Time consistent behavioral portfolio policy for dynamic mean–variance formulation

Abstract

When one considers an optimal portfolio policy under a mean-risk formulation, it is essential to correctly model investors’ risk aversion which may be time variant or even state dependent. In this paper, we propose a behavioral risk aversion model, in which risk aversion is a piecewise linear function of the current excess wealth level with a reference point at the discounted investment target (either surplus or shortage), to reflect a behavioral pattern with both house money and break-even effects. Due to the time inconsistency of the resulting multi-period mean–variance model with adaptive risk aversion, we investigate the time consistent behavioral portfolio policy by solving a nested mean–variance game formulation. We derive a semi-analytical time consistent behavioral portfolio policy which takes a piecewise linear feedback form of the current excess wealth level with respect to the discounted investment target. Finally, we extend the above results to time consistent behavioral portfolio selection for dynamic mean–variance formulation with a cone constraint.

Appendices

Appendix 1: The Proof of Proposition 3.1

Proof

Define $\xi =\Vert \mathbf {K}\Vert ,\,\mathbf {L}=\mathbf {K}\xi ^{-1}$ (which implies $\Vert \mathbf {L}\Vert =1$) and $y_t=\mathbf {P}_t'\mathbf {L}$. Then, for any $\mathbf {L}$, we have $M\ge \text{ Var }(y_t)=\mathbf {L}'\text{ Cov }(\mathbf {P}_t)\mathbf {L}>0$, where M is the largest eigenvalue of $\text{ Cov }(\mathbf {P}_t)$.

If $y_t1_{\{y_t\ge 0\}}$ is zero, (i.e., $y_t\le 0$ almost surely), we can construct an arbitrage portfolio by shorting $\mathbf {L}$ and holding $\mathbf {L}'\mathbf {1}$ risk-free asset. Similarly, if $y_t1_{\{y_t< 0\}}$ is zero, (i.e., $y_t\ge 0$ almost surely), we also can construct an arbitrage portfolio by holding $\mathbf {L}$ and shorting $\mathbf {L}'\mathbf {1}$ risk-free asset. Thus, we conclude that $y_t1_{\{y_t\ge 0\}}$ and $y_t1_{\{y_t< 0\}}$ are nontrivial random variables with finite second moment.

Moreover, $\mathbf {P}_t$ is absolutely integrable, so do $y_t,\,y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}$ and $y_t1_{\{y_t< \frac{-s_t}{\xi }\}}$. Then, for given $\mathbf {L}$, we have

$$\begin{aligned} F_t^+(\mathbf {K})=\tilde{F}_t^+(\xi ), \end{aligned}$$

where

$$\begin{aligned} \tilde{F}_t^+(\xi )&= \, \rho _{t+1}^2\text{ Var }(y_t)\xi ^2+{\mathbb{E}}\left[ (2\rho _{t+1}a_{t+1}^+ + b_{t+1}^+)(s_t+\xi y_t)^21_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] \\&\quad +\,{\mathbb{E}}\left[ (2\rho _{t+1}a_{t+1}^- + b_{t+1}^-)(s_t+\xi y_t)^21_{\{y_t< \frac{-s_t}{\xi }\}}\right] \\&\quad -\,\left( {\mathbb{E}}\left[ a_{t+1}^+(s_t+\xi y_t)1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] +{\mathbb{E}}\left[ a_{t+1}^-(s_t+\xi y_t)1_{\{y_t< \frac{-s_t}{\xi }\}}\right] \right) ^2\\&\quad -\,2\rho _{t+1} \left( {\mathbb{E}}\left[ a_{t+1}^+(s_t+\xi y_t)1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] +{\mathbb{E}}\left[ a_{t+1}^-(s_t+\xi y_t)1_{\{y_t< \frac{-s_t}{\xi }\}}\right] \right) (s_t+{\mathbb{E}}[y_t]\xi )\\&\quad -\,\gamma _t^+\left( {\mathbb{E}}\left[ a_{t+1}^+(s_t+\xi y_t)1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] +{\mathbb{E}}\left[ a_{t+1}^-(s_t+\xi y_t)1_{\{y_t<\frac{-s_t}{\xi }\}}\right] \right) \\&\quad -\,\rho _{t+1}\gamma _t^+(s_t+{\mathbb{E}}[y_t]\xi ). \end{aligned}$$

