Investment–consumption–insurance optimisation problem with multiple habit formation and non-exponential discounting

Abstract

This paper is devoted to an investment–consumption and life insurance problem with habit formation and non-exponential discounting. General utility functions are employed to evaluate non-habitual consumption and bequest. Distinct from Liu et al. in (Math. Control Relat. Fields 10:761–783, 2020) for consumption habit and feedback control, we assume that past consumption and bequest amounts have an interaction in formulating their endogenous reference levels, and we seek open-loop controls for both the pre-commitment solution and the time-consistent solution. Since the model coefficients are allowed to be random, we use the stochastic maximum principle to solve our problems. For both the pre-commitment and the time-consistent solution, an analytical expression is obtained via a system of forward-backward stochastic differential equations. Additionally, when the model coefficients are Markovian, we show that our problem for open-loop control can also be reduced to solving a Hamilton–Jacobi–Bellman equation, and then we introduce a transformation method for solving the equation. In particular, we provide a semi-analytical solution with numerical results based on simulations for the constant relative risk aversion (CRRA) utility with hyperbolic discounting.

Appendices

Appendix A: Proof of Lemma 4.7

For any fixed $\bar{u}, u \in \mathcal{U} ( 0, {x}_{0} )$, by the continuity of $\upsilon$, we obtain

$$\begin{aligned} & \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \bigg| \mathbb{E} \bigg[ \int _{t}^{ t + \varepsilon } {e}^{ - \int _{t}^{s} { \mu }_{v} dv } \bigg( \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } - \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } \bigg) \big( {U}_{1} ( s, \bar{C}_{s} ) - {U}_{1} ( s, {C}_{s} ) \big) ds \bigg| \mathcal{F}_{t} \bigg] \bigg| \\ & \le \bigg( \lim _{\varepsilon \downarrow 0} \sup _{ s \in [ t, t + \varepsilon ] } \bigg| \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } - \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } \bigg| \bigg) \\ & \phantom{=:} \times \bigg( \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \int _{t}^{ t + \varepsilon } \mathbb{E} \big[ | {U}_{1} ( s, \bar{C}_{s} ) | + | {U}_{1} ( s, {C}_{s} ) | \big| \mathcal{F}_{t} \big] ds \bigg) \\ & \le \bigg( \limsup _{ s \downarrow t } \bigg| \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } - \frac{ \upsilon ( t,t ) }{ \upsilon ( t,T ) } \bigg| + \limsup _{ s \downarrow t } \bigg| \frac{ \upsilon ( t,t ) }{ \upsilon ( t,T ) } - \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } \bigg| \bigg) \\ & \phantom{=:} \times \big( | {U}_{1} ( t, \bar{C}_{t} ) | + | {U}_{1} ( t, {C}_{t} ) | \big) \\ & = 0. \end{aligned}$$

Note that the right-continuity of $\{ \mathbb{E} [ | {U}_{1} ( s, \bar{C}_{s} ) | + | {U}_{1} ( s, {C}_{s} ) | | \mathcal{F}_{t} ] \}_{ s \in [ t,T ] }$ follows from the dominated convergence theorem with the right-continuity of $(C, \bar{C})$ and the condition $\mathbb{E} [ \sup _{ s \in [ t,T ] } | {U}_{1} ( s, \bar{C}_{s} ) | + \sup _{ s \in [ t,T ] } | {U}_{1} ( s, {C}_{s} ) | | \mathcal{F}_{t} ] < \infty$ from Definition 3.1. In the same manner,

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \bigg| \mathbb{E} \bigg[ &\int _{t}^{ t + \varepsilon } {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \bigg( \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } - \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } \bigg) \\ & \phantom{=:} \!\qquad \times {\mu }_{s} \big( {U}_{2} ( s, \bar{B}_{s} ) - {U}_{2} ( s, {B}_{s} ) \big) ds \bigg| \mathcal{F}_{t} \bigg] \bigg| = 0. \end{aligned}$$

It follows that

$$\begin{aligned} & \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \big| \big( J ( t; \bar{u} ) - J ( t; \bar{u}^{ t, \varepsilon , u } ) \big) - \big( \tilde{J} ( t; \bar{u} ) - \tilde{J} ( t; \bar{u}^{ t, \varepsilon , u } ) \big) \big| \\ & \begin{aligned} \le \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \bigg| \mathbb{E} \bigg[ & \int _{t}^{ t + \varepsilon } {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \bigg( \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } - \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } \bigg) \\ & \phantom{\int _{t}^{ t + \varepsilon } } \times \big( {U}_{1} ( s, \bar{C}_{s} ) - {U}_{1} ( s, {C}_{s} ) \big) ds \bigg| \mathcal{F}_{t} \bigg] \bigg| \end{aligned} \\ & \quad \begin{aligned} + \lim _{\varepsilon \downarrow 0} \frac{1}{\varepsilon } \bigg| \mathbb{E} \bigg[ & \int _{t}^{ t + \varepsilon } {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \bigg( \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } - \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } \bigg) \\ & \phantom{=:} \quad \, \times {\mu }_{s} \big( {U}_{2} ( s, \bar{B}_{s} ) - {U}_{2} ( s, {B}_{s} ) \big) ds \bigg| \mathcal{F}_{t} \bigg] \bigg| = 0. \end{aligned} \end{aligned}$$

If $\bar{u}$ realises the maximum of $\tilde{J} ( 0; \cdot )$ over $\mathcal{U} ( 0, {x}_{0} )$, due to the dynamic programming principle, we conclude that $\tilde{J} ( t; \bar{u} ) \ge \tilde{J} ( t; \bar{u}^{ t, \varepsilon , u } )$ ℙ-a.s. Hence,

$$ \liminf _{ \varepsilon \downarrow 0 } \frac{1}{\varepsilon } \big( J ( t; \bar{u} ) - J ( t; \bar{u}^{ t, \varepsilon , u } ) \big) = \liminf _{ \varepsilon \downarrow 0 } \frac{1}{\varepsilon } \big( \tilde{J} ( t; \bar{u} ) - \tilde{J} ( t; \bar{u}^{ t, \varepsilon , u } ) \big) \ge 0 \qquad \text{$\mathbb{P}$-a.s.}, $$

which means that $\bar{u}$ is an open-loop Nash equilibrium control for Problem 4.3. □

Appendix B: Proof of (4.4) and (4.5)

We follow the same line of proof as for Yong and Zhou [55, Lemma 3.4.2]. Fix $t \in [ 0,T )$ and an integer $k > 0$. To prove (4.4) and (4.5), it suffices to show that there exists a constant $K > 0$ such that

$$\begin{aligned} \sup _{ s \in [ t,T ] } \mathbb{E} \big[ | {x}_{s} |^{2k} \big| \mathcal{F}_{t} \big] \le K \bigg( & | {x}_{t} |^{2k} + \Big( \int _{t}^{T} \big( \mathbb{E} \big[ | {a}_{s} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } ds \Big)^{2k} \\ & + \Big( \int _{t}^{T} \big( \mathbb{E} \big[ | {b}_{s} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } ds \Big)^{k} \bigg) \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

(B.1)

for $d {x}_{s} = ( {A}_{s} {x}_{s} + {a}_{s} ) ds + ( {B}_{s} {x}_{s} + {b}_{s} ) d {W}_{s}$, where

• $A$ and $B$ are $\mathbb{F}$-adapted, and $| {A}_{s} |, | {B}_{s} | \le L$ for a.e. $s \in [ 0,T ]$, ℙ-a.s.;

• $a, b \in \mathbb{L}_{\mathbb{F}}^{2k} ( [ t,T ] \times \Omega ; \mathbb{R} )$.

