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Robust state-dependent mean–variance portfolio selection: a closed-loop approach

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Abstract

This paper studies a class of robust mean–variance portfolio selection problems with state-dependent risk aversion. Model uncertainty, in the sense of considering alternative dominated models, is introduced to the problem to reflect the investor’s uncertainty-averse preference. To characterise the robust portfolios, we consider closed-loop equilibrium control and spike variation approaches. Moreover, we show that a closed-loop equilibrium strategy exists and is unique under some technical conditions. This partially addresses open problems left in Björk et al. (Finance Stoch. 21:331–360, 2017) and Pun (Automatica 94:249–257, 2018). By using a necessary and sufficient condition for the equilibrium, we manage to derive the analytical form of the equilibrium strategy via the unique solution to a nonlinear ordinary differential equation system. To validate the proposed closed-loop control framework, we show that when there is no uncertainty, our equilibrium strategy is reduced to the strategy in Björk et al. (Math. Finance 24:1–24, 2014), which cannot be deduced under the open-loop control framework.

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Acknowledgements

The authors should like to thank the anonymous referees, the Associate Editor and the Editor Martin Schweizer for their valuable comments and suggestions which greatly improved this manuscript.

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Correspondence to Chi Seng Pun.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Bingyan Han is supported by UIC Start-up Research Fund (Reference No: R72021109). Chi Seng Pun gratefully acknowledges the Ministry of Education (MOE), AcRF Tier 2 grant (Reference No: MOE2017-T2-1-044) for the funding of this research. Hoi Ying Wong acknowledges the support from the Research Grants Council of Hong Kong via GRF 14303915.

Appendix: Proofs of some results

Appendix: Proofs of some results

1.1 A.1 Proof of Lemma 3.2

Proof

By the definition of \(\Lambda ^{\varepsilon }\) in Lemma 3.1 and the ansatz (3.4), we have

$$\begin{aligned} & \tilde{\mathbb{E}}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \Lambda ^{\varepsilon }(s;t) '\eta ds\bigg] \\ &= \tilde{\mathbb{E}}^{*}_{t}\bigg[\int _{t}^{t+\varepsilon } \bigg( \alpha ^{t, \varepsilon , v}_{s} \tilde{X}^{t,\varepsilon ,v,*}_{s} \big( M({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t, \varepsilon ,v,*}, s) \tilde{X}^{t,\varepsilon ,v,*}_{s} \\ &\phantom{=::}\qquad \qquad \qquad \qquad \qquad - \Gamma ({}^{s}{ \alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t,\varepsilon ,v,*}, s) x \\ &\phantom{=::}\qquad \qquad \qquad \qquad \qquad - N({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t,\varepsilon ,v,*}, s) \tilde{\mathbb{E}}^{*}_{t}[ \tilde{X}^{t,\varepsilon ,v,*}_{s}] \big) \\ &\qquad \qquad \qquad - \frac{1}{\xi } (\tilde{X}^{t, \varepsilon ,v,*}_{s})^{2} \tilde{h}^{t,\varepsilon ,v,*}_{s} \bigg) ' \eta ds \bigg]. \end{aligned}$$

We deal with these terms separately in the sequel. As \(\varepsilon \downarrow 0\),

$$\begin{aligned} & \frac{1}{\varepsilon }\bigg| \tilde{\mathbb{E}}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \Big( \alpha ^{t, \varepsilon , v}_{s} \big( M({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t, \varepsilon ,v,*}, s) |\tilde{X}^{t,\varepsilon ,v,*}_{s}|^{2} - M({}^{s}{ \alpha }^{*}, {}^{s}{h}^{*}, s) x^{2} \big) \Big)ds \bigg]\bigg| \\ &\leq \frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } | \alpha ^{t, \varepsilon , v}_{s}| \big| M({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{ \tilde{h}}^{t,\varepsilon ,v,*}, s) \tilde{\mathbb{E}}^{*}_{t}[| \tilde{X}^{t,\varepsilon ,v,*}_{s}|^{2} ]- M({}^{s}{\alpha }^{*}, {}^{s}{h}^{*}, s) x^{2} \big|ds \\ &\leq \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } | \alpha ^{t, \varepsilon , v}_{s} M({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{ \tilde{h}}^{t,\varepsilon ,v,*}, s) - \alpha ^{t, \varepsilon , v}_{s} M({}^{s}{ \alpha }^{*}, {}^{s}{h}^{*}, s) |^{2}ds} \\ &\phantom{=:} \times \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } x^{4}ds} \\ &\phantom{=:} + \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } ( \tilde{\mathbb{E}}^{*}_{t}[|\tilde{X}^{t,\varepsilon ,v,*}_{s}|^{2} ] - x^{2} )^{2}ds} \\ & \phantom{=:} \quad \times \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } | \alpha ^{t, \varepsilon , v}_{s} M({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t,\varepsilon ,v,*}, s) |^{2}ds} \\ &\longrightarrow 0, \end{aligned}$$

