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Robust time-consistent mean–variance portfolio selection problem with multivariate stochastic volatility

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Abstract

This paper solves for the robust time-consistent mean–variance portfolio selection problem on multiple risky assets under a principle component stochastic volatility model. The model uncertainty is introduced to the drifts of the risky assets prices and the stochastic eigenvalues of the covariance matrix of asset returns. Using an extended dynamic programming approach, we manage to derive a semi-closed form solution of the desired portfolio via the solution to a coupled matrix Riccati equation. We provide the conditions, under which we prove the existence and the boundedness of the solution to the coupled matrix Riccati equation and derive the value function of the control problem. Moreover, we conduct numerical and empirical studies to perform sensitivity analyses and examine the losses due to ignoring model uncertainty or volatility information.

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Correspondence to Tingjin Yan.

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This work was supported by Ministry of Education, Singapore under AcRF Tier 2 Grant (Number: MOE2017-T2-1-044).

Appendices

Appendix A. Proof of Lemma 1

Proof

We first prove the second inequality. By Lemma 3.2 in [14], we have \({\mathbb {E}}^{{\mathbb {P}}}\left[ \sup _{0\le t \le T}\Vert {\varvec{\varLambda }}_t\Vert ^{4p} \right] < \infty \) for any \(p\ge 1\). Then, by the Hölder’s inequality, we have

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}} \left[ \sup _{0\le t \le T} \Vert {\varvec{\varLambda }}_t\Vert ^{2p} \right]\le & {} {\mathbb {E}}^{{\mathbb {P}}} \left[ \sup _{0\le t \le T} \Vert {\varvec{\varLambda }}_t\Vert ^{2p} \;\mathcal {E}_T\left( {\varvec{q}}^{\varvec{S}}\cdot {\varvec{W}}^{\varvec{S}}\right) \mathcal {E}_T\left( {\varvec{q}}^{{\varvec{\varLambda }}} \cdot {\varvec{W}}^{\varvec{\varLambda }}\right) \right] \nonumber \\\le & {} \left( {\mathbb {E}}^{{\mathbb {P}}} \left[ \sup _{0\le t \le T} \Vert {\varvec{\varLambda }}_t\Vert ^{4p} \right] \right) ^{\frac{1}{2}} \left( {\mathbb {E}}^{{\mathbb {P}}} \left[ \exp \left( 6 \int _0^T\Vert {\varvec{q}}_s\Vert ^2 ds\right) \right] \right) ^{\frac{1}{4}}\nonumber \\&\times \left( {\mathbb {E}}^{{\mathbb {P}}}\left[ \mathcal {E}_T\left( 4{\varvec{q}}^{\varvec{S}}\cdot {\varvec{W}}^{\varvec{S}}\right) \mathcal {E}_T\left( 4{\varvec{q}}^{{\varvec{\varLambda }}} \cdot {\varvec{W}}^{\varvec{\varLambda }}\right) \right] \right) ^{\frac{1}{4}}\nonumber \\= & {} \left( {\mathbb {E}}^{{\mathbb {P}}} \left[ \sup _{0\le t \le T} \Vert {\varvec{\varLambda }}_t\Vert ^{4p} \right] \right) ^{\frac{1}{2}} \left( {\mathbb {E}}^{{\mathbb {P}}}\left[ \exp \left( 6 \int _0^T \Vert {\varvec{q}}_s\Vert ^2 ds\right) \right] \right) ^{\frac{1}{4}}, \end{aligned}$$
(26)

where the last equality holds since we can define a new measure using the Radon-Nikodym derivative \(\mathcal {E}_T\left( 4{\varvec{q}}^{\varvec{S}}\cdot {\varvec{W}}^{\varvec{S}}\right) \mathcal {E}_T\left( 4{\varvec{q}}^{{\varvec{\varLambda }}} \cdot {\varvec{W}}^{\varvec{\varLambda }}\right) \) based on the Novikov’s condition (7).