Consequently, with $A := r + \eta$, $B \equiv 0$ and $L = \operatorname*{{\mathrm{ess}\sup}}_{ [ 0,T ] \times \Omega } | r + \eta |$, we obtain

$$ \frac{1}{ {\varepsilon }^{k} } \sup _{ s \in [ t,T ] } \mathbb{E} \big[ | {X}^{(1)}_{s} |^{2k} \big| \mathcal{F}_{t} \big] \le K \bigg( \frac{1}{\varepsilon } \int _{v}^{ v + \varepsilon } \big( \mathbb{E} \big[ | {\Pi }_{s} - \tilde{\Pi }_{s} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } ds \bigg)^{k} $$

(B.2)

and

$$\begin{aligned} & \frac{1}{ {\varepsilon }^{2k} } \sup _{ s \in [ t,T ] } \mathbb{E} \big[ | {X}^{(2)}_{s} |^{2k} \big| \mathcal{F}_{t} \big] \\ & \le K \bigg( \frac{1}{\varepsilon } \int _{v}^{ v + \varepsilon } \big( \mathbb{E} \big[ ( | {\Pi }_{s} - \tilde{\Pi }_{s} | + | {C}_{s} - \tilde{C}_{s} | + | {B}_{s} - \tilde{B}_{s} | )^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } ds \bigg)^{2k} \\ & \quad \times \max \bigg\{ \operatorname*{{\mathrm{ess}\sup}}_{ [ 0,T ] \times \Omega } | \theta |^{2k}, \operatorname*{{\mathrm{ess}\sup}}_{ [ 0,T ] \times \Omega } | \phi |^{2k}, \operatorname*{{\mathrm{ess}\sup}}_{ [ 0,T ] \times \Omega } | \varphi |^{2k} \bigg\} . \end{aligned}$$

(B.3)

Using the assumption that $( \Pi - \tilde{\Pi } ) |_{ {S}_{\varepsilon } \times \Omega } =:\zeta \in \mathbb{L}^{2k}_{ \mathcal{F}_{v} } ( \Omega ; \mathbb{R} )$, the right-continuity of $( C,B )$ and the dominated convergence theorem, we obtain (4.4) and (4.5).

Now we show the validity of (B.1). Applying Itô’s rule to ${x}_{s}^{2k}$, we obtain

$$\begin{aligned} \mathbb{E} \big[ | {x}_{s} |^{2k} \big| \mathcal{F}_{t} \big] & = | {x}_{t} |^{2k} + 2k \mathbb{E} \bigg[ \int _{t}^{s} {x}_{v}^{2k-1} ( {A}_{v} {x}_{v} + {a}_{v} ) dv \bigg| \mathcal{F}_{t} \bigg] \\ & \phantom{=:} + k (2k-1) \mathbb{E} \bigg[ \int _{t}^{s} {x}_{v}^{2k-2} ( {B}_{v} {x}_{v} + {b}_{v} )^{2} dv \bigg| \mathcal{F}_{t} \bigg] \\ & \le | {x}_{t} |^{2k} + 2k \mathbb{E} \bigg[ \int _{t}^{s} | {x}_{v} |^{2k-1} ( L | {x}_{v} | + | {a}_{v} | ) dv \bigg| \mathcal{F}_{t} \bigg] \\ & \phantom{=:} + 2k (2k-1) \mathbb{E} \bigg[ \int _{t}^{s} | {x}_{v} |^{2k-2} ( {L}^{2} | {x}_{v} |^{2} + | {b}_{v} |^{2} ) dv \bigg| \mathcal{F}_{t} \bigg] \\ & \le | {x}_{t} |^{2k} + 4 {k}^{2} (L^{2}+1) \mathbb{E} \bigg[ \int _{t}^{s} | {x}_{v} |^{2k} dv \bigg| \mathcal{F}_{t} \bigg] \\ & \phantom{=:} + 2k \mathbb{E} \bigg[ \int _{t}^{s} | {x}_{v} |^{2k-1} | {a}_{v} | dv \bigg| \mathcal{F}_{t} \bigg] \\ & \phantom{=:} + 2k ( 2k-1 ) \mathbb{E} \bigg[ \int _{t}^{s} | {x}_{v} |^{2k-2} | {b}_{v} |^{2} dv \bigg| \mathcal{F}_{t} \bigg] \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

Let ${K}_{0} := 4 {k}^{2} ( L^{2} + 1 ) > 4k \ge 4$. Then

$$\begin{aligned} \mathbb{E} \big[ | {x}_{s} |^{2k} \big| \mathcal{F}_{t} \big] \le | {x}_{t} |^{2k} + {K}_{0} \mathbb{E} \bigg[ \int _{t}^{s} ( & | {x}_{v} |^{2k} + | {x}_{v} |^{2k-1} | {a}_{v} | \\ & \!\!+ | {x}_{v} |^{2k-2} | {b}_{v} |^{2} ) dv \bigg| \mathcal{F}_{t} \bigg] \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

(B.4)

Applying Young’s and Gronwall’s inequalities, we show the essential boundedness of $( \mathbb{E} [ | {x}_{s} |^{2k} | \mathcal{F}_{t} ] )_{ s \in [ t,T ] }$ by

$$\begin{aligned} & \mathbb{E} \big[ | {x}_{s} |^{2k} \big| \mathcal{F}_{t} \big] \\ & \le | {x}_{t} |^{2k} + {K}_{0} \mathbb{E} \bigg[ \int _{t}^{s} \bigg( \frac{6k-3}{2k} | {x}_{v} |^{2k} + \frac{1}{2k} | {a}_{v} |^{2k} + \frac{1}{k} | {b}_{v} |^{2k} \bigg) dv \bigg| \mathcal{F}_{t} \bigg] \\ & \le | {x}_{t} |^{2k} {e}^{ \frac{6k-3}{2k} {K}_{0} ( s-t ) } + {K}_{0} \int _{t}^{s} {e}^{ \frac{6k-3}{2k} {K}_{0} ( v-t ) } \mathbb{E} \bigg[ \frac{1}{2k} | {a}_{v} |^{2k} + \frac{1}{k} | {b}_{v} |^{2k} \bigg| \mathcal{F}_{t} \bigg] dv \\ & \le {K}_{0} {e}^{ 3 {K}_{0} T } \bigg( | {x}_{t} |^{2k} + \int _{t}^{T} \mathbb{E} \big[ | {a}_{v} |^{2k} + | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] dv \bigg), \qquad \text{$\mathbb{P}$-a.s., $\forall s \in [ t,T ]$.} \end{aligned}$$