where the last equality is due to (3.6) and continuity of \(| \tilde{\mathbb{E}}^{*}_{t}[|\tilde{X}^{t,\varepsilon ,v,*}_{s}|^{2} ] - x^{2} |^{2}\) with respect to time \(s\). Similarly, we have

$$\begin{aligned} & \frac{1}{\varepsilon }\bigg| \tilde{\mathbb{E}}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \Big(\alpha ^{t, \varepsilon , v}_{s} \big( \Gamma ({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t, \varepsilon ,v,*}, s) \tilde{X}^{t,\varepsilon ,v,*}_{s} x - \Gamma ({}^{s}{ \alpha }^{*}, {}^{s}{h}^{*}, s) x^{2} \big) \Big)ds \bigg] \bigg| \\ &\leq \frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } | \alpha ^{t, \varepsilon , v}_{s} \Gamma ({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{ \tilde{h}}^{t,\varepsilon ,v,*}, s) \tilde{\mathbb{E}}^{*}_{t}[ \tilde{X}^{t,\varepsilon ,v,*}_{s}] x - \alpha ^{t, \varepsilon , v}_{s} \Gamma ({}^{s}{\alpha }^{*}, {}^{s}{h}^{*}, s) x^{2} | ds \\ &\leq x \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } | \alpha ^{t, \varepsilon , v}_{s} \Gamma ({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t,\varepsilon ,v,*}, s) x - \alpha ^{t, \varepsilon , v}_{s} \Gamma ({}^{s}{\alpha }^{*}, {}^{s}{h}^{*}, s) x |^{2}ds} \\ & \phantom{=:}+ x \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } ( \tilde{\mathbb{E}}^{*}_{t}[|\tilde{X}^{t,\varepsilon ,v,*}_{s}|] - x )^{2}ds} \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } |\alpha ^{t, \varepsilon , v}_{s} \Gamma ({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{ \tilde{h}}^{t,\varepsilon ,v,*}, s) |^{2}ds} \\ & \longrightarrow 0 \end{aligned}$$

and

$$\begin{aligned} &\frac{1}{\varepsilon }\bigg| \tilde{\mathbb{E}}^{*}_{t}\bigg[\int _{t}^{t+ \varepsilon }\Big( \alpha ^{t, \varepsilon , v}_{s} \big( N({}^{s}{ \alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t,\varepsilon ,v,*}, s) \tilde{X}^{t,\varepsilon ,v,*}_{s} \tilde{\mathbb{E}}^{*}_{t}[ \tilde{X}^{t,\varepsilon ,v,*}_{s}] \\ &\qquad \qquad \qquad - N({}^{s}{\alpha }^{*}, {}^{s}{h}^{*}, s) x^{2} \big) \Big)ds \bigg]\bigg| \\ & \leq \frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } | \alpha ^{t, \varepsilon , v}_{s} N({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{ \tilde{h}}^{t,\varepsilon ,v,*}, s) (\tilde{\mathbb{E}}^{*}_{t}[ \tilde{X}^{t,\varepsilon ,v,*}_{s}])^{2} - \alpha ^{t, \varepsilon , v}_{s} N({}^{s}{\alpha }^{*}, {}^{s}{h}^{*}, s) x^{2} |ds \\ & \leq x^{2} \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } | \alpha ^{t, \varepsilon , v}_{s} N({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t,\varepsilon ,v,*}, s) - \alpha ^{t, \varepsilon , v}_{s} N({}^{s}{\alpha }^{*}, {}^{s}{h}^{*}, s) |^{2}ds} \\ & \phantom{=:} + \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } \big( ( \tilde{\mathbb{E}}^{*}_{t}[\tilde{X}^{t,\varepsilon ,v,*}_{s}])^{2} - x^{2} \big)^{2}ds} \\ &\phantom{=:}\quad \times \sqrt{\frac{1}{\varepsilon } \int _{t}^{t+\varepsilon } | \alpha ^{t, \varepsilon , v}_{s} N({}^{s}{\alpha }^{t, \varepsilon , v}, {}^{s}{\tilde{h}}^{t,\varepsilon ,v,*}, s) |^{2}ds} \\ & \longrightarrow 0. \end{aligned}$$