Furthermore, we have

$$\begin{aligned} \tilde{F}_t^+(\xi )&\ge \,\rho _{t+1}^2\text{ Var }(y_t)\xi ^2+(a_{t+1}^+)^2{\mathbb{E}}\left[ y_t^21_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] \xi ^2+(a_{t+1}^-)^2{\mathbb{E}}\left[ y_t^21_{\{y_t< \frac{-s_t}{\xi }\}}\right] \xi ^2\\&\quad -\,\left( a_{t+1}^+{\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] \xi +a_{t+1}^-{\mathbb{E}}\left[ y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] \xi \right) ^2\\&\quad +\,2\rho _{t+1}\left( a_{t+1}^+{\mathbb{E}}\left[ y_t^21_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] \xi ^2+a_{t+1}^-{\mathbb{E}}\left[ y_t^21_{\{y_t< \frac{-s_t}{\xi }\}}\right] \xi ^2\right) \\&\quad -\,2\rho _{t+1}\left( a_{t+1}^+{\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] +a_{t+1}^-{\mathbb{E}}\left[ y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] \right) {\mathbb{E}}[y_t]\xi ^2+O(\xi )\\&= \, \rho _{t+1}^2\text{ Var }\left( y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right) \xi ^2+\rho _{t+1}^2\text{ Var }\left( y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right) \xi ^2\\&\quad +\,2\rho _{t+1}^2\left( {\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}} y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] -{\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] {\mathbb{E}}\left[ y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] \right) \xi ^2\\&\quad +\,(a_{t+1}^+)^2\text{ Var }\left( y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right) \xi ^2+(a_{t+1}^-)^2\text{ Var }\left( y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right) \xi ^2\\&\quad +\,2a_{t+1}^+a_{t+1}^-\left( {\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}} y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] -{\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] {\mathbb{E}}\left[ y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] \right) \xi ^2\\&\quad +\,2\rho _{t+1}\left( a_{t+1}^+\text{ Var }\left( y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right) \xi ^2+a_{t+1}^-\text{ Var }\left( y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right) \xi ^2\right) \\&\quad +\,2\rho _{t+1}a_{t+1}^+\left( {\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}} y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] -{\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] {\mathbb{E}}\left[ y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] \right) \xi ^2\\&\quad +\,2\rho _{t+1}a_{t+1}^-\left( {\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}} y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] -{\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\right] {\mathbb{E}}\left[ y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] \right) \xi ^2 \\&\quad +\,O(\xi )\\&=\left[ \rho _{t+1}+a_{t+1}^+,\rho _{t+1}+a_{t+1}^-\right] \text{ Cov }\left[ \begin{array}{ll} y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}}\\ y_t1_{\{y_t< \frac{-s_t}{\xi }\}} \end{array}\right] \left[ \begin{array}{ll} \rho _{t+1}+a_{t+1}^+\\ \rho _{t+1}+a_{t+1}^- \end{array}\right] \xi ^2 +O(\xi ), \end{aligned}$$

where $O(\xi )$ is the infinity of the same order as $\xi$ and the second equality holds due to the fact of ${\mathbb{E}}\left[ y_t1_{\{y_t\ge \frac{-s_t}{\xi }\}} y_t1_{\{y_t< \frac{-s_t}{\xi }\}}\right] =0$. Hence,

$$\begin{aligned}&\lim _{\xi \rightarrow +\infty }\tilde{F}_t^+(\xi )=\lim _{\xi \rightarrow +\infty }\left[ \rho _{t+1}+a_{t+1}^+,\rho _{t+1}+a_{t+1}^-\right] \text{ Cov }\left[ \begin{array}{ll} y_t1_{\{y_t\ge 0\}}\\ y_t1_{\{y_t< 0\}} \end{array}\right] \left[ \begin{array}{ll} \rho _{t+1}+a_{t+1}^+\\ \rho _{t+1}+a_{t+1}^- \end{array}\right] \xi ^2 \\&\quad +\, O(\xi )=+\infty . \end{aligned}$$

Based on the discussion for all possible $\mathbf {L}$, we make our conclusion for $F_t^+(\mathbf {K})$. Similarly we can prove the result of $F_t^-(\mathbf {K})$. $\square$