Let $\delta := 1 / ( 6 {K}_{0} )$ and ${y}_{s} := ( \operatorname*{{\mathrm{ess}\sup}}_{ v \in [ t,s ] } \mathbb{E} [ | {x}_{v} |^{2k} | \mathcal{F}_{t} ] )^{ \frac{1}{2k} }$. For any $s \in [ t, t + \delta ]$, it follows from (B.4) that

$$\begin{aligned} | {y}_{s} |^{2k} & \le | {y}_{t} |^{2k} + {K}_{0} \bigg( | {y}_{s} |^{2k} \delta + \int _{t}^{s} \big( \mathbb{E} \big[ | {x}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{2k-1}{2k} } \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \\ & \phantom{\le | {y}_{t} |^{2k} + {K}_{0} \bigg( } + \int _{t}^{s} \big( \mathbb{E} \big[ | {x}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{k-1}{k} } \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg) \\ & \le | {y}_{t} |^{2k} + {K}_{0} \bigg( | {y}_{s} |^{2k} \delta + | {y}_{s} |^{2k-1} \int _{t}^{s} \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \\ & \phantom{\le | {y}_{t} |^{2k} + {K}_{0} \bigg( } + | {y}_{s} |^{2k-2} \int _{t}^{s} \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg) \\ & \le | {y}_{t} |^{2k} + \bigg( {K}_{0} \delta + \frac{2k-1}{2k} \frac{1}{ {6}^{ \frac{2k}{2k-1} } } + \frac{k-1}{k} \frac{1}{ {6}^{ \frac{k}{k-1} } } \bigg) | {y}_{s} |^{2k} \\ & \phantom{=:} + \frac{1}{2k} \bigg( 6 {K}_{0} \int _{t}^{s} \!\! \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \bigg)^{2k} + \frac{1}{k} \bigg( 6 {K}_{0} \int _{t}^{s} \!\! \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg)^{k} \\ & \le | {y}_{t} |^{2k} + \frac{1}{2} | {y}_{s} |^{2k} + \frac{1}{2k} ( 6 {K}_{0} )^{2k} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \bigg)^{2k} \\ & \phantom{=:} + \frac{1}{k} ( 6 {K}_{0} )^{k} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg)^{k} \qquad \text{$\mathbb{P}$-a.s.}, \end{aligned}$$

where the first and third inequalities follow from Hölder’s inequality and Young’s inequality, respectively. Let ${K}_{1} := ( 6 {K}_{0} )^{2k} / k = {24}^{2k} {k}^{4k-1} ( L^{2} + 1 )^{2k}$. Consequently, for $s \in [ t, t + \delta ]$,

$$\begin{aligned} | {y}_{s} |^{2k} & \le 2 | {y}_{t} |^{2k} + {K}_{1} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \bigg)^{2k} \\ & \phantom{=:} + {K}_{1} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg)^{k} \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

In the same manner, for $s \in [ t + \delta , t + 2 \delta ]$, we can obtain

$$\begin{aligned} \mathbb{E} \big[ | {x}_{s} |^{2k} \big| \mathcal{F}_{t} \big] & \le {K}_{0} \mathbb{E} \bigg[ \int _{ t + \delta }^{s} ( | {x}_{v} |^{2k} + | {x}_{v} |^{2k-1} | {a}_{v} | + | {x}_{v} |^{2k-2} | {b}_{v} |^{2} ) dv \bigg| \mathcal{F}_{t} \bigg] \\ & \phantom{=:} + \mathbb{E} \big[ | {x}_{ t + \delta } |^{2k} \big| \mathcal{F}_{t} \big] \qquad \text{$\mathbb{P}$-a.s.}, \end{aligned}$$

$$\begin{aligned} | {y}_{s} |^{2k} & \le | {y}_{ t + \delta } |^{2k} + {K}_{0} \delta | {y}_{s} |^{2k} + {K}_{0} \mathbb{E} \bigg[ \int _{ t + \delta }^{s} \big( | {x}_{v} |^{2k-1} | {a}_{v} | + | {x}_{v} |^{2k-2} | {b}_{v} |^{2} \big) dv \bigg| \mathcal{F}_{t} \bigg] \\ & \le | {y}_{ t + \delta } |^{2k} + \frac{1}{2} | {y}_{s} |^{2k} + \frac{1}{2k} ( 6 {K}_{0} )^{2k} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \bigg)^{2k} \\ & \phantom{=:} + \frac{1}{k} ( 6 {K}_{0} )^{k} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg)^{k} \qquad \text{$\mathbb{P}$-a.s.}, \end{aligned}$$

and hence

$$\begin{aligned} | {y}_{s} |^{2k} & \le 2 | {y}_{ t + \delta } |^{2k} + {K}_{1} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \bigg)^{2k} \\ & \phantom{=:} + {K}_{1} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg)^{k} \\ & \le 4 | {y}_{t} |^{2k} + 3 {K}_{1} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \bigg)^{2k} \\ & \phantom{=:} + 3 {K}_{1} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg)^{k} \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

We can do the same thing on $[ t + 2 \delta , t + 3 \delta ]$ and on $[ t + 3 \delta , t + 4 \delta ]$, and so on. Finally, we end up with

$$\begin{aligned} | {y}_{s} |^{2k} & \le {2}^{N} | {y}_{t} |^{2k} + ( {2}^{N} - 1 ) {K}_{1} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {a}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2k} } dv \bigg)^{2k} \\ & \phantom{=:} + ( {2}^{N} - 1 ) {K}_{1} \bigg( \int _{t}^{T} \big( \mathbb{E} \big[ | {b}_{v} |^{2k} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{k} } dv \bigg)^{k}, \qquad \text{$\mathbb{P}$-a.s., $\forall s \in [ t,T ]$,} \end{aligned}$$

where $N := \min \{ n \in \mathbb{N}: n \ge 6 {K}_{0} (T-t) \}$ so that $(N-1) \delta < T-t \le N \delta$. Define $K := {2}^{ 1 + 6k + 24 {k}^{2} ( L^{2} + 1 ) T } {3}^{2k} {k}^{4k-1} ( L^{2} + 1 )^{2k} = {2}^{ 1 + 6 {K}_{0} T } {K}_{1} > {2}^{N} {K}_{1}$. By the definition of ${y}_{s}$, we obtain the desired estimate (B.1).  □

Appendix C: Proof of Lemma 4.8

By the definition of $J ( t; \cdot )$, we obtain

$$\begin{aligned} & J ( t; \tilde{u}^{ \varepsilon , u } ) - J ( t; \tilde{u} ) \\ & \begin{aligned} = \mathbb{E} \bigg[ & \int _{ {S}_{\varepsilon } } {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } \big( {U}_{1} ( s, {C}_{s} ) + {\mu }_{s} {U}_{2} ( s, {B}_{s} ) \\ & \phantom{\int _{ {S}_{\varepsilon } } {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } \big( } - {U}_{1} ( s, \tilde{C}_{s} ) - {\mu }_{s} {U}_{2} ( s, \tilde{B}_{s} ) \big) ds \\ & + {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \big( {U}_{3} ( T, {X}^{ t,x; \tilde{u}^{ \varepsilon , u } }_{T} ) - {U}_{3} ( T, {X}^{ t,x; \tilde{u} }_{T} ) \big) \bigg| \mathcal{F}_{t} \bigg] \qquad \text{$\mathbb{P}$-a.s.,} \end{aligned} \end{aligned}$$

where

$$\begin{aligned} & {U}_{3} ( T, {X}^{ t,x; \tilde{u}^{ \varepsilon , u } }_{T} ) - {U}_{3} ( T, {X}^{ t,x; \tilde{u} }_{T} ) \\ & = {U}_{3;x} ( T, {X}^{ t,x; \tilde{u} }_{T} ) ( {X}^{(1)}_{T} + {X}^{(2)}_{T} ) + \frac{1}{2} {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) ( {X}^{(1)}_{T} + {X}^{(2)}_{T} )^{2} \\ & \quad + \frac{1}{2} ( {X}^{(1)}_{T} + {X}^{(2)}_{T} )^{3} \int _{0}^{1} {\xi }^{2} {U}_{3;xxx} \big( T, \xi {X}^{ t,x; \tilde{u} }_{T} + ( 1 - \xi ) {X}^{ t,x; \tilde{u}^{ \varepsilon , u } }_{T} \big) d \xi . \end{aligned}$$