Because \(\eta \) is essentially bounded, (3.7) follows. □

1.2 A.2 Proof of Lemma 3.4

Proof

Under the measure \(\mathbb{Q}^{*}\), we define

$$\begin{aligned} d L^{t,\varepsilon ,v}_{s} &= \Big(r_{s} + \big(\theta _{s} + \tilde{h}^{*}({}^{s}{\alpha }^{t,\varepsilon ,v}, s)\big)'\alpha ^{t, \varepsilon ,v}_{s}\Big) L^{t,\varepsilon ,v}_{s}ds + (\alpha ^{t, \varepsilon ,v}_{s} L^{t,\varepsilon ,v}_{s})' d W^{*}_{s}, \\ L^{t,\varepsilon ,v}_{t} &= x, \end{aligned}$$
(A.1)

and

$$\begin{aligned} \textstyle\begin{array}{rcl} d\ell ^{\varepsilon }(s;t) & = & -\bigg( \big(r_{s} + (\theta _{s} + \tilde{h}^{t,\varepsilon ,v,*}_{s})'\alpha ^{t, \varepsilon , v}_{s} \big) \ell ^{\varepsilon }(s;t) + (\alpha ^{t, \varepsilon , v}_{s})' z^{ \varepsilon }(s;t) \\ & &\displaystyle\,\,\, \quad -\frac{1}{\xi } (\tilde{h}^{t,\varepsilon ,v,*}_{s})' \tilde{h}^{t,\varepsilon ,v,*}_{s} L^{t,\varepsilon ,v}_{s} \bigg)ds + z^{ \varepsilon }(s;t)'dW^{*}_{s},\qquad s\in [t,T], \\ \ell ^{\varepsilon }(T;t) &= & L^{t,\varepsilon ,v}_{T} - \mathbb{E}^{*}_{t}[L^{t, \varepsilon ,v}_{T}] - \mu _{1} x. \end{array}\displaystyle \end{aligned}$$

By similar arguments as in Lemma 3.1, the expectations of \(\tilde{X}^{t,\varepsilon ,v,*}_{s}p^{\varepsilon }(s;t)\) and \((\tilde{X}^{t,\varepsilon ,v,*})^{2}\) under \(\tilde{\mathbb{Q}}^{*}\) are equal to the expectations of \(L^{t,\varepsilon ,v}_{s}\ell ^{\varepsilon }(s;t)\) and \((L^{t,\varepsilon ,v})^{2}\) under \(\mathbb{Q}^{*}\). Thus we have

$$\begin{aligned} & \tilde{\mathbb{E}}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \Lambda ^{\varepsilon }(s;t) {'\eta } ds \bigg] \\ & = \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \bigg( \alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s} \ell ^{\varepsilon }(s;t)- \frac{1}{\xi } ( L^{t,\varepsilon ,v}_{s})^{2} \tilde{h}^{t,\varepsilon ,v,*}_{s} {\bigg) '\eta } ds \bigg]. \end{aligned}$$

