s} ) + {\lambda _P}{\sigma _{D2}} } )\,{{\rm d}}t\\ & \quad+ {\sigma _{D1}}( {t,s} )\,{{\rm d}}{W_1}(t) + {\sigma _{D2}}\,{{\rm d}}{W_2}(t),\\ D( {s,s} ) & = C( s ), \end{aligned} \end{equation}

with ${\sigma _{D1}}( {t,s} ) = ( {{\sigma _{C1}} + {f_1}( {t,s} ){\sigma _R}})\sqrt {R(t)}$ and ${\sigma _{D2}} = {\sigma _{C2}}$.

The discounted contribution process $F(t)$ can be expressed as an integral of $D( {t,s} )$. In fact, by the conditional Fubini's theorem,

(42) \begin{align} F(t)& := \frac{1}{{H(t)}}{\mathbb{E}}\left[ {\left. {\int_t^T {C( s )H( s )\,{{\rm ds}}} } \,\right|{{\mathcal{G}}_t}} \right]\nonumber\\ & = \int_t^T {\frac{1}{{H(t)}}{\mathbb{E}} {{C( s )H( s )} \,|\,{{\mathcal{G}}_t}} ]} \,{{\rm d}}s\nonumber\\ & = \int_t^T {D( {t,s} )} \,{{\rm d}}s. \end{align}

Based on (42), we can derive the specific expression of ${\mathbf {u}}(t)$ in Proposition 1.

Proposition 2. The $\mathbb {G}$-predictable process ${\mathbf {u}}(t) = {( {{u_1}(t),{u_2}(t),{u_3}(t)} )^{{\rm T}}}$ in Proposition 1 is

(43) \begin{align} {u_1}(t) & = H(t)F(t)\left( {\int_t^T {\frac{{D( {t,s} ){\sigma _{D1}}( {t,s} )}}{{F(t)}}\,{{\rm d}}s - {\lambda _R}\sqrt {R(t)} } } \right), \end{align}

(44) \begin{align} {u_2}(t) & = H(t)F(t)( {{\sigma _{D2}} - {\lambda _P}} ), \end{align}

(45) \begin{align} {u_3}(t) & = H(t)F(t)( { - \widehat {{\lambda _S}}(t)} ). \end{align}

Now we turn to specify the explicit form of the minimum guarantee $G( T )$. Referring to Chen et al. [Reference Chen, Li, Zeng and Sun5], we define a $\mathbb {G}$-adapted process $L(t)$, which is used to describe the basic standard of living of the member at time $t$ and is governed by

(46) \begin{equation} \begin{aligned} \frac{{{{\rm d}}L(t)}}{{L(t)}} & = {\mu _L}\,{{\rm d}}t + {\sigma _{L1}}\sqrt {R(t)} \,{{\rm d}}{W_1}(t)+ {\sigma _{L2}}\,{{\rm d}}{W_2}(t),\\ L( 0 ) & = {L_0}, \end{aligned} \end{equation}

where ${\mu _L} > 0$ is the expected growth rate, and ${\sigma _{L1}}$ and ${\sigma _{L2}}$ are the volatility parameters. The minimum guarantee at time $T$ is defined as

(47) \begin{equation} G( T ):= {\mathbb{E}}\left[ {\left. {\int_T^{T'} {L( u )\frac{{H( u )}}{{H( T )}}\,{{\rm d}}u} } \,\right|{{\mathcal{G}}_T}} \right], \end{equation}

where $T'$ is the death moment of the plan member, which should be uncertain in real life. However, in our model, $T'$ is assumed to be a deterministic constant in order to facilitate the analysis of the impact of the life span of the member on the investment strategy. We only consider the case where the member dies after retirement, i.e. $T' > T$. When the member dies before retirement ($T \le T'$), the minimum guarantee is no longer needed in the absence of bequest motivation, and the model degenerates to the case without the minimum guarantee constraint. $G(t)$ represents the present value of the member's basic requirement of living after retirement. Similar to the method for the replication of the contribution, we construct a fictitious bond with a payoff of $L( s )$ at maturing time $s$ whose price at time $t$ is $K( {t,s} )$. Then we have