Appendix 2: The Proof of Theorem 3.1

Proof

Let $Y_t = X_t - \rho _t^{-1}W$. Then,

$$\begin{aligned} Y_{t+1}&= X_{t+1} - \rho _{t+1}^{-1}W \\&= s_tX_t + \mathbf {P}_t'\mathbf {u}_t - \rho _{t+1}^{-1}W \\&= s_t(X_t - \rho _t^{-1}W) + \mathbf {P}_t'\mathbf {u}_t \\&= s_tY_t + \mathbf {P}_t'\mathbf {u}_t, \end{aligned}$$

and $\gamma _t(X_t)$ can be rewritten into

$$\begin{aligned} \gamma _t(X_t) = \hat{\gamma }_t(Y_t)=\left\{ \begin{array}{ll} \gamma _t^+ Y_t, &{}\quad \text{ if } Y_t \ge 0, \\ -\gamma _t^- Y_t, &{}\quad \text{ if } Y_t < 0. \end{array}\right. \end{aligned}$$

Also, we have ${\mathrm{Var}}_t(X_T)=\text{Var}_t(Y_T)$ according to the variance property. Hence, problem $({\mathrm{MV}}_t(\gamma _t(X_t)))$ in (2) can be reduced into the following equivalent problem,

$$\begin{aligned} \min&\quad {\mathrm{Var}}_t(Y_T)- \hat{\gamma }_t(Y_t) {\mathbb{E}}_t[Y_T] - \hat{\gamma }_t(Y_t)W, \nonumber \\ \text{ s.t. }&~~ Y_{j+1} = s_jY_j+\mathbf {P}_j'\mathbf {u}_j, \quad j=t,t+1,\ldots ,T-1, \end{aligned}$$

(15)

where ${\mathrm{Var}}_t(Y_T)=\text{ Var }(Y_T|Y_t)$ and ${\mathbb{E}}_t[Y_T]={\mathbb{E}}[Y_T|Y_t]$.

At time t ($t=0,1,\ldots , T$), the investor faces the following optimization problem,

$$\begin{aligned} \min _{\mathbf {u}_t}~~J_t(Y_t;\mathbf {u}_t) =\Big ({\mathbb{E}}_t[Y_T^2] - ({\mathbb{E}}_t[Y_T])^2\Big ) - \hat{\gamma }_t(Y_t){\mathbb{E}}_t[Y_T] - \hat{\gamma }_t(Y_t)W, \end{aligned}$$

(16)

where the conditional expectations ${\mathbb{E}}_t[Y_T]={\mathbb{E}}[Y_T|Y_t]$ and ${\mathbb{E}}_t[Y_T^2]={\mathbb{E}}[Y_T^2|Y_t]$ are computed along the policy $\{\mathbf {u}_t,\mathbf {u}_{t+1}^{TC},\ldots , \mathbf {u}_{T-1}^{TC}\}$.

We now prove by induction that the following two expressions,

$$\begin{aligned} {\mathbb{E}}_t[Y_T]&= \rho _tY_t+a_t^+Y_t1_{\{Y_t \ge 0\}}+a_t^-Y_t1_{\{Y_t < 0\}}, \end{aligned}$$

(17)

$$\begin{aligned} {\mathbb{E}}_t[Y_T^2]&= \rho _t^2Y_t^2+(2\rho _ta_t^+ + b_t^+)Y_t^21_{\{Y_t \ge 0\}}+(2\rho _ta_t^- + b_t^-)Y_t^21_{\{Y_t < 0\}}, \end{aligned}$$

(18)

hold along the time consistent policy, $\{\mathbf {u}_t^{TC},\mathbf {u}_{t+1}^{TC},\ldots , \mathbf {u}_{T-1}^{TC}\}$, at time t.

At time T, we have

$$\begin{aligned} {\mathbb{E}}_T[Y_T]=Y_T, \quad {\mathbb{E}}_T[Y_T^2]=Y_T^2, \end{aligned}$$

with $a_T^+=a_T^- = 0$ and $b_T^+=b_T^- = 0$. Assume that expressions of the first moment and the second moment in (17) and (18), respectively, hold at time $t+1$ along the time consistent policy $\{\mathbf {u}_{t+1}^{TC},\ldots, \mathbf {u}_{T-1}^{TC}\}$. We will prove that these two expressions still hold at time t and the corresponding time consistent policy is given by (9).