We notice that

$$\begin{aligned} & \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \bigg| \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \bigg( \alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s} \ell ^{\varepsilon }(s;t) - \frac{1}{\xi } ( L^{t,\varepsilon ,v}_{s})^{2} \tilde{h}^{*}({}^{s}{ \alpha }^{t,\varepsilon ,v} ,s) \bigg) '{\eta } ds\bigg] \\ & - \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \bigg( \alpha ^{*}_{s} X^{*}_{s} p(s;t)- \frac{1}{\xi } (X^{*}_{s})^{2} \tilde{h}^{*}({}^{s}{\alpha }^{*} ,s) \bigg) '{\eta } ds \bigg]\bigg| \\ &\leq \lim _{\varepsilon \downarrow 0} \frac{K}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \bigg| \alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s} \ell ^{\varepsilon }(s;t)- \frac{1}{\xi } ( L^{t,\varepsilon ,v}_{s})^{2} \tilde{h}^{*}({}^{s}{ \alpha }^{t,\varepsilon ,v} ,s) \\ & \hspace{3cm} - \alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s} p(s;t) + \frac{1}{\xi } ( L^{t,\varepsilon ,v}_{s})^{2} \tilde{h}^{*}({}^{s}{ \alpha }^{*} ,s) \\ & \hspace{3cm} + \alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s} p(s;t) - \frac{1}{\xi } ( L^{t,\varepsilon ,v}_{s})^{2} \tilde{h}^{*}({}^{s}{ \alpha }^{*} ,s) \\ & \hspace{3cm} - \alpha ^{*}_{s} X^{*}_{s} p(s;t) + \frac{1}{\xi } (X^{*}_{s})^{2} \tilde{h}^{*}({}^{s}{\alpha }^{*} ,s) \bigg| ds \bigg] \\ & \leq K \lim _{\varepsilon \downarrow 0} \bigg( \sqrt{ \frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+ \varepsilon } |\alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s}|^{2} ds \bigg]} \sqrt{\frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } |\ell ^{\varepsilon }(s;t) - p(s;t)|^{2} ds \bigg] } \\ & \phantom{=:}\quad \qquad + \sqrt{\frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } |\tilde{h}^{*}({}^{s}{\alpha }^{*}, s) |^{2} ds \bigg]} \\ & \phantom{=:} \quad \qquad \quad \times \sqrt{\frac{1}{\varepsilon } \mathbb{E}^{*}_{t} \bigg[\int _{t}^{t+\varepsilon } \bigg|\frac{1}{\xi } ( L^{t, \varepsilon ,v}_{s})^{2} - \frac{1}{\xi } (X^{*}_{s})^{2}\bigg|^{2} ds \bigg] } \\ & \phantom{=:} \quad \qquad + \frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+ \varepsilon } \bigg| \frac{1}{\xi } ( L^{t,\varepsilon ,v}_{s})^{2} \tilde{h}^{*}({}^{s}{\alpha }^{t,\varepsilon ,v} ,s) - \alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s} p(s;t) \bigg| ds \bigg] \\ & \phantom{=:}\quad \qquad + \frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+ \varepsilon } \bigg| \frac{1}{\xi } ( L^{t,\varepsilon ,v}_{s})^{2} \tilde{h}^{*}({}^{s}{\alpha }^{*} ,s) - \alpha ^{*}_{s} X^{*}_{s} p(s;t) \bigg| ds \bigg] \bigg). \end{aligned}$$

By the stability results for BSDEs (see e.g. Yong and Zhou [38, Theorem 7.3.3]), the first term tends to 0. The second term also converges to 0. For the third term,

$$\begin{aligned} & \frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+ \varepsilon } \bigg| \frac{1}{\xi } ( L^{t,\varepsilon ,v}_{s})^{2} \tilde{h}^{*}({}^{s}{\alpha }^{t,\varepsilon ,v} ,s) - \alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s} p(s;t) \bigg| ds \bigg] \\ & = \frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+ \varepsilon } \Big(\big| ( L^{t,\varepsilon ,v}_{s})^{2} \alpha ^{t, \varepsilon ,v}_{s} \big( M({}^{s}{\alpha }^{t,\varepsilon ,v}, s) - N({}^{s}{ \alpha }^{t,\varepsilon ,v}, s) - \Gamma ({}^{s}{\alpha }^{t, \varepsilon ,v}, s)\big) \\ & \phantom{=:} \qquad \qquad \quad \,\,\, \, - \alpha ^{t, \varepsilon , v}_{s} L^{t, \varepsilon ,v}_{s} \\ &\phantom{=:}\,\, \qquad \qquad \quad \, \,\quad \times \big( M({}^{s}{\alpha }^{*}, s) X^{*}_{s} - N({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[ X^{*}_{s}] - \Gamma ({}^{s}{\alpha }^{*}, s) x\big) \big| \Big) ds\bigg] \\ & \leq \sqrt{\frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+ \varepsilon } |\alpha ^{t, \varepsilon , v}_{s} L^{t,\varepsilon ,v}_{s}|^{2} ds \bigg]} \\ & \phantom{=:} \times \bigg( \sqrt{\frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+\varepsilon } \big( |M({}^{s}{\alpha }^{t,\varepsilon ,v}, s) L^{t,\varepsilon ,v}_{s} - M({}^{s}{\alpha }^{*}, s) X^{*}_{s}|^{2} \big) ds \bigg]} \\ & \phantom{=:}\quad \ \ + \sqrt{\frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+ \varepsilon } \big( |N({}^{s}{\alpha }^{t,\varepsilon ,v}, s) L^{t, \varepsilon ,v}_{s} - N({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[ X^{*}_{s}] |^{2} \big) ds \bigg]} \\ & \phantom{=:} \quad\ \ + \sqrt{\frac{1}{\varepsilon } \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{t+ \varepsilon } \big( |\Gamma ({}^{s}{\alpha }^{t,\varepsilon ,v}, s) L^{t, \varepsilon ,v}_{s} - \Gamma ({}^{s}{\alpha }^{*}, s) x|^{2} \big) ds \bigg] } \bigg) \\ & \longrightarrow 0. \end{aligned}$$