Moreover, due to the condition (1) in Definition 1, we have \({\mathbb {E}}^{\mathbb {P}}\left[ \left( \int _0^T \Vert {\varvec{q}}_s\Vert ^2ds \right) ^p\right] < \infty \) for any \(p\ge 1\). By mimicking the steps in (26), we have \({\mathbb {E}}^{\mathbb {Q}}\left[ \left( \int _0^T \Vert {\varvec{q}}_s\Vert ^2ds \right) ^p\right] < \infty \) for any \(p\ge 1\). Therefore, by (6), (8), (9) and Doob’s martingale maximal inequality, there exist constants \(C_1, C_2, C_3 > 0\) such that

$$\begin{aligned} {\mathbb {E}}^{{\mathbb {Q}}}\left[ \sup _{0\le t \le T}|X_t^{\varvec{u}}|^4 \right]\le & {} C_1+ C_1{\mathbb {E}}^{\mathbb {Q}}\left[ \left( \int _0^T\left| {\varvec{u}}_s' {\varvec{A}}{{\mathbb {D}}\left[ {{\varvec{\varLambda }}_s}\right] } {\varvec{\delta }}\right| + \left| {\varvec{u}}_s'{\varvec{A}}{{\mathbb {D}}\left[ {\sqrt{{\varvec{\varLambda }}_s}}\right] } {\varvec{q}}_s^{{\varvec{S}}}\right| ds\right) ^4\right] \\&+\,C_1{\mathbb {E}}^{\mathbb {Q}}\left[ \left( \sup _{0\le t \le T} \int _0^t e^{\int _s^Tr_vdv}{\varvec{u}}_s'{\varvec{A}}{{\mathbb {D}}\left[ {\sqrt{{\varvec{\varLambda }}_s}}\right] }d{\varvec{W}}_s^{{\varvec{S}}}\right) ^4\right] \\\le & {} C_2+ C_2 {\mathbb {E}}^{\mathbb {Q}}\left[ \left( \int _0^T\Vert {\varvec{u}}_s\Vert ^2\Vert {\varvec{\varLambda }}_s\Vert + \Vert {\varvec{\varLambda }}_s\Vert +\Vert {\varvec{q}}_s^{{\varvec{S}}}\Vert ^2ds \right) ^4\right] \\\le & {} C_2+ C_3 {\mathbb {E}}^{\mathbb {Q}}\left[ \left( \int _0^T\Vert {\varvec{u}}_s\Vert ^2\Vert {\varvec{\varLambda }}_s\Vert ds \right) ^4\right] +C_3{\mathbb {E}}^{\mathbb {Q}}\left[ \left( \int _0^T \Vert {\varvec{\varLambda }}_s\Vert ds \right) ^4\right] \\&+\, C_3{\mathbb {E}}^{\mathbb {Q}}\left[ \left( \int _0^T \Vert {\varvec{q}}_s^{{\varvec{S}}}\Vert ^2ds \right) ^4\right] <\infty . \end{aligned}$$

\(\square \)

Appendix B. Proof of Lemma 4

Proof

We only prove that \(\mathcal {A}^{{\varvec{u}}, {\varvec{q}}} g^2(t,X_t,{\varvec{\varLambda }}_t) \in {\mathbb {L}}^1_{\mathcal {F}}(0,\, T;\, {\mathbb {R}}, \, {\mathbb {Q}})\), \({{\varvec{\sigma }}^{X}({\varvec{u}}_t, {\varvec{\varLambda }}_t)}' \frac{\partial g^2}{\partial x}(t, X_t, {\varvec{\varLambda }}_t) \in {\mathbb {L}}^2_{\mathcal {F}}(0,\, T; \, {\mathbb {R}}^n,\, {\mathbb {Q}})\); others can be proved with similar arguments. For simplicity, we omit the argument t of functions g and \(g_1\) in our proof.