As the dynamics of $Y_t$ at period t is given by

$$\begin{aligned} Y_{t+1}=s_tY_t+\mathbf {P}_t'\mathbf {u}_t. \end{aligned}$$

It follows from the policy $\{\mathbf {u}_t,\mathbf {u}_{t+1}^{TC},\ldots , \mathbf {u}_{T-1}^{TC}\}$ that we have

$$\begin{aligned}&{\mathbb{E}}_t[Y_T] = {\mathbb{E}}_t\big [{\mathbb{E}}_{t+1}[Y_T]\big ] \nonumber \\&\quad = {\mathbb{E}}_t \left[ \rho _{t+1}Y_{t+1}+a_{t+1}^+Y_{t+1}1_{\{Y_{t+1} \ge 0\}}+a_{t+1}^-Y_{t+1}1_{\{Y_{t+1}< 0\}}\right] \nonumber \\&\quad = {\mathbb{E}}_t[\rho _{t+1}(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)]+{\mathbb{E}}_t\left[ a_{t+1}^+(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)1_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t \ge 0\}}\right] \nonumber \\&\qquad +\,{\mathbb{E}}_t \left[ a_{t+1}^-(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)1_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t < 0\}}\right] \end{aligned}$$

(19)

and

$$\begin{aligned}&{\mathbb{E}}_t[Y_T^2] = {\mathbb{E}}_t\big [{\mathbb{E}}_{t+1}[Y_T^2]\big ] \nonumber \\&\quad = {\mathbb{E}}_t\!\Big [\rho _{t+1}^2Y_{t+1}^2+(2\rho _{t+1}a_{t+1}^+ + b_{t+1}^+)Y_{t+1}^21_{\{Y_{t+1} \ge 0\}} \nonumber \\&\qquad +\,(2\rho _{t+1}a_{t+1}^- + b_{t+1}^-)Y_{t+1}^21_{\{Y_{t+1}< 0\}}\Big ] \nonumber \\&\quad = {\mathbb{E}}_t[\rho _{t+1}^2(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)^2]\nonumber \\&\qquad +\,{\mathbb{E}}_t \left[ (2\rho _{t+1}a_{t+1}^+ + b_{t+1}^+)(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)^21_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t \ge 0\}}\right] \nonumber \\&\qquad +\,{\mathbb{E}}_t\left[ (2\rho _{t+1}a_{t+1}^- + b_{t+1}^-)(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)^21_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t < 0\}}\right] . \end{aligned}$$

(20)

For $Y_t > 0$, we denote any admissible policy as $\mathbf {u}_t=\mathbf {K}Y_t$ with $\mathbf {K}\in {\mathbb {R}}^n$. Then, the cost functional can be expressed as

$$\begin{aligned}&J_t(Y_t;\mathbf {u}_t) = \Big ({\mathbb{E}}_t[Y_T^2] - ({\mathbb{E}}_t[Y_T])^2\Big ) - \gamma _t^+Y_t{\mathbb{E}}_t[Y_T] - \gamma _t^+Y_tW \\&\quad = {\mathbb{E}}_t[\rho _{t+1}^2(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)^2]- \big ({\mathbb{E}}_t[\rho _{t+1}(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)]\big )^2 \\&\qquad +\,{\mathbb{E}}_t \left[ (2\rho _{t+1}a_{t+1}^+ + b_{t+1}^+)(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)^21_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t \ge 0\}}\right] \\&\qquad +\,{\mathbb{E}}_t \left[ (2\rho _{t+1}a_{t+1}^- + b_{t+1}^-)(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)^21_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t< 0\}}\right] \\&\qquad -\,\left( {\mathbb{E}}_t \left[ a_{t+1}^+(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)1_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t\ge 0\}}\right] +{\mathbb{E}}_t \left[ a_{t+1}^-(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)1_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t< 0\}}\right] \right) ^2 \\&\qquad -\,2\rho _{t+1}\Big (\!{\mathbb{E}}_t \left[ a_{t+1}^+(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)1_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t \ge 0\}}\right] \\&\qquad +\,{\mathbb{E}}_t \left[ a_{t+1}^-(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)1_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t< 0\}}\right] \Big ){\mathbb{E}}_t[s_tY_t+\mathbf {P}_t'\mathbf {u}_t] \\&\qquad -\,\gamma _t^+Y_t\Big (\!{\mathbb{E}}_t\left[ a_{t+1}^+(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)1_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t \ge 0\}}\right] +{\mathbb{E}}_t \left[ a_{t+1}^-(s_tY_t+\mathbf {P}_t'\mathbf {u}_t)1_{\{s_tY_t+\mathbf {P}_t'\mathbf {u}_t< 0\}}\right] \Big ) \\&\qquad -\,\rho _{t+1}\gamma _t^+Y_t{\mathbb{E}}_t[s_tY_t+\mathbf {P}_t'\mathbf {u}_t]-\gamma _t^+Y_tW \\&\quad = Y_t^2 F_t^+(\mathbf {K})-\gamma _t^+Y_tW. \end{aligned}$$