Similarly, we can also prove that the fourth term tends to 0. The lemma is proved. □

1.3 A.3 Proof of Lemma 4.2

Proof

By definition,

$$\begin{aligned} &\mathbb{E}^{*}_{t}\bigg[ \int _{t}^{T} \delta H(s, {}^{s}{\alpha }^{*}, X^{*}) ds\bigg] \\ &=\mathbb{E}^{*}_{t}\bigg[ \int _{t}^{T} \bigg( \Big( \big(\theta _{s} + \tilde{h}^{*}({}^{s}{\alpha }^{t,\varepsilon ,v}, s) \big)' \alpha ^{t, \varepsilon , v}_{s} X^{*}_{s} - \big(\theta _{s} + \tilde{h}^{*}({}^{s}{ \alpha }^{*}, s) \big)' \alpha ^{*}_{s} X^{*}_{s} \Big) { p(s;t)} \\ &\phantom{=} \qquad \qquad\ \ -\frac{1}{2\xi } \big( |\tilde{h}^{*}({}^{s}{ \alpha }^{t,\varepsilon ,v}, s)|^{2} - |\tilde{h}^{*}({}^{s}{\alpha }^{*}, s)|^{2} \big) (X^{*}_{s})^{2} \\ &\phantom{=} \qquad \qquad\ \ + X^{*}_{s} k(s;t)' v{\mathbf{1}}_{[t, t+ \varepsilon )}(s) \bigg) ds \bigg] \\ & = \mathbb{E}^{*}_{t}\bigg[ \int _{t}^{T} \bigg( \Big( \big(\theta _{s} + \tilde{h}^{*}({}^{s}{\alpha }^{*}, s)\big) p(s;t) X^{*}_{s} + k(s;t) X^{*}_{s} {\Big)' v} {\mathbf{1}}_{[t, t+\varepsilon )} (s) \\ &\phantom{=} \qquad \qquad\ \ + \big(\tilde{h}^{*}({}^{s}{\alpha }^{t, \varepsilon ,v}, s) - \tilde{h}^{*}({}^{s}{\alpha }^{*}, s)\big)' \\ & \qquad \qquad \quad \quad\ \,\times \Big( \alpha ^{t, \varepsilon , v}_{s} X^{*}_{s} p(s;t) -\frac{1}{2\xi } \big(\tilde{h}^{*}({}^{s}{ \alpha }^{t,\varepsilon ,v}, s) + \tilde{h}^{*}({}^{s}{\alpha }^{*}, s) \big) (X^{*}_{s})^{2} \Big) \bigg) ds \bigg] . \end{aligned}$$

The first term in the second equality can be handled in a straightforward manner. Now we deal with the second term. For simplicity, we introduce the notations