By Young’s inequality, (B.2) and (B.3) with $k = 1,2$, we obtain

$$\begin{aligned} & \mathbb{E} \big[ | {U}_{3;x} ( T, {X}^{ t,x; \tilde{u} }_{T} ) | ( | {X}^{(1)}_{T} | + | {X}^{(2)}_{T} | ) + | {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) | | {X}^{(1)}_{T} |^{2} \big| \mathcal{F}_{t} \big] \\ & \le \mathbb{E} \big[ | {U}_{3;x} ( T, {X}^{ t,x; \tilde{u} }_{T} ) |^{2} \big| \mathcal{F}_{t} \big] + \frac{1}{2} \mathbb{E} \big[ | {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) |^{2} \big| \mathcal{F}_{t} \big] \\ & \quad + \frac{1}{2} \mathbb{E} \big[ | {X}^{(1)}_{T} |^{2} \big| \mathcal{F}_{t} \big] + \frac{1}{2} \mathbb{E} \big[ | {X}^{(2)}_{T} |^{2} \big| \mathcal{F}_{t} \big] + \frac{1}{2} \mathbb{E} \big[ | {X}^{(1)}_{T} |^{4} \big| \mathcal{F}_{t} \big] < \infty \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

Applying Hölder’s inequality, (4.4) and (4.5) with $k = 2$, we obtain

$$\begin{aligned} & \lim _{ \varepsilon \downarrow 0 } \frac{1}{\varepsilon } \big| \mathbb{E} [ {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) {X}^{(1)}_{T} {X}^{(2)}_{T} | \mathcal{F}_{t} ] \big| \\ & \le \big( \mathbb{E} \big[ | {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) |^{2} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2} } \Big( \limsup _{ \varepsilon \downarrow 0 } \mathbb{E} \big[ | {X}^{(1)}_{T} |^{4} \big| \mathcal{F}_{t} \big] \Big)^{ \frac{1}{4} } \\ & \phantom{=:} \times \bigg( \limsup _{ \varepsilon \downarrow 0 } \frac{1}{ {\varepsilon }^{4} } \mathbb{E} \big[ | {X}^{(2)}_{T} |^{4} \big| \mathcal{F}_{t} \big] \bigg)^{ \frac{1}{4} } \\ & = 0 \qquad \text{$\mathbb{P}$-a.s.}, \end{aligned}$$

and

$$\begin{aligned} & \lim _{ \varepsilon \downarrow 0 } \frac{1}{\varepsilon } \big| \mathbb{E} [ {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) ( {X}^{(2)}_{T} )^{2} | \mathcal{F}_{t} ] \big| \\ & \le \big( \mathbb{E} \big[ | {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) |^{2} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2} } \bigg( \limsup _{ \varepsilon \downarrow 0 } \frac{1}{ {\varepsilon }^{2} } \mathbb{E} \big[ | {X}^{(2)}_{T} |^{4} \big| \mathcal{F}_{t} \big] \bigg)^{ \frac{1}{2} } = 0 \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

Applying the convexity of ${U}_{3;x} ( T, \cdot )$, Hölder’s and Minkowski’s inequalities, (4.4) and (4.5) with $k = 2$, and the dominated convergence theorem, we have

$$\begin{aligned} & \lim _{ \varepsilon \downarrow 0 } \frac{1}{\varepsilon } \bigg| \mathbb{E} \bigg[ ( {X}^{(1)}_{T} + {X}^{(2)}_{T} )^{3} \int _{0}^{1} { \xi }^{2} {U}_{3;xxx} \big( T, \xi {X}^{ t,x; \tilde{u} }_{T} + ( 1 - \xi ) {X}^{ t,x; \tilde{u}^{ \varepsilon , u } }_{T} \big) d \xi \bigg| \mathcal{F}_{t} \bigg] \bigg| \\ & \begin{aligned} \le \lim _{ \varepsilon \downarrow 0 } \frac{1}{\varepsilon } \mathbb{E} \bigg[ & | {X}^{(1)}_{T} + {X}^{(2)}_{T} |^{2} \\ & \times \Big| ( {X}^{(1)}_{T} + {X}^{(2)}_{T} ) \int _{0}^{1} {U}_{3;xxx} \big( T, \xi {X}^{ t,x; \tilde{u} }_{T} + ( 1 - \xi ) {X}^{ t,x; \tilde{u}^{ \varepsilon , u } }_{T} \big) d \xi \Big| \bigg| \mathcal{F}_{t} \bigg] \end{aligned} \\ & \begin{aligned} \le \lim _{ \varepsilon \downarrow 0 } \bigg( & \frac{1}{\varepsilon } \big( \mathbb{E} \big[ | {X}^{(1)}_{T} + {X}^{(2)}_{T} |^{4} \big| \mathcal{F}_{t} \big] \big)^{ \frac{1}{2} } \\ &\! \times \Big( \mathbb{E} \big[ | {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) - {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u}^{ \varepsilon , u } }_{T} ) |^{2} \big| \mathcal{F}_{t} \big] \Big)^{ \frac{1}{2} } \bigg) \end{aligned} \\ & \le \bigg( \Big( \limsup _{ \varepsilon \downarrow 0 } \frac{1}{ {\varepsilon }^{2} } \mathbb{E} \big[ | {X}^{(1)}_{T} |^{4} \big| \mathcal{F}_{t} \big] \Big)^{ \frac{1}{4} } + \Big( \limsup _{ \varepsilon \downarrow 0 } \frac{1}{ {\varepsilon }^{2} } \mathbb{E} \big[ | {X}^{(2)}_{T} |^{4} \big| \mathcal{F}_{t} \big] \Big)^{ \frac{1}{4} } \bigg)^{2} \\ & \phantom{=:} \times \Big( \mathbb{E} \Big[ \lim _{ \varepsilon \downarrow 0 } | {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) - {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u}^{ \varepsilon , u } }_{T} ) |^{2} \Big| \mathcal{F}_{t} \Big] \Big)^{ \frac{1}{2} } \\ & = 0 \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

Collecting the results, we obtain the first equality in Lemma 4.8.