Applying Proposition 3.1 yields the optimal time consistent policy at time t,

$$\begin{aligned} \mathbf {u}_t^{TC}={\mathop {\mathrm{argmin}}\limits _{\mathbf {u}_t\in {\mathbb {R}}^n}}\,\, J_t(Y_t;\mathbf {u}_t)=\mathbf {K}_t^+Y_t. \end{aligned}$$

Then, substituting the above optimal time consistent policy back into (19) and (20) gives rise to

$$\begin{aligned} {\mathbb{E}}_t[Y_T]&= \,\rho _tY_t+Y_t\Big (\rho _{t+1}{\mathbb{E}}[\mathbf {P}_t']\mathbf {K}_t^++{\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K}_t^+)1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^+ \ge 0\}}\right] \\&\quad +\,{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K}_t^+)1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^+ < 0\}}\right] \Big ) \\&=\,\rho _tY_t+a_t^+Y_t \end{aligned}$$

and

$$\begin{aligned} {\mathbb{E}}_t[Y_T^2]&= \, \rho _t^2Y_t^2+2\rho _tY_t^2\Big (\rho _{t+1}{\mathbb{E}}[\mathbf {P}_t']\mathbf {K}_t^++{\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K}_t^+)1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^+ \ge 0\}}\right] \\&\quad +\,{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K}_t^+)1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^+< 0\}}\right] \Big ) \\&\quad +\,\Big (\rho _{t+1}^2 (\mathbf {K}_t^+)' {\mathbb{E}}[\mathbf {P}_t\mathbf {P}_t']\mathbf {K}_t^++2\rho _{t+1}{\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K}_t^+)\mathbf {P}_t'\mathbf {K}_t^+1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^+\ge 0\}}\right] \\&\quad +\,2\rho _{t+1}{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K}_t^+)\mathbf {P}_t'\mathbf {K}_t^+1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^+< 0\}}\right] \\&\quad +\,{\mathbb{E}}\left[ b_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K}_t^+)^21_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^+\ge 0\}}\right] +{\mathbb{E}}\left[ b_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K}_t^+)^21_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^+< 0\}}\right] \Big )Y_t^2 \\&=\rho _t^2Y_t^2+(2\rho _ta_t^++b_t^+)Y_t^2. \end{aligned}$$

Furthermore,

$$\begin{aligned} \text{Var}_t(Y_T)={\mathbb{E}}_t[Y_T^2]-({\mathbb{E}}_t[Y_T])^2 =(b_t^+ - ( a_t^+)^2)Y_t^2 \ge 0, \end{aligned}$$

which implies $b_t^+ - ( a_t^+)^2\ge 0$.

For $Y_t < 0$, we denote any admissible policy as $\mathbf {u}_t=\mathbf {K}Y_t$ with $\mathbf {K}\in {\mathbb {R}}^n$. Then, the cost functional can be expressed as