When conditions in Lemma 3 hold, by Lemmas 1 and 3, there exist three constants \(C_1, C_2, C_3>0\) such that

$$\begin{aligned}&{\mathbb {E}}^{\mathbb {Q}}\left[ \int _0^T\big | 2g\,g_1\big |\left| r_sX_s +{\varvec{u}}_s' {\varvec{A}}{{\mathbb {D}}\left[ {{\varvec{\varLambda }}_s}\right] } {\varvec{\delta }}+ {\varvec{u}}_s' {\varvec{A}}{{\mathbb {D}}\left[ {\sqrt{{\varvec{\varLambda }}_s}}\right] } {\varvec{q}}^{{\varvec{S}}}\right| ds\right] \\&\quad \le C_1 + C_2{\mathbb {E}}^{\mathbb {Q}}\left[ \int _0^T\left( \Vert {\varvec{u}}_s\Vert ^2\Vert {\varvec{\varLambda }}_s\Vert +\Vert {\varvec{q}}_s\Vert ^2\right) \left( |X_s|+\Vert {\varvec{\varLambda }}_s\Vert \right) ds\right] \\&\quad \le C_1 + C_3 \left( {\mathbb {E}}^{\mathbb {Q}}\left[ \left( \int _0^T\Vert {\varvec{u}}_s\Vert ^2 \Vert {\varvec{\varLambda }}_s\Vert ds\right) ^2 +\left( \int _0^T\Vert {\varvec{q}}_s\Vert ^2\Vert ds\right) ^2 \right] \right) ^{\frac{1}{2}} \\&\qquad \times \left( {\mathbb {E}}^{\mathbb {Q}}\left[ \sup _{0\le t \le T}|X_t|^2 +\sup _{0\le t \le T} \Vert {\varvec{\varLambda }}_t\Vert ^2 \right] \right) ^{\frac{1}{2}}<\infty . \end{aligned}$$

Again, by Lemma 2.2, it is easy to verify that \(\mathcal {A}^{{\varvec{u}}, {\varvec{q}}} g^2(t,X_t,{\varvec{\varLambda }}_t) \in {\mathbb {L}}^1_{\mathcal {F}}(0,\, T;\, {\mathbb {R}}, \, {\mathbb {Q}})\).

Similarly, there exist two constants \(C_4, C_5>0\) such that

$$\begin{aligned}&{\mathbb {E}}^{\mathbb {Q}}\left[ \int _0^Tg^2\,g_1^2 \,{\varvec{u}}_s' {\varvec{A}}{{\mathbb {D}}\left[ {{\varvec{\varLambda }}_s}\right] } {\varvec{A}}'{\varvec{u}}_s ds\right] \\&\quad \le C_4 {\mathbb {E}}^{\mathbb {Q}}\left[ \int _0^T \left( |X_s^{{\varvec{u}}}|^2 +\Vert {\varvec{\varLambda }}_s\Vert ^2+ 1 \right) \Vert {\varvec{u}}_s\Vert ^2\Vert {\varvec{\varLambda }}_s\Vert ds \right] \\&\quad \le C_5 {\mathbb {E}}^{\mathbb {Q}}\left[ \sup _{0\le t \le T} {|X_t^{{\varvec{u}}}}|^4 \right] +C_5 {\mathbb {E}}^{\mathbb {Q}}\left[ \sup _{0\le t \le T} \Vert {\varvec{\varLambda }}_t\Vert ^4 \right] +C_5 {\mathbb {E}}^{\mathbb {Q}}\left[ \left( \int _0^T\Vert {\varvec{u}}_s\Vert ^2\Vert {\varvec{\varLambda }}_s\Vert ds\right) ^2 \right] \\&\qquad +\,C_5, \end{aligned}$$

whose upper bound is finite by noting the construction of the admissible control-measure policy and Lemma 1. \(\square \)

Appendix C. Proof of Proposition 1

Proof

We ought to prove that under the assumptions in this proposition, \(({\varvec{u}}^*, {\varvec{q}}^*)\) given by (22) is an admissible control-measure policy in Definition 1. The rest is a direct consequence of Theorem 1 and Lemmas 2, 3, and 4.