(48) \begin{align} K( {t,s} ) & = \frac{1}{{H(t)}}{\mathbb{E}}[ {{L( s )H( s )} \,|\,{{\mathcal{G}}_t}}] \nonumber\\ & = L(t)\exp [ {{m_0}( {t,s} ) - {m_1}( {t,s} )R(t)} ], \end{align}

where

\begin{align*} {m_1}( {t,s} ) & = \frac{{2( {1 + {\lambda _R}{\sigma _{L1}}} )( {{{{\rm e}}^{2y( {s - t} )}} - 1} )}}{{( {x + 2y} )( {{{{\rm e}}^{2y( {s - t} )}} - 1} ) +4y}},\\ {m_0}( {t,s} ) & ={-} \frac{a}{{\sigma _R^2}}\left[ {2\log \left( {\frac{{( {x + 2y} )( {{{{\rm e}}^{2y( {s - t} )}} - 1} ) + 4y}}{{4y}}} \right) - ( {x + 2y} )( {s - t} )} \right]\\ & \quad +{\mu _L}( {s - t} ) - {\sigma _{L2}}{\lambda _P}( {s - t} ),\\ x & = b - {\sigma _R}{\lambda _R} + {\sigma _R}{\sigma _{L1}},\\ y & = \frac{1}{2}\sqrt {{{( {b - {\sigma _R}{\lambda _R} + {\sigma _R}{\sigma _{L1}}} )}^2} + 2\sigma _R^2( {1 + {\lambda _R}{\sigma _{L1}}} )} . \end{align*}

$K( {t,s} )$ also satisfies the following stochastic differential equation:

(49) \begin{equation} \begin{aligned} \frac{{{{\rm d}}K( {t,s} )}}{{K( {t,s} )}} & = ( {R(t) + {\lambda _R}\sqrt {R(t)} {\sigma _{K1}}( {t,s} ) + {\lambda _P}{\sigma _{K2}} } )\,{{\rm d}}t\\ & \quad+ {\sigma _{K1}}( {t,s} )\,{{\rm d}}{W_1}(t) + {\sigma _{K2}}\,{{\rm d}}{W_2}(t),\\ K( {s,s} ) & = L( s ), \end{aligned} \end{equation}

where ${\sigma _{K1}}( {t,s} ) = ( {{\sigma _{L1}} + {m_1}( {t,s} ){\sigma _R}} )\sqrt {R(t)}$ and ${\sigma _{K2}} = {\sigma _{L2}}$.

Similar to (42), $G(t)$ can be expressed as an integral of $K( {t,s} )$. In fact, by the conditional Fubini's theorem and the tower property of the conditional expectation, we have

(50) \begin{align} G(t)& := \dfrac{1}{{H(t)}}{\mathbb{E}}[ { {G( T )H( T )} \,|\,{{\mathcal{G}}_t}}]\nonumber\\ & = \dfrac{1}{{H(t)}}{\mathbb{E}}\left[ {\left. {{\mathbb{E}}\left[ {\left. {\int_T^{T'} {L( s )\frac{{H( s )}}{{H( T )}}\,{{\rm d}}s} } \,\right|{{\mathcal{G}}_T}} \right]H( T )} \,\right|{{\mathcal{G}}_t}} \right]\nonumber\\ & = {\mathbb{E}}\left[ {\left. {\int_T^{T'} {L( s )\frac{{H( s )}}{{H(t)}}{{\rm d}}s} } \,\right|{{\mathcal{G}}_t}} \right]\nonumber\\ & = \int_T^{T'} {\frac{1}{{H(t)}}{\mathbb{E}}[{{L( s )H( s )} \,|\,{{\mathcal{G}}_t}} ]\,{{\rm d}}s} \nonumber\\ & = \int_T^{T'} {K( {t,s})\,{{\rm d}}s} . \end{align}

By (50), we can obtain the concrete expression of ${\mathbf {v}}(t)$ in Proposition 1.

Proposition 3. The $\mathbb {G}$-predictable process ${\mathbf {v}}(t) = {( {{v_1}(t),{v_2}(t),{v_3}(t)} )^{{\rm T}}}$ in Proposition 1 is

(51) \begin{align} {v_1}(t)& = G(t)H(t)\left( {\int_T^{T'} {\frac{{K( {t,s} ){\sigma _{K1}}( {t,s} )}}{{G(t)}}\,{{\rm d}}s - {\lambda _R}\sqrt {R(t)} } } \right), \end{align}

(52) \begin{align} {v_2}(t)& = G(t)H(t)( {{\sigma _{K2}} - {\lambda _P}}), \end{align}

(53) \begin{align} {v_3}(t) & = G(t)H(t)( { - \widehat {{\lambda _S}}(t)}). \end{align}

Finally, we give the specific expression of the optimal investment strategy of problem (10).