$$\begin{aligned} J_t(Y_t;\mathbf {u}_t)&= Y_t^2 \Big \{\rho _{t+1}^2 \mathbf {K}' ({\mathbb{E}}[\mathbf {P}_t\mathbf {P}_t']-{\mathbb{E}}[\mathbf {P}_t']{\mathbb{E}}[\mathbf {P}_t])\mathbf {K}\\&\quad +\,{\mathbb{E}}\left[ (2\rho _{t+1}a_{t+1}^+ + b_{t+1}^+)(s_t+\mathbf {P}_t'\mathbf {K})^21_{\{s_t+\mathbf {P}_t'\mathbf {K}\le 0\}}\right] \\&\quad +\,{\mathbb{E}}\left[ (2\rho _{t+1}a_{t+1}^- + b_{t+1}^-)(s_t+\mathbf {P}_t'\mathbf {K})^21_{\{s_t+\mathbf {P}_t'\mathbf {K}> 0\}}\right] \\&\quad -\,\left( {\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K})1_{\{s_t+\mathbf {P}_t'\mathbf {K}\le 0\}}\right] +{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K})1_{\{s_t+\mathbf {P}_t'\mathbf {K}> 0\}}\right] \right) ^2 \\&\quad -\,2\rho _{t+1} {\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K})1_{\{s_t+\mathbf {P}_t'\mathbf {K}\le 0\}}\right] (s_t+{\mathbb{E}}[\mathbf {P}_t']\mathbf {K}) \\&\quad -\,2\rho _{t+1}{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K})1_{\{s_t+\mathbf {P}_t'\mathbf {K}> 0\}}\right] (s_t+{\mathbb{E}}[\mathbf {P}_t']\mathbf {K}) \\&\quad +\,\gamma _t^-\left( {\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K})1_{\{s_t+\mathbf {P}_t'\mathbf {K}\le 0\}}\right] +{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K})1_{\{s_t+\mathbf {P}_t'\mathbf {K}> 0\}}\right] \right) \\&\quad +\,\rho _{t+1}\gamma _t^-(s_t+{\mathbb{E}}[\mathbf {P}_t']\mathbf {K})\Big \}+\gamma _t^-Y_tW \\&= Y_t^2 F_t^-(\mathbf {K})+\gamma _t^-Y_tW. \end{aligned}$$

Applying Proposition 3.1 yields the optimal time consistent policy at time t,

$$\begin{aligned} \mathbf {u}_t^{TC}={\mathop {\mathrm{argmin}}\limits _{\mathbf {u}_t\in {\mathbb {R}}^n}}\,J_t(Y_t;\mathbf {u}_t)=\mathbf {K}_t^-Y_t. \end{aligned}$$

Then, substituting the above optimal time consistent policy back into (19) and (20) gives rise to

$$\begin{aligned} {\mathbb{E}}_t[Y_T]&=\rho _tY_t+Y_t\Big (\rho _{t+1}{\mathbb{E}}[\mathbf {P}_t']\mathbf {K}_t^-+{\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K}_t^-)1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^-\le 0\}}\right] \\&\quad +\,{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K}_t^-)1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^-> 0\}}\right] \Big ) \\&=\rho _tY_t+a_t^-Y_t \end{aligned}$$

and

$$\begin{aligned} {\mathbb{E}}_t[Y_T^2]&= \rho _t^2Y_t^2+2\rho _tY_t^2\Big (\rho _{t+1}{\mathbb{E}}[\mathbf {P}_t']\mathbf {K}_t^-+{\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K}_t^-)1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^-\le 0\}}\right] \\&\quad +\,{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K}_t^-)1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^-> 0\}}\right] \Big ) \\&\quad +\,\Big (\rho _{t+1}^2 (\mathbf {K}_t^-)' {\mathbb{E}}[\mathbf {P}_t\mathbf {P}_t']\mathbf {K}_t^-+2\rho _{t+1}{\mathbb{E}}\left[ a_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K}_t^-)\mathbf {P}_t'\mathbf {K}_t^-1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^-\le 0\}}\right] \\&\quad +\,2\rho _{t+1}{\mathbb{E}}\left[ a_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K}_t^-)\mathbf {P}_t'\mathbf {K}_t^-1_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^-> 0\}}\right] \\&\quad +\,{\mathbb{E}}\left[ b_{t+1}^+(s_t+\mathbf {P}_t'\mathbf {K}_t^-)^21_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^-\le 0\}}\right] +{\mathbb{E}}\left[ b_{t+1}^-(s_t+\mathbf {P}_t'\mathbf {K}_t^-)^21_{\{s_t+\mathbf {P}_t'\mathbf {K}_t^-> 0\}}\right] \Big )Y_t^2 \\&= \rho _t^2Y_t^2+(2\rho _ta_t^-+b_t^-)Y_t^2. \end{aligned}$$

Furthermore,

$$\begin{aligned} \text{Var}_t(Y_T)={\mathbb{E}}_t[Y_T^2]-({\mathbb{E}}_t[Y_T])^2 =(b_t^- - ( a_t^-)^2)Y_t^2 \ge 0, \end{aligned}$$

which implies $b_t^- - ( a_t^-)^2\ge 0$.