By Itô’s rule, combining (4.6) and (4.2) yields

$$\begin{aligned} \mathbb{E} \big[ {e}^{ - \int _{t}^{T} {\mu }_{v} dv } {U}_{3;x} ( T, {X}^{ t,x; \tilde{u} }_{T} ) {X}^{(1)}_{T} \big| \mathcal{F}_{t} \big] = \mathbb{E} \bigg[ \int _{ {S}_{\varepsilon } } {e}^{ - \int _{t}^{s} { \mu }_{v} dv } \mathfrak{Y}^{(t)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} ) ds \bigg| \mathcal{F}_{t} \bigg]. \end{aligned}$$

Combining (4.6) and (4.3) yields

$$\begin{aligned} & \mathbb{E} \big[ {e}^{ - \int _{t}^{T} {\mu }_{v} dv } {U}_{3;x} ( T, {X}^{ t,x; \tilde{u} }_{T} ) {X}^{(2)}_{T} \big| \mathcal{F}_{t} \big] \\ & = \mathbb{E} \bigg[ \int _{ {S}_{\varepsilon } } {e}^{ - \int _{t}^{s} {\mu }_{v} dv } {Y}^{(t)}_{s} \big( ( {\Pi }_{s} - \tilde{\Pi }_{s} ) { \theta }_{s} - ( {C}_{s} - \tilde{C}_{s} ) {\phi }_{s} - ( {B}_{s} - \tilde{B}_{s} ) {\varphi }_{s} \big) ds \bigg| \mathcal{F}_{t} \bigg]. \end{aligned}$$

Combining (4.7) and (4.2) yields

$$\begin{aligned} & \mathbb{E} \big[ {e}^{ - \int _{t}^{T} {\mu }_{v} dv } {U}_{3;xx} ( T, {X}^{ t,x; \tilde{u} }_{T} ) ( {X}^{(1)}_{T} )^{2} \big| \mathcal{F}_{t} \big] \\ & = \mathbb{E} \bigg[ \int _{ {S}_{\varepsilon } } {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \big( {Q}^{(t)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} )^{2} + 2 \mathfrak{Q}^{(t)}_{s} {X}^{(1)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} ) \big) ds \bigg| \mathcal{F}_{t} \bigg] \\ & = \mathbb{E} \bigg[ \int _{ {S}_{\varepsilon } } {e}^{ - \int _{t}^{s} {\mu }_{v} dv } {Q}^{(t)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} )^{2} ds \bigg| \mathcal{F}_{t} \bigg] + o ( \varepsilon ), \end{aligned}$$

of which the last equality arises from

$$\begin{aligned} & \lim _{ \varepsilon \downarrow 0 } \frac{1}{\varepsilon } \bigg| \mathbb{E} \bigg[ \int _{ {S}_{\varepsilon } } {e}^{ - \int _{t}^{s} { \mu }_{v} dv } \mathfrak{Q}^{(t)}_{s} {X}^{(1)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} ) ds \bigg| \mathcal{F}_{t} \bigg] \bigg| \\ & \begin{aligned} \le \lim _{ \varepsilon \downarrow 0 } \bigg( & \Big( \int _{ {S}_{ \varepsilon } } \mathbb{E} \big[ | \mathfrak{Q}^{(t)}_{s} |^{2} \big| \mathcal{F}_{t} \big] ds \Big)^{ \frac{1}{2} } \Big( \frac{1}{ {\varepsilon }^{3} } \int _{ {S}_{\varepsilon } } \mathbb{E} \big[ | {X}^{(1)}_{s} |^{4} \big| \mathcal{F}_{t} \big] ds \Big)^{ \frac{1}{4} } \\ & \times \Big( \frac{1}{\varepsilon } \int _{ {S}_{\varepsilon } } \mathbb{E} \big[ | {\Pi }_{s} - \tilde{\Pi }_{s} |^{4} \big| \mathcal{F}_{t} \big] ds \Big)^{ \frac{1}{4} } \bigg) \end{aligned} \\ & \le \bigg( \lim _{ \varepsilon \downarrow 0 } \int _{ {S}_{ \varepsilon } } \mathbb{E} \big[ | \mathfrak{Q}^{(t)}_{s} |^{2} \big| \mathcal{F}_{t} \big] ds \bigg)^{ \frac{1}{2} } \bigg( \limsup _{ \varepsilon \downarrow 0 } \frac{1}{ {\varepsilon }^{2} } \sup _{ s \in [ t,T ] } \mathbb{E} \big[ | {X}^{(1)}_{s} |^{2} \big| \mathcal{F}_{t} \big] \bigg)^{ \frac{1}{4} } ( \mathbb{E} [ {\zeta }^{4} | \mathcal{F}_{t} ] )^{ \frac{1}{4} } \\ & = 0. \end{aligned}$$

Collecting the above equalities, we obtain

$$\begin{aligned} & J \big( t; \tilde{u}^{ \varepsilon , u} \big) - J ( t; \tilde{u} ) \\ & \begin{aligned} = \mathbb{E} \bigg[ \int _{ {S}_{\varepsilon } } \bigg( & \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } \big( {U}_{1} ( s, {C}_{s} ) + {\mu }_{s} {U}_{2} ( s, {B}_{s} ) - {U}_{1} ( s, \tilde{C}_{s} ) - {\mu }_{s} {U}_{2} ( s, \tilde{B}_{s} ) \big) \\ & + {Y}^{(t)}_{s} \big( ( {\Pi }_{s} - \tilde{\Pi }_{s} ) {\theta }_{s} - ( {C}_{s} - \tilde{C}_{s} ) {\phi }_{s} - ( {B}_{s} - \tilde{B}_{s} ) {\varphi }_{s} \big) \\ & + \mathfrak{Y}^{(t)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} ) + \frac{1}{2} {Q}^{(t)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} )^{2} \bigg) {e}^{ - \int _{t}^{s} {\mu }_{v} dv } ds \bigg| \mathcal{F}_{t} \bigg] + o ( \varepsilon ). \end{aligned} \end{aligned}$$

Hence, the second equality in Lemma 4.8 follows.  □

Appendix D: Proof of Theorem 4.9

Since (4.8) holds and ${U}_{1} ( s, \cdot )$ and ${U}_{2} ( s, \cdot )$ are strictly concave, the equivalence between (4.9) and the combination of (4.10)–(4.12) can be shown by considering the first-order conditions. We prove the connection between (4.9) and a $( t,x )$-pre-commitment control as follows.