Theorem 3. When $C(t)$ and $G( T )$ satisfy (39) and (47) respectively, the optimal investment strategy ${{\mathbf {x}}^*}(t) = {( {x_B^*(t),x_I^*(t),x_S^*(t)} )^{{\rm T}}}$ of problem (10) is

(54) \begin{align} x_B^*(t) & = x_{B - {{\rm spec}}}^*(t) + x_{B - F}^*(t) - x_{B - G}^*(t)\nonumber\\ & = \underbrace{\frac{\gamma }{{1 - \gamma }}\left( {{\lambda _R} - \frac{{{\lambda _P}( {{\sigma _{P1}} + {\sigma _R}{q_1}( {{\tau _2}} )} )}}{{{\sigma _{I2}}}}} \right)\frac{1}{{{\sigma _R}{h_1}( {{\tau _1}} )}}}_{x_{B - {{\rm spec}}}^*(t)}\nonumber\\ & \quad+\underbrace {\frac{{F(t)}}{{{V^*}(t)}}\left[ {\frac{1}{{1 - \gamma }}\left( {{\lambda _R}\sqrt {R(t)} - \frac{{{\lambda _P}{\sigma _{I1}}( {{\tau _2}} )}}{{{\sigma _{I2}}}}} \right) - \int_t^T {\frac{{D( {t,s} ){\sigma _{D1}}( {t,s} )}}{{F(t)}}\,{{\rm d}}s} + \frac{{{\sigma _{I1}}( {{\tau _2}} ){\sigma _{D2}}}}{{{\sigma _{I2}}}}} \right]\frac{1}{{{\sigma _B}( {{\tau _1}} )}}}_{x_{B - F}^*(t)}\nonumber\\ & \quad- \underbrace{\frac{{G(t)}}{{{V^*}(t)}}\left[ {\frac{1}{{1 - \gamma }}\left( {{\lambda _R}\sqrt {R(t)} - \frac{{{\lambda _P}{\sigma _{I1}}( {{\tau _2}} )}}{{{\sigma _{I2}}}}} \right) - \int_T^{T'} {\frac{{K( {t,s} ){\sigma _{K1}}( {t,s} )}}{{G(t)}}\,{{\rm d}}s} + \frac{{{\sigma _{I1}}( {{\tau _2}} ){\sigma _{K2}}}}{{{\sigma _{I2}}}}} \right]\frac{1}{{{\sigma _B}( {{\tau _1}} )}}}_{x_{B - G}^*(t)}, \end{align}

(55) \begin{align} x_I^*(t) & = x_{I - {{\rm spec}}}^*(t) + x_{I - F}^*(t) - x_{I - G}^*(t)\nonumber\\ & = \underbrace{\frac{\gamma }{{1 - \gamma }}{\lambda _P}\frac{1}{{{\sigma _{I2}}}}}_{x_{I - {{\rm spec}}}^*(t)} + \underbrace {\frac{{F(t)}}{{{V^*}(t)}}\left( {\frac{1}{{1 - \gamma }}{\lambda _P} - {\sigma _{D2}}} \right)\frac{1}{{{\sigma _{I2}}}}}_{x_{I - F}^*(t)}\nonumber\\ & \quad- \underbrace{\frac{{G(t)}}{{{V^*}(t)}}\left( {\frac{1}{{1 - \gamma }}{\lambda _P} - {\sigma _{K2}}} \right)\frac{1}{{{\sigma _{I2}}}}}_{x_{I - G}^*(t)}, \end{align}

(56) \begin{align}x_S^*(t) & = x_{S - {{\rm spec}}}^*(t) + x_{S - F}^*(t) - x_{S - G}^*(t)\nonumber\\ & = \underbrace{\frac{\gamma }{{1 - \gamma }}\widehat {{\lambda _S}}(t)\frac{1}{{{\sigma _S}}}}_{x_{S - {{\rm spec}}}^*(t)} + \underbrace {\frac{{F(t)}}{{{V^*}(t)}}\left( {\frac{1}{{1 - \gamma }}\widehat {{\lambda _S}}(t) } \right)\frac{1}{{{\sigma _S}}}}_{x_{S - F}^*(t)}\nonumber\\ & \quad-\underbrace{\frac{{G(t)}}{{{V^*}(t)}}\left( {\frac{1}{{1 - \gamma }}\widehat {{\lambda _S}}(t) } \right)\frac{1}{{{\sigma _S}}}}_{x_{S - G}^*(t)}. \end{align}