For $Y_t=0$, the cost functional reduces to the conditional variance of the terminal wealth along policy $\{\mathbf {u}_t,\mathbf {u}_{t+1}^{TC},\ldots , \mathbf {u}_{T-1}^{TC}\}$, which can be expressed as

$$\begin{aligned} J_t(Y_t;\mathbf {u}_t)&= \rho _{t+1}^2 \mathbf {u}_t' ({\mathbb{E}}[\mathbf {P}_t\mathbf {P}_t']-{\mathbb{E}}[\mathbf {P}_t']{\mathbb{E}}[\mathbf {P}_t])\mathbf {u}_t\\&\quad +\,{\mathbb{E}}\left[ b_{t+1}^+(\mathbf {P}_t'\mathbf {u}_t)^21_{\{\mathbf {P}_t'\mathbf {u}_t\ge 0\}}\right] +{\mathbb{E}}\left[ b_{t+1}^-(\mathbf {P}_t'\mathbf {u}_t)^21_{\{\mathbf {P}_t'\mathbf {u}_t< 0\}}\right] \\&\quad -\,\Big ({\mathbb{E}}\left[ a_{t+1}^+\mathbf {P}_t'\mathbf {u}_t1_{\{\mathbf {P}_t'\mathbf {u}_t\ge 0\}}\right] +{\mathbb{E}}\left[ a_{t+1}^-\mathbf {P}_t'\mathbf {u}_t1_{\{\mathbf {P}_t'\mathbf {u}_t< 0\}}\right] \Big )^2\\&\quad +\,2\rho _{t+1} \left( {\mathbb{E}}\left[ a_{t+1}^+(\mathbf {P}_t'\mathbf {u}_t)^21_{\{\mathbf {P}_t'\mathbf {u}_t\ge 0\}}\right] + {\mathbb{E}}\left[ a_{t+1}^-(\mathbf {P}_t'\mathbf {u}_t)^21_{\{\mathbf {P}_t'\mathbf {u}_t< 0\}}\right] \right) \\&\quad -\,2\rho _{t+1}\Big ({\mathbb{E}}\left[ a_{t+1}^+\mathbf {P}_t'\mathbf {u}_t1_{\{\mathbf {P}_t'\mathbf {u}_t\ge 0\}}\right] +{\mathbb{E}}\left[ a_{t+1}^-\mathbf {P}_t'\mathbf {u}_t1_{\{\mathbf {P}_t'\mathbf {u}_t< 0\}}\right] \Big ) {\mathbb{E}}[\mathbf {P}_t']\mathbf {u}_t\\&\ge 0. \end{aligned}$$

It is not difficult to conclude that $\mathbf {u}_t^{TC}={\mathop {\mathrm{argmin}}\limits _{\mathbf {u}_t\in {\mathbb {R}}^n}} J_t(Y_t;\mathbf {u}_t)=\mathbf {0}$.

Therefore, along the time consistent policy $\{\mathbf {u}_t^{TC},\mathbf {u}_{t+1}^{TC},\ldots , \mathbf {u}_{T-1}^{TC}\}$, expressions (17) and (18) hold at time t, which completes our proof. $\square$

Appendix 3: The Proof of Theorem 4.1

Proof

Following the technique in the proof of Theorem 3.1, we can derive the main results directly with the following specifics.

• (1) For $X_t>\rho _t^{-1}W$, we denote any admissible policy as $\mathbf {u}_t=\mathbf {K}(X_t-\rho _t^{-1}W)$ with $\mathbf {K}\in {\mathcal {A}}_t$.

• (2) For $X_t<\rho _t^{-1}W$, we denote any admissible policy as $\mathbf {u}_t=\mathbf {K}(X_t-\rho _t^{-1}W)$ with $\mathbf {K}\in -{\mathcal {A}}_t$, where $-{\mathcal {A}}_t$ is the negative cone of ${\mathcal {A}}_t$.

• (3) For $X_t=\rho _t^{-1}W$, we can similarly prove $\mathbf {u}_t^{TC}=\mathbf {0}$.

Therefore, we have

$$\begin{aligned} \widetilde{\mathbf{K}}_{t}^+={\mathop {\text{argmin}}_{\mathbf{K}\in\mathcal{A}_{t}}}\quad {F_{t}^+} {(\mathbf{K})},\quad \widetilde{\mathbf{K}}_{t}^-={\mathop {\text{argmin}}_{\mathbf{K}\in\mathcal{A}_{t}}}\quad {F_{t}^-} {(\mathbf{K})},\end{aligned}$$

This completes the proof. $\square$