$$\begin{aligned} A^{\varepsilon }_{s} & = \tilde{h}^{*}({}^{s}{\alpha }^{t,\varepsilon ,v}, s) - \tilde{h}^{*}({}^{s}{\alpha }^{*}, s), \\ A_{s} & = \xi v{\mathbf{1}}_{[t,t+\varepsilon )} (s) \big( M({}^{s}{ \alpha }^{*}, s) - N({}^{s}{\alpha }^{*}, s) - \Gamma ({}^{s}{\alpha }^{*}, s) \big), \\ B^{\varepsilon }_{s} &= \alpha ^{t, \varepsilon , v}_{s} X^{*}_{s} p(s;t) -\frac{1}{2\xi } \big(\tilde{h}^{*}({}^{s}{\alpha }^{t,\varepsilon ,v}, s) + \tilde{h}^{*}({}^{s}{\alpha }^{*}, s)\big) (X^{*}_{s})^{2}, \\ B_{s} &= \frac{1}{2} v{\mathbf{1}}_{[t,t+\varepsilon )} (s) \big( M({}^{s}{ \alpha }^{*}, s) - N({}^{s}{\alpha }^{*}, s) - \Gamma ({}^{s}{\alpha }^{*}, s) \big) x^{2}. \end{aligned}$$

Then

$$\begin{aligned} & \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \bigg| \mathbb{E}^{*}_{t}\bigg[\int _{t}^{T} \big((A^{\varepsilon }_{s})' B^{\varepsilon }_{s} - A'_{s} B_{s} \big) ds \bigg]\bigg| \\ & = \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \bigg| \int _{t}^{T} \big((A^{\varepsilon }_{s})' \mathbb{E}^{*}_{t}[B^{\varepsilon }_{s}] - A'_{s} \mathbb{E}^{*}_{t}[B^{\varepsilon }_{s}] + A'_{s} \mathbb{E}^{*}_{t}[B^{\varepsilon }_{s}] - A'_{s} B_{s} \big) ds \bigg| \\ &\leq \lim _{\varepsilon \downarrow 0} \bigg( \sqrt{ \frac{1}{\varepsilon } \int _{t}^{T} |\mathbb{E}^{*}_{t}[B^{\varepsilon }_{s}] |^{2} ds} \sqrt{\frac{1}{\varepsilon } \int _{t}^{T} |A^{\varepsilon }_{s} - A_{s}|^{2} ds} \\ & \phantom{=:} \qquad \ + \sqrt{\frac{1}{\varepsilon } \int _{t}^{T} |A_{s}|^{2} ds} \sqrt{\frac{1}{\varepsilon } \int _{t}^{T} |\mathbb{E}^{*}_{t}[B^{\varepsilon }_{s}] - B_{s}|^{2} ds}\bigg). \end{aligned}$$

Recall the expressions for \(p(s;t)\), \(k(s;t)\) and \(\tilde{h}^{*}({}^{s}{\alpha }^{*}, s)\) in (4.1), (4.2) and (3.9). Note that \(M({}^{s}{\alpha }^{t,\varepsilon , v}, s)\), \(N({}^{s}{\alpha }^{t,\varepsilon , v}, s)\) and \(\Gamma ({}^{s}{\alpha }^{t,\varepsilon , v}, s)\) converge uniformly to \(M({}^{s}{\alpha }^{*}, s)\), \(N({}^{s}{\alpha }^{*}, s)\) and \(\Gamma ({}^{s}{\alpha }^{*}, s)\) by Lemma 3.3. Therefore