5. Numerical analysis

This section conducts a numerical analysis for the optimal investment strategy derived in Theorem 3, focusing on the impacts of some model parameters on the investment strategy. We mainly adopt the Milstein method to simulate the trajectories of most stochastic processes in our model except for the CIR process and the Wonham filter process. Because the Milstein method has some limitations, the algorithm for stochastic volatility processes proposed by Broadie and Kaya [Reference Broadie and Kaya4] and the Balanced Implicit Method proposed by Milstein et al. [Reference Milstein, Platen and Schurz26] are used to simulate the CIR process and the Wonham filter process, respectively. Without loss of generality, we suppose that $N = 2$, i.e., the market has two states—“bullish market” and “bearish market” corresponding to Regime 1 and Regime 2. For the transition rate $q_{ij}$, we have $- {q_{11}} = {q_{12}} = {q_1}$ and ${q_{21}} = - {q_{22}} = {q_2}$. The basic values for the parameters of the hidden Markov chain are specified as ${p_1}( 0 ) = 0.3$, ${p_2}( 0 ) = 0.7$, ${q_1} = 0.3$ and ${q_2} = 0.6$. In addition, we set ${\mu _1} = 0.15$, ${\mu _2} = 0.07$, reflecting the fact that the return rate of the stock in the bullish market is usually higher than that in the bearish market. The default values for other parameters are shown in Table 1, which are mainly set on the basis of Han and Hung [Reference Han and Hung19] and Deelstra et al. [Reference Deelstra, Grasselli and Koehl7]. The long-term mean of the interest rate is $a/b$, which is about 0.068 and lower than the return rate of the stock. We assume that two rolling bonds have the same maturity date, i.e. $\tau _1 = \tau _2 = \tau$.

Table 1. Default values of model parameters.

Figure 1 depicts the evolution of the optimal investment strategy under the default parameter settings. We find that the optimal wealth proportions invested in the stock and the zero-coupon bond are initially high and decrease over time, while those invested in the inflation-indexed bond and the risk-free asset increase gradually on the whole. In addition, we note that the plan member hedges the inflation risk by short selling the inflation-indexed bond in the beginning years of the investment phase. At the initial time, the proportion of wealth invested in the stock is the largest among all assets, which is about 160%. The proportion of the risk-free asset exceeds that of the stock at about $t = 12$, and ultimately accounts for the largest proportion of the total wealth which is about 51%. This indicates that the member tends to be defensive in investment when time approaches the retirement date. In the early years of the investment phase, the wealth is not enough to reach the minimum guarantee. Therefore, the plan member invests heavily in risky assets to boost the pension's wealth. As the wealth reaches a high level, the aim of risk minimization forces the member to shift the wealth to safe assets. The movements of the optimal portfolio weights are consistent with the empirical criterion.

Figure 1. The evolution of the optimal investment strategy under default parameters.

Figure 2 shows the effects of the minimum guarantee on the optimal investment strategy $( {x_B^*, x_I^*, x_S^*} )$ when $t=0$. As is shown in Figure 2(a), the initial present value of the minimum guarantee $G( 0 )$ increases with the expected growth rate of living standard ${\mu _L}$ and the life span $T'$. Figure 2(b) shows that when $t = 0$, $x_B^*$ and $x_I^*$ are positively correlated with ${\mu _L}$, but $x_S^*$ is opposite. The effect of $T'$ on the investment proportions of three risky assets $( {x_B^*, x_I^*, x_S^*})$ is similar to that of ${\mu _L}$, as shown in Figure 2(c). Combined with Figure 2(a), it can be seen that $G( 0 )$ increases with ${\mu _L}$ and $T'$, which subsequently results in the increase in $x_B^*$ and $x_I^*$ but a decrease in $x_S^*$. A high $G( 0 )$ means a high level of guarantee, which raises concerns about possible losses on the high-risk investment and prompts the member to adopt conservative investment strategies to reduce risk, especially in an aging society. The stock is the riskiest asset of the three, and hence it bears the brunt of the cuts. This result differs from the conclusion obtained by Chen et al. [Reference Chen, Li, Zeng and Sun5] that “a high level of the guarantee constraint leads to the increase of the investment in stock.” Although the form of the minimum guarantee in this paper is similar to that in Chen et al. [Reference Chen, Li, Zeng and Sun5], the latter chooses the maximization of S-shaped utility as the optimization objective and does not consider the interest rate risk.