The “only if” part is from a perturbation argument. Let ${S}_{\varepsilon } = [ s, s + \varepsilon )$ with a sufficiently small $\varepsilon > 0$ for fixed $s \in [ t,T )$. By Lemma 4.8, for any spike modification $\tilde{u}^{\varepsilon , u}$ of the pre-commitment control $\tilde{u}$, using $( \Pi - \tilde{\Pi } ) |_{ {S}_{\varepsilon } \times \Omega } =:\zeta \in \mathbb{L}^{4}_{ \mathcal{F}_{s} } ( \Omega ; \mathbb{R} )$, dividing both sides of $J ( t; \tilde{u}^{\varepsilon , u} ) - J ( t; \tilde{u} ) \le 0$ by $\varepsilon$ and letting $\varepsilon$ tend to 0, we obtain

$$\begin{aligned} \mathbb{E} \bigg[ \bigg( & \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } \big( {U}_{1} ( s, {C}_{s} ) + {\mu }_{s} {U}_{2} ( s, {B}_{s} ) - {U}_{1} ( s, \tilde{C}_{s} ) - {\mu }_{s} {U}_{2} ( s, \tilde{B}_{s} ) \big) \\ & + {Y}^{(t)}_{s} \big( ( {\Pi }_{s} - \tilde{\Pi }_{s} ) {\theta }_{s} - ( {C}_{s} - \tilde{C}_{s} ) {\phi }_{s} - ( {B}_{s} - \tilde{B}_{s} ) {\varphi }_{s} \big) \\ & + \mathfrak{Y}^{(t)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} ) + \frac{1}{2} {Q}^{(t)}_{s} ( {\Pi }_{s} - \tilde{\Pi }_{s} )^{2} \bigg) {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \bigg| \mathcal{F}_{t} \bigg] \le 0 \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

Consequently,

$$\begin{aligned} \mathbb{E} \big[ \big( & \mathbb{H} ( t, s, {X}^{ t,x; \tilde{u} }_{s}, {\Pi }_{s}, {C}_{s}, {B}_{s}, {Y}^{(t)}_{s}, {Q}^{(t)}_{s} ) \\ & - \mathbb{H} ( t, s, {X}^{ t,x; \tilde{u} }_{s}, \tilde{\Pi }_{s}, \tilde{C}_{s}, \tilde{B}_{s}, {Y}^{(t)}_{s}, {Q}^{(t)}_{s} ) \\ & + ( \mathfrak{Y}^{(t)}_{s} - {Q}^{(t)}_{s} \tilde{\Pi }_{s} ) ( { \Pi }_{s} - \tilde{\Pi }_{s} ) \big) {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \big] \le 0. \end{aligned}$$

For any $A \in \mathcal{F}_{s}$ and $( \Pi , C, B ) \in \mathbb{R}^{3}$ in a neighbourhood of $( \tilde{\Pi }_{s}, \tilde{C}_{s}, \tilde{B}_{s} )$, we let $( {\Pi }_{s}, {C}_{s}, {B}_{s} ) = ( \Pi , C, B ) {\mathbf{1}}_{A} + ( \tilde{\Pi }_{s}, \tilde{C}_{s}, \tilde{B}_{s} ) {\mathbf{1}}_{\Omega \setminus A}$. Then

$$\begin{aligned} \mathbb{E} \big[ \big( & \mathbb{H} ( t, s, {X}^{ t,x; \tilde{u} }_{s}, \Pi , C, B, {Y}^{(t)}_{s}, {Q}^{(t)}_{s} ) \\ & - \mathbb{H} ( t, s, {X}^{ t,x; \tilde{u} }_{s}, \tilde{\Pi }_{s}, \tilde{C}_{s}, \tilde{B}_{s}, {Y}^{(t)}_{s}, {Q}^{(t)}_{s} ) \\ & + ( \mathfrak{Y}^{(t)}_{s} - {Q}^{(t)}_{s} \tilde{\Pi }_{s} ) ( { \Pi } - \tilde{\Pi }_{s} ) \big) {e}^{ - \int _{t}^{s} {\mu }_{v} dv } {\mathbf{1}}_{A} \big] \le 0. \end{aligned}$$

Because $A \in \mathcal{F}_{s}$ is arbitrary, it follows that

$$\begin{aligned} & \mathbb{H} ( t, s, {X}^{ t,x; \tilde{u} }_{s}, \tilde{\Pi }_{s}, \tilde{C}_{s}, \tilde{B}_{s}, {Y}^{(t)}_{s}, {Q}^{(t)}_{s} ) + ( \mathfrak{Y}^{(t)}_{s} - {Q}^{(t)}_{s} \tilde{\Pi }_{s} \big) \tilde{\Pi }_{s} \\ & \ge \mathbb{H} ( t, s, {X}^{ t,x; \tilde{u} }_{s}, \Pi , C, B, {Y}^{(t)}_{s}, {Q}^{(t)}_{s} ) + ( \mathfrak{Y}^{(t)}_{s} - {Q}^{(t)}_{s} \tilde{\Pi }_{s} ) \Pi \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

Since the right-hand side of the above inequality is strictly concave in $( \Pi , C, B )$, the local maximiser $( \tilde{\Pi }_{s}, \tilde{C}_{s}, \tilde{B}_{s} )$ is also the global maximiser. Notably, when ${u}_{s} = \tilde{u}_{s}$, the inequality holds as an equality. Thus (4.9) follows.

In terms of the “if” part, let us proceed with the concavity of ${U}_{3} ( T, \cdot )$. For any $u \in \mathcal{U} ( t,x )$,

$$ {U}_{3} ( T, {X}^{t,x;u}_{T} ) \le {U}_{3} ( T, {X}^{ t,x; \hat{u} }_{T} ) + {U}_{3;x} ( T, {X}^{ t,x; \hat{u} }_{T} ) ( {X}^{t,x;u}_{T} - {X}^{ t,x; \hat{u} }_{T} ). $$

Combining the BSDE (4.6) with ${X}^{t,x;u}_{t} - {X}^{ t,x; \hat{u} }_{t} = 0$ and

$$\begin{aligned} d ( {X}^{t,x;u}_{s} - {X}^{ t,x; \hat{u} }_{s} ) = \big( & ( {X}^{t,x;u}_{s} - {X}^{ t,x; \hat{u} }_{s} ) ( {r}_{s} + {\eta }_{s} ) + ( {\Pi }_{s} - \hat{\Pi }_{s} ) {\theta }_{s} \\ & - ( {C}_{s} - \hat{C}_{s} ) {\phi }_{s} - ( {B}_{s} - \hat{B}_{s} ) { \varphi }_{s} \big) ds + ( {\Pi }_{s} - \hat{\Pi }_{s} ) d {W}_{s}, \end{aligned}$$

we obtain

$$\begin{aligned} & \mathbb{E} \big[ {e}^{ - \int _{t}^{T} {\mu }_{v} dv } {U}_{3;x} ( T, {X}^{ t,x; \hat{u} }_{T} ) ( {X}^{t,x;u}_{T} - {X}^{ t,x; \hat{u} }_{T} ) \big| \mathcal{F}_{t} \big] \\ & \begin{aligned} = \mathbb{E} \bigg[ \int _{t}^{T} {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \Big( & {Y}^{(t)}_{s} \big( ( {\Pi }_{s} - \hat{\Pi }_{s} ) {\theta }_{s} - ( {C}_{s} - \hat{C}_{s} ) {\phi }_{s} - ( {B}_{s} - \hat{B}_{s} ) { \varphi }_{s} \big) \\ & + \mathfrak{Y}^{(t)}_{s} ( {\Pi }_{s} - \hat{\Pi }_{s} ) \Big) ds \bigg| \mathcal{F}_{t} \bigg] \end{aligned} \\ & \begin{aligned} \le - \mathbb{E} \bigg[ \int _{t}^{T} {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } \big( & {U}_{1} ( s, {C}_{s} ) + {\mu }_{s} {U}_{2} ( s, {B}_{s} ) \\ & - {U}_{1} ( s, \hat{C}_{s} ) - {\mu }_{s} {U}_{2} ( s, \hat{B}_{s} ) \big) ds \bigg| \mathcal{F}_{t} \bigg] \qquad \text{$\mathbb{P}$-a.s.} \end{aligned} \end{aligned}$$