We note that

$$\begin{aligned}&{\mathbb {E}}^{{\mathbb {P}}}\left[ \exp \left( 8\int _0^T \Vert {\varvec{q}}_s^*\Vert ^2ds\right) \right] \\&\quad = {\mathbb {E}}^{{\mathbb {P}}} \left[ \exp \left( \sum _{i=1}^{n} \frac{8\xi ^2\delta _i^2}{(\xi +\gamma )^2}\int _0^T\varLambda _i(s)ds + 8\xi ^2\sigma _i^2(1-\rho _i^2) \int _0^TV_{2i}^2(s)\varLambda _i(s)ds\right) \right] \\&\quad = \prod _{i=1}^{n} {\mathbb {E}}^{{\mathbb {P}}} \left[ \exp \left( \frac{8\xi ^2\delta _i^2}{(\xi +\gamma )^2}\int _0^T\varLambda _i(s)ds +8\xi ^2\sigma _i^2(1-\rho _i^2)\int _0^TV_{2i}^2(s)\varLambda _i(s)ds\right) \right] , \end{aligned}$$

which is the product of the Laplace transform of the integrated CIR process. We first discuss the boundedness of \(V_{2i}(t)\) on [0, T]. By the Proposition in [25], we obtain the following results under the conditions in Lemma 3:

  1. (1)

    Under condition (i), \(|V_{2i}(t)| \le \eta ^i_1 +\frac{1}{\beta ^i(\frac{2}{\alpha ^i}-T)}\);

  2. (2)

    Under condition (ii), \(|V_{2i}(t)| \le \eta ^i_2 +\frac{\sqrt{\varDelta ^i}}{\beta ^i(1-\frac{\eta ^i_1}{\eta ^i_2}e^{\sqrt{\varDelta ^i}T})}\);

  3. (3)

    Under condition (iii), \(|V_{2i}(t)| \le -\frac{\alpha ^i}{2\beta ^i} + \frac{\sqrt{|\varDelta ^i|}}{2\beta ^i} \tan \left( \frac{\sqrt{|\varDelta ^i|}}{2}T + \arctan \frac{\alpha ^i}{\sqrt{|\varDelta ^i|}}\right) \),

for any \(t\in [0,T]\) and \(i = 1, 2, \ldots , n\). We use function \(U_i(T)\) to denote the bound of \(V_{2i}(t)\) on [0, T]. Define \(\bar{U}_i(T)=\frac{8\xi ^2\delta _i^2}{(\xi +\gamma )^2} + 8\xi ^2\sigma _i^2(1-\rho _i^2)U_i^2(T)\). By [2], \({\mathbb {E}}^{{\mathbb {P}}}\left[ \exp \left( 8\int _0^T \Vert {\varvec{q}}_s^*\Vert ^2 ds\right) \right] <\infty \) if \(k_i^2 - 2 \bar{U}_i(T)\sigma _i^2 \ge 0\) for \(i=1, 2, \ldots , n\).

The assumptions on T in this proposition are imposed such that both conditions in Lemma 3 and \(k_i^2 - 2 \bar{U}_i(T)\sigma _i^2 \ge 0\) are satisfied and thus the first condition of Definition 1 is satisfied. We remark here that one can easily verify that all the upper bounds of T in the assumptions are positive.

Since \({\varvec{u}}^*\) is a deterministic continuous function on \(t \in [0,T]\), by Lemma 1, \(({\varvec{u}}^*, {\varvec{q}}^*)\) is an admissible control-measure policy. \(\square \)

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Yan, T., Han, B., Pun, C.S. et al. Robust time-consistent mean–variance portfolio selection problem with multivariate stochastic volatility. Math Finan Econ 14, 699–724 (2020). https://doi.org/10.1007/s11579-020-00271-0

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