Figure 2. The effects of ${\mu _L}$ and $T'$ on $G( 0 )$ and the investment strategy when $t = 0$.

Figure 3 demonstrates the effects of the future contribution on the optimal investment strategy $( {x_B^*, x_I^*, x_S^*} )$ when $t=0$. As shown in Figure 3(a), the increase in the expected growth rate of contribution rate ${\mu _C}$ and the retirement time $T$ both lead to the increase in $F( 0 )$. Figure 3(b) shows that ${\mu _C}$ has negative influences on $x_B^*$ and $x_I^*$, but has a positive influence on $x_S^*$. In Figure 3(c), we find that the effects of $T$ on the optimal proportions of risky assets $( {x_B^*, x_I^*, x_S^*} )$ is similar to that of ${\mu _C}$. Combining Remark 2 and Figure 3(a), it can be inferred that $x_B^*$ and $x_I^*$ move downwards with $F( 0 )$, while the opposite is true for $x_S^*$. These results can be explained by the following reason. When the contribution grows, the member becomes more confident in achieving the minimum guarantee at retirement and has stronger tolerance for risk. Therefore, they prefer to put more money in the assets with high-risk premiums in the beginning years of the investment horizon, e.g. the stock. As the result of the substitution effect, $x_B^*$ and $x_I^*$ decrease. Figures 2 and 3 show that $G( 0 )$ and $F( 0 )$ have opposite effects on the optimal investment strategy.

Figure 3. The effects of ${\mu _C}$ and $T$ on $F( 0 )$ and the investment strategy when $t = 0$.

Figure 4 reveals that the impacts of the risk aversion parameter $\gamma$, the interest rate volatility parameter ${\sigma _R}$ and the inflation volatility parameter ${\sigma _P}$ on the optimal investment strategy $( {x_B^*, x_I^*, x_S^*} )$ when $t = 0$. As shown in Figure 4(a), $x_I^*$ and $x_S^*$ move up when $\gamma$ decreases. This is because the higher $\gamma$ is, the lower the relative risk aversion level of the member is. Then for the sake of increasing the speed of wealth accumulation, the member tends to increase the investment in risky assets in the initial time. The phenomenon is very evident in the investment of stock. By contrast, $x_B^*$ is less sensitive to the change of $\gamma$, and just decreases slightly when $\gamma$ increases. Figure 4(b) shows that $x_S^*$ remains almost flat as ${\sigma _R}$ changes, so does $x_I^*$. It makes sense that the investment of the zero-coupon bond is most sensitive to the changes in ${\sigma _R}$ among the three risky assets. When ${\sigma _R}$ increases by 29%, $x_B^*$ decreases by about 131%. The larger ${\sigma _R}$ is, the greater the fluctuation of the interest rate is, which hints at the higher risk for the investment in the zero-coupon bond. In order to reduce risk, the DC plan member will reduce the proportion of investment in the zero-coupon bond and, due to the substitution effect, invest more in other assets (mainly the risk-free asset in this paper). As is exhibited in Figure 4(c), when ${\sigma _{P2}}$ moves up, $x_B^*$ moves down but ${\sigma _{P2}}$ increases. However, $x_S^*$ is almost unaffected by ${\sigma _{P2}}$. A higher volatility ${\sigma _{P2}}$ means that the price of the inflation-indexed bond is more unstable and the return is of higher risk. Therefore, the short-selling proportion of the inflation-indexed bond diminishes when ${\sigma _{P2}}$ grows.

Figure 4. The effects of ${\gamma }$, $\sigma _R$ and $\sigma _{P2}$ on the investment strategy when $t = 0$.