of which the inequality is derived from (4.9), or the equivalent statement that

$$\begin{aligned} 0 & \ge {Y}^{(t)}_{s} \big( ( {\Pi }_{s} - \hat{\Pi }_{s} ) {\theta }_{s} - ( {C}_{s} - \hat{C}_{s} ) {\phi }_{s} - ( {B}_{s} - \hat{B}_{s} ) { \varphi }_{s} \big) \\ & \quad + \mathfrak{Y}^{(t)}_{s} ( {\Pi }_{s} - \hat{\Pi }_{s} ) + \frac{1}{2} {Q}^{(t)}_{s} ( {\Pi }_{s} - \hat{\Pi }_{s} )^{2} \\ & \quad + \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } \big( {U}_{1} ( s, {C}_{s} ) + {\mu }_{s} {U}_{2} ( s, {B}_{s} ) - {U}_{1} ( s, \hat{C}_{s} ) - {\mu }_{s} {U}_{2} ( s, \hat{B}_{s} ) \big) \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

Hence we have

$$\begin{aligned} J ( t;u ) &= \mathbb{E} \bigg[ \int _{t}^{T} {e}^{ - \int _{t}^{s} { \mu }_{v} dv } \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } \big( {U}_{1} ( s, {C}_{s} ) + {\mu }_{s} {U}_{2} ( s, {B}_{s} ) \big) ds \\ & \phantom{= \mathbb{E} \bigg[} + {e}^{ - \int _{t}^{T} {\mu }_{v} dv } {U}_{3} \big( T, {X}^{t,x;u}_{T} \big) \bigg| \mathcal{F}_{t} \bigg] \\ & \le \mathbb{E} \bigg[ \int _{t}^{T} {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \frac{ \upsilon ( t,s ) }{ \upsilon ( t,T ) } \big( {U}_{1} ( s, \hat{C}_{s} ) + {\mu }_{s} {U}_{2} ( s, \hat{B}_{s} ) \big) ds \\ & \phantom{= \mathbb{E} \bigg[} + {e}^{ - \int _{t}^{T} {\mu }_{v} dv } {U}_{3} ( T, {X}^{ t,x; \hat{u} }_{T} ) \bigg| \mathcal{F}_{t} \bigg] \\ &= J ( t; \hat{u} ) \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

 □

Appendix E: Concavity of the value function

First, $V ( T,x ) = {U}_{3} ( T,x )$, which is strictly concave in $x$. In the following discussion, we consider $t \in [ 0,T )$, ${x}_{1}, {x}_{2} \in \mathbb{R}_{+}$ and ${\lambda }_{1}, {\lambda }_{2} \in ( 0,1 )$ satisfying ${\lambda }_{1} + {\lambda }_{2} = 1$. For any ${u}^{(k)} = ( {\Pi }^{(k)}, {C}^{(k)}, {B}^{(k)} ) \in \mathcal{U} ( t, {x}_{k} )$, $k=1,2$, it follows from (3.4) that

$$ {X}^{ t, {\lambda }_{1} {x}_{1} + {\lambda }_{2} {x}_{2}; {\lambda }_{1} {u}^{(1)} + {\lambda }_{2} {u}^{(2)} } = {\lambda }_{1} {X}^{ t, {x}_{1}; {u}^{(1)} } + {\lambda }_{2} {X}^{ t, {x}_{2}; {u}^{(2)} }. $$

Moreover, it can be easily verified that

$$ {\lambda }_{1} {u}^{(1)} + {\lambda }_{2} {u}^{(2)} \in \mathbb{L}^{4}_{ \mathbb{F}} ( [ t,T ] \times \Omega ; \mathbb{R} ) \times \mathbb{L}^{1}_{ \mathbb{F}} ( [ t,T ] \times \Omega ; \mathbb{R}_{+} ) \times \mathbb{L}^{1}_{\mathbb{F}} ( [ t,T ] \times \Omega ; \mathbb{R}_{+} ), $$

$$\begin{aligned} \mathbb{E} \Big[ \sup _{ s \in [ t,T ] } | {\lambda }_{1} {C}^{(1)}_{s} + {\lambda }_{2} {C}^{(2)}_{s} |^{4} \Big| \mathcal{F}_{t} \Big] & \le 8 \mathbb{E} \Big[ {\lambda }_{1}^{4} \sup _{ s \in [ t,T ] } | {C}^{(1)}_{s} |^{4} + {\lambda }_{2}^{4} \sup _{ s \in [ t,T ] } | {C}^{(2)}_{s} |^{4} \Big| \mathcal{F}_{t} \Big] \\ &< \infty \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

and

$$\begin{aligned} & \mathbb{E} \Big[ \sup _{ s \in [ t,T ] } | {U}_{1} ( s, {\lambda }_{1} {C}^{(1)}_{s} + {\lambda }_{2} {C}^{(2)}_{s} ) | \Big| \mathcal{F}_{t} \Big] \\ & \le \mathbb{E} \Big[ \sup _{ s \in [ t,T ] } | {U}_{1} ( s, {C}^{(1)}_{s} \vee {C}^{(2)}_{s} ) | \vee \sup _{ s \in [ t,T ] } | {U}_{1} ( s, {C}^{(1)}_{s} \wedge {C}^{(2)}_{s} ) | \Big| \mathcal{F}_{t} \Big] \\ & \le \mathbb{E} \Big[ \sup _{ s \in [ t,T ] } | {U}_{1} ( s, {C}^{(1)}_{s} ) | \Big| \mathcal{F}_{t} \Big] + \mathbb{E} \Big[ \sup _{ s \in [ t,T ] } | {U}_{1} ( s, {C}^{(2)}_{s} ) | \Big| \mathcal{F}_{t} \Big] < \infty \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

By a similar argument, we can obtain

$$\begin{aligned} \mathbb{E} \Big[ & \sup _{ s \in [ t,T ] } | {U}_{1} ( s, {\lambda }_{1} {C}^{(1)}_{s} + {\lambda }_{2} {C}^{(2)}_{s} ) | + \sup _{ s \in [ t,T ] } | {U}_{2} ( s, {\lambda }_{1} {B}^{(1)}_{s} + {\lambda }_{2} {B}^{(2)}_{s} ) | \\ & + | {U}_{3} ( T, {X}^{ t, {\lambda }_{1} {x}_{1} + {\lambda }_{2} {x}_{2}; {\lambda }_{1} {u}^{(1)} + {\lambda }_{2} {u}^{(2)} }_{T} ) | + | {U}_{3;x} ( T, {X}^{ t, {\lambda }_{1} {x}_{1} + {\lambda }_{2} {x}_{2}; { \lambda }_{1} {u}^{(1)} + {\lambda }_{2} {u}^{(2)} }_{T} ) |^{2} \\ & + | {U}_{3;xx} ( T, {X}^{ t, {\lambda }_{1} {x}_{1} + {\lambda }_{2} {x}_{2}; {\lambda }_{1} {u}^{(1)} + {\lambda }_{2} {u}^{(2)} }_{T} ) |^{2} + \sup _{ s \in [ t,T ] } | {\lambda }_{1} {\Pi }^{(1)}_{s} + { \lambda }_{2} {\Pi }^{(2)}_{s} |^{4} \\ & + \sup _{ s \in [ t,T ] } | {\lambda }_{1} {C}^{(1)}_{s} + { \lambda }_{2} {C}^{(2)}_{s} |^{4} + \sup _{ s \in [ t,T ] } | { \lambda }_{1} {B}^{(1)}_{s} + {\lambda }_{2} {B}^{(2)}_{s} |^{4} \Big| \mathcal{F}_{t} \Big] < \infty \qquad \text{$\mathbb{P}$-a.s.} \end{aligned}$$

