Optimal convergence trading with unobservable pricing errors

Abstract

We study a dynamic portfolio optimization problem related to convergence trading, which is an investment strategy that exploits temporary mispricing by simultaneously buying relatively underpriced assets and selling short relatively overpriced ones with the expectation that their prices converge in the future. We build on the model of Liu and Timmermann (Rev Financ Stud 26(4):1048–1086, 2013) and extend it by incorporating unobservable Markov-modulated pricing errors into the price dynamics of two co-integrated assets. We characterize the optimal portfolio strategies in full and partial information settings under the assumption of unrestricted and beta-neutral strategies. By using the innovations approach, we provide the filtering equation which is essential for solving the optimization problem under partial information. Finally, in order to illustrate the model capabilities, we provide an example with a two-state Markov chain.

Proofs

Proof of Theorem 1

Existence We denote by ${\mathcal {L}}^h_{\mathbb {G}}$ the generator of the process (t, W, X, Y), that is

$$\begin{aligned} {\mathcal {L}}^h_{\mathbb {G}}F(t,w,x,i)&=F_t(t,w,x,i) + \left( \varGamma _1 +\lambda _1^i\alpha _1^i+\lambda _2^i\alpha _2^i- (\lambda _1^i + \lambda _2^i)x \right) F_x(t,w,x,i)\\&+w \left( r+ \mu _m \left( h^{(m)}+ h^{(1)}\beta _1 + h^{(2)}\beta _2 \right) \right. \\&+ \left. h^{(2)}\lambda _2^i(x-\alpha _2^i) - h^{(1)}\lambda _1^i(x-\alpha _1^i) \right) F_{w}(t,w,x,i) \\&+\frac{1}{2} w^2 \left( \sigma _m^2 \left( h^{(m)}+ h^{(1)}\beta _1 + h^{(2)}\beta _2 \right) ^2 + \sigma ^2 \left( h^{(1)}+ h^{(2)}\right) ^2 \right. \\&+\left. (h^{(1)}b_1)^2 + (h^{(2)}b_2)^2 \right) F_{ww}(t,w,x,i)\\&+ w \left( \sigma _m^2 ( \beta _1 - \beta _2 ) \left( h^{(m)}+ h^{(1)}\beta _1 + h^{(2)}\beta _2 \right) + h^{(1)}b_1 - h^{(2)}b_2 \right) F_{wx}(t,w,x,i) \\&+ \frac{1}{2}\varGamma _2F_{xx}(t,w,x,i)+\sum _{j=1}^K F (t,w,x,j)q^{ij}, \end{aligned}$$

for every function $F(\cdot , i)\in C^{1,2,2}([0,T]\times {\mathbf {R}}_{+}\times {\mathbf {R}})$, i.e. bounded, differentiable with respect to t and twice differentiable with respect to w and x, for every $i\in \{1, \dots , K\}$.

Suppose that the value function $V(\cdot , i)\in C^{1,2,2}([0,T]\times {\mathbf {R}}_{+}\times {\mathbf {R}})$ for every $i\in \{1,\dots ,K\}$. Then it solves the HJB equation given by

$$\begin{aligned} 0=\underset{h\in {\mathcal {A}