As stated in Remark 3, the hidden Markov chain ${\mathbf {X}}$ mainly affects the optimal portfolio weight of the stock. Therefore, we only display how state parameters ${q_1}$, ${q_2}$, ${\mu _1}$ and ${\mu _2}$ affect $x_S^*(t)$ in Figure 5. Comparing the strategies under $( {{q_1},{q_2}} ) = ( {0.3,0.6} )$ and $( {{q_1},{q_2}} ) = ( {0.6,0.6} )$ in Figure 5(a), a smaller ${q_1}$ corresponds to a larger wealth proportion invested in the stock. When ${q_2}$ is fixed, the smaller ${q_1}$ means that the average time of the Markov chain staying in Regime 1 (bullish market) is longer, and its stationary distribution has larger probability to stay in Regime 1 (bullish market). Moreover, the expected return rate of the stock is higher in Regime 1 (bullish market). Similarly, we can give an intuitive explanation on what happens to $x_S^*(t)$ when $( {{q_1},{q_2}} ) = ( {0.3,0.3} )$ and $( {{q_1},{q_2}} ) = ( {0.3,0.6} )$. Figure 5(b) shows that the portfolio weight of stock increases with the stock return rate. When the volatility ${\sigma _S}$ and the interest rate $R(t)$ are fixed, the market price of risk $\widehat {{\lambda _S}}(t)$ increases with $\mu _1$ or $\mu _2$. The above results show that even though the DC pension plan members’ goal is not to maximize the utility of the terminal wealth, when they have good expectations for the stock market, the risky assets still have strong attractions for them.

Figure 5. The effects of ${q_{1}}$, $q_{2}$, $\mu _{1}$ and $\mu _{2}$ on $x^*_S(t)$.

Instead of using the filtering theory based on sample information to estimate the distribution of the hidden Markov chain, we consider the case where the mean value, i.e. $\overline {\mathbf {p}} (t) = {{\rm E}}[ {{\mathbf {X}}(t)} ]$ is used as an estimate for the distribution. Note that $\overline {\mathbf {p}} (t) = {( {{{\overline p }_1}(t),{{\overline p }_2}(t), \ldots,{{\overline p }_N}(t)} )^{{\rm T}}}$ is a deterministic function of $t$ and satisfies the following ordinary differential equation:

(57) \begin{equation} \begin{aligned} {{\rm d}}\overline {\mathbf{p}} (t) & = {\mathbf{Q}}'\overline {\mathbf{p}} (t)\,{{\rm d}}t,\\ {\overline p _k}( 0 ) & = {\mathbb{P}}( {{\mathbf{X}}( 0 ) = {{\mathbf{e}}_k}} ),\quad k = 1, \ldots ,N. \end{aligned} \end{equation}

The solution to (57) is $\overline {\mathbf {p}}(t)=\overline {\mathbf {p}}(0)\exp [{\mathbf {Q}}'t]$. Then the estimates of the market price of risk and the optimal portfolio weight of stock in Theorem 3 are

(58) \begin{equation} \begin{aligned} \overline {{\lambda _S}} (t) & = \frac{{\langle {{\mathbf{\mu }},\overline {\mathbf{p}} (t)} \rangle - R(t)}}{{{\sigma _S}}},\\ \overline {x_S^*} (t) & = \frac{\gamma }{{1 - \gamma }}\overline {{\lambda _S}} (t)\frac{1}{{{\sigma _S}}}\\ & \quad+ \frac{{F(t)}}{{{V^*}(t)}}\left( {\frac{1}{{1 - \gamma }}\overline {{\lambda _S}} (t) } \right)\frac{1}{{{\sigma _S}}}\\ & \quad- \frac{{G(t)}}{{{V^*}(t)}}\left( {\frac{1}{{1 - \gamma }}\overline {{\lambda _S}} (t) } \right)\frac{1}{{{\sigma _S}}}. \end{aligned} \end{equation}

Figure 6(a) plots the paths of the filtered estimate and the mean estimate of the stock return rate $\mu (t)$ when $( {{q_1},{q_2}} ) = ( {0.3,0.6} )$, while the case when $( {{q_1},{q_2}} ) = ( {0.6,0.3} )$ is illustrated in Figure 6(b). Figure 6(c) and (d) show the optimal proportion of wealth invested in the stock when $( {{q_1},{q_2}} ) = ( {0.3,0.6} )$ and $( {{q_1},{q_2}} ) = ( {0.6,0.3} )$, respectively. We can see that in the first half of the investment phase, the proportion of wealth invested in the stock obtained by the filtered estimate is larger but, when retirement approaches, there is no significant difference between the optimal investment strategies obtained by the two estimation methods. Compared with the mean estimate, the filtered estimate enables the DC pension plan member to acquire more information about the market states. In this case, the uncertainty of stock return is less, which spurs more investment in stock in the early stage of the investment phase to promote wealth accumulation. As the distribution of the Markov chain tends to be stationary, the information obtained from the stock price process by using two different estimates tends to be identical, hence the investment strategies adopted on the stock also converge.

Figure 6. The paths of the stock return rate and the effects of $( {q_{1}}, q_{2})$ on $x^*_S(t)$ under two estimation methods.