According to Definition 3.1, ${\lambda }_{1} {u}^{(1)} + {\lambda }_{2} {u}^{(2)} \in \mathcal{U} ( t, {\lambda }_{1} {x}_{1} + {\lambda }_{2} {x}_{2} )$. We assumed that ${u}^{(1)}$ (resp. ${u}^{(2)}$) realises the maximum of the right-hand side of (4.31) with $x = {x}_{1}$ (resp. $x = {x}_{2}$). Due to the strict concavity of the utility functions,

$$\begin{aligned} & {\lambda }_{1} V ( t, {x}_{1} ) + {\lambda }_{2} V ( t, {x}_{2} ) \\ & \begin{aligned} = \mathbb{E} \bigg[ & \int _{t}^{T} {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } \big( {\lambda }_{1} {U}_{1} ( s, {C}^{(1)}_{s} ) + {\lambda }_{2} {U}_{1} ( s, {C}^{(2)}_{s} ) \big) ds \\ & + \int _{t}^{T} {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } {\mu }_{s} \big( { \lambda }_{1} {U}_{2} ( s, {B}^{(1)}_{s} ) + {\lambda }_{2} {U}_{2} ( s, {B}^{(2)}_{s} ) \big) ds \\ & + {e}^{ - \int _{t}^{T} {\mu }_{v} dv } \big( {\lambda }_{1} {U}_{3} ( T, {X}^{ t, {x}_{1}; {u}^{(1)} }_{T} ) + {\lambda }_{2} {U}_{3} ( T, {X}^{ t, {x}_{2}; {u}^{(2)} }_{T} ) \big) \bigg| \mathcal{F}_{t} \bigg] \end{aligned} \\ & \begin{aligned} < \mathbb{E} \bigg[ & \int _{t}^{T} {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } {U}_{1} ( s, { \lambda }_{1} {C}^{(1)}_{s} + {\lambda }_{2} {C}^{(2)}_{s} ) ds \\ & + \int _{t}^{T} {e}^{ - \int _{t}^{s} {\mu }_{v} dv } \frac{ \upsilon ( s,s ) }{ \upsilon ( s,T ) } {\mu }_{s} {U}_{2} ( s, { \lambda }_{1} {B}^{(1)}_{s} + {\lambda }_{2} {B}^{(2)}_{s} ) ds \\ & + {e}^{ - \int _{t}^{T} {\mu }_{v} dv } {U}_{3} ( T, {X}^{ t, { \lambda }_{1} {x}_{1} + {\lambda }_{2} {x}_{2}; {\lambda }_{1} {u}^{(1)} + {\lambda }_{2} {u}^{(2)} }_{T} ) \bigg| \mathcal{F}_{t} \bigg] \end{aligned} \\ & \le V ( t, {\lambda }_{1} {x}_{1} + {\lambda }_{2} {x}_{2} ). \end{aligned}$$

 □

Appendix F: Proof of Theorem 5.1

If $( {\mathrm{{Tr}}} ( X ) )^{2} \ne 4 \det X$, we just need to prove that

$$ X = \big( X - {\mathrm{diag}}( {\chi }_{2}, {\chi }_{1} ) \big)^{-1} {\mathrm{diag}}( {\chi }_{1}, {\chi }_{2} ) \big( X - {\mathrm{diag}}( {\chi }_{2}, {\chi }_{1} ) \big). $$

Let $X^{*}$ denote the adjugate matrix of $X$, i.e., ${X}^{*} X =( \det X) {\mathrm{diag}}( 1,1 )$. Then we have $( X - { \chi }_{k} {\mathrm{diag}}( 1,1 ) ) ( {X}^{*} - {\chi }_{k} {\mathrm{diag}}( 1,1 ) ) = 0$ according to the eigenvalue equation. This implies that the $k$th column of ${X}^{*} - {\mathrm{diag}}( {\chi }_{1}, {\chi }_{2} )$ gives an eigenvector corresponding to the eigenvalue ${\chi }_{k}$. Notably, since $\det ( X - {\mathrm{diag}}( {\chi }, {\chi }_{1} ) ) = 0$ as a linear equation for $\chi$ admits a unique solution $\chi = {\chi }_{1} \ne {\chi }_{2}$, we have

$$ \det \big( {X}^{*} - {\mathrm{diag}}( {\chi }_{1}, {\chi }_{2} ) \big) = \det \big( X - {\mathrm{diag}}( {\chi }_{2}, {\chi }_{1} )\big) \ne 0. $$

Hence

$$\begin{aligned} X & = \big( X^{*} - {\mathrm{diag}}( {\chi }_{1}, {\chi }_{2} ) \big) { \mathrm{diag}}( {\chi }_{1}, {\chi }_{2} ) \big( X^{*} - {\mathrm{diag}}( { \chi }_{1}, {\chi }_{2} ) \big)^{-1} \\ & = \big( X - {\mathrm{diag}}( {\chi }_{2}, {\chi }_{1} ) \big)^{-1} {\mathrm{diag}}( {\chi }_{1}, {\chi }_{2} ) \big( X - {\mathrm{diag}}( {\chi }_{2}, {\chi }_{1} ) \big). \end{aligned}$$

If $( {\mathrm{{Tr}}} ( X ) )^{2} = 4 \det X$ and ${\chi }_{0} = \frac{1}{2} {\mathrm{{Tr}}} ( X )$, the Hamilton–Cayley theorem gives ${X}^{2} - 2 {\chi }_{0} X + {\chi }_{0}^{2} {\mathrm{diag}}( 1,1 ) = 0$. It follows that

$$\begin{aligned} & \frac{d}{dt} \Big( {e}^{ t {\chi }_{0} } {\mathrm{diag}}( 1,1 ) + t {e}^{ t {\chi }_{0} } \big( X - {\chi }_{0} {\mathrm{diag}}( 1,1 ) \big) \Big) \\ &= {e}^{ t {\chi }_{0} } X + t {e}^{ t {\chi }_{0} } \big( {\chi }_{0} X - {\chi }_{0}^{2} {\mathrm{diag}}( 1,1 ) \big) \\ &= \Big( {e}^{ t {\chi }_{0} } {\mathrm{diag}}( 1,1 ) + t {e}^{ t {\chi }_{0} } \big( X - {\chi }_{0} {\mathrm{diag}}( 1,1 ) \big) \Big) X. \end{aligned}$$

Hence we conclude that $\mathfrak{E} ( t; X ) = {e}^{ t {\chi }_{0} } {\mathrm{diag}}( 1,1 ) + t {e}^{ t {\chi }_{0} } ( X - {\chi }_{0} {\mathrm{diag}}( 1,1 ) )$.  □