$$\begin{aligned} & \frac{1}{\varepsilon } \int _{t}^{T} |\mathbb{E}^{*}_{t}[B^{\varepsilon }_{s}] - B_{s}|^{2} ds \\ &= \frac{1}{\varepsilon } \int _{t}^{T} \bigg|\alpha ^{t, \varepsilon , v}_{s} \big( M({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[ (X^{*}_{s})^{2}] - N({}^{s}{\alpha }^{*}, s) (\mathbb{E}^{*}_{t}[ X^{*}_{s}])^{2} - \Gamma ({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[ X^{*}_{s}] x \big) \\ &\hspace{1.5cm} -\frac{1}{2\xi } \Big( \xi \alpha ^{t, \varepsilon , v}_{s} \big( M({}^{s}{ \alpha }^{t, \varepsilon , v}, s) - N({}^{s}{\alpha }^{t, \varepsilon , v}, s) - \Gamma ({}^{s}{\alpha }^{t, \varepsilon , v}, s) \big) \\ &\hspace{2.5cm} + \xi \alpha ^{*}_{s} \big( M({}^{s}{\alpha }^{*}, s) - N({}^{s}{ \alpha }^{*}, s) - \Gamma ({}^{s}{\alpha }^{*}, s) \big) \Big) \mathbb{E}^{*}_{t}[(X^{*}_{s})^{2}] \\ &\hspace{1.5cm} - \frac{1}{2} v{\mathbf{1}}_{[t,t+\varepsilon )} (s) \big( M({}^{s}{ \alpha }^{*}, s) - N({}^{s}{\alpha }^{*}, s) - \Gamma ({}^{s}{\alpha }^{*}, s) \big) x^{2} \bigg|^{2} ds \\ & \leq \frac{1}{\varepsilon } \int _{t}^{T} \bigg|\alpha ^{t, \varepsilon , v}_{s} M({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[(X^{*}_{s})^{2}] \\ & \hspace{1.5cm} - \frac{1}{2} \big( \alpha ^{t, \varepsilon , v}_{s} M({}^{s}{\alpha }^{t, \varepsilon , v}, s) \mathbb{E}^{*}_{t}[(X^{*}_{s})^{2}] + \alpha ^{*}_{s} M({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[(X^{*}_{s})^{2}] \\ & \hspace{2.2cm} + v{\mathbf{1}}_{[t,t+\varepsilon )} (s) M({}^{s}{\alpha }^{*}, s) x^{2} \big) \bigg|^{2} ds \\ &\quad + \frac{1}{\varepsilon } \int _{t}^{T} \bigg|\alpha ^{t, \varepsilon , v}_{s} N({}^{s}{\alpha }^{*}, s) (\mathbb{E}^{*}_{t}[X^{*}_{s}])^{2} \\ &\hspace{1.7cm} - \frac{1}{2} \big( \alpha ^{t, \varepsilon , v}_{s} N({}^{s}{\alpha }^{t, \varepsilon , v}, s) \mathbb{E}^{*}_{t}[(X^{*}_{s})^{2}] + \alpha ^{*}_{s} N({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[(X^{*}_{s})^{2}] \\ & \hspace{2.5cm} + v{\mathbf{1}}_{[t,t+\varepsilon )} (s) N({}^{s}{\alpha }^{*}, s) x^{2} \big) \bigg|^{2} ds \\ &\quad + \frac{1}{\varepsilon } \int _{t}^{T} \bigg|\alpha ^{t, \varepsilon , v}_{s} \Gamma ({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[X^{*}_{s}] x \\ & \hspace{1.7cm} - \frac{1}{2} \big( \alpha ^{t, \varepsilon , v}_{s} \Gamma ({}^{s}{ \alpha }^{t, \varepsilon , v}, s) \mathbb{E}^{*}_{t}[(X^{*}_{s})^{2}] + \alpha ^{*}_{s} \Gamma ({}^{s}{\alpha }^{*}, s) \mathbb{E}^{*}_{t}[(X^{*}_{s})^{2}] \\ & \hspace{2.5cm} + v{\mathbf{1}}_{[t,t+\varepsilon )} (s) \Gamma ({}^{s}{\alpha }^{*}, s) x^{2} \big) \bigg|^{2} ds \\ & \longrightarrow 0. \end{aligned}$$

The remaining part about \(|A^{\varepsilon }_{s} - A_{s}|^{2}\) can be proved similarly. Therefore,

$$\begin{aligned} & \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \mathbb{E}^{*}_{t} \bigg[\int _{t}^{T} \bigg(\big(\tilde{h}^{*}({}^{s}{\alpha }^{t, \varepsilon ,v}, s) - \tilde{h}^{*}({}^{s}{\alpha }^{*}, s)\big)' \\ & \qquad \qquad \qquad \times \Big( \alpha ^{t, \varepsilon , v}_{s} X^{*}_{s} p(s;t) -\frac{1}{2\xi } \big(\tilde{h}^{*}({}^{s}{\alpha }^{t, \varepsilon ,v}, s) + \tilde{h}^{*}({}^{s}{\alpha }^{*}, s)\big) (X^{*}_{s})^{2} \Big) \bigg)ds \bigg] \!\!\!\!\!\\ & = \frac{\xi }{2} \big( M({}^{t}{\alpha }^{*}, t) - N({}^{t}{\alpha }^{*}, t) - \Gamma ({}^{t}{\alpha }^{*}, t) \big)^{2} |v|^{2} x^{2}. \end{aligned}$$

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Han, B., Pun, C.S. & Wong, H.Y. Robust state-dependent mean–variance portfolio selection: a closed-loop approach. Finance Stoch 25, 529–561 (2021). https://doi.org/10.1007/s00780-021-00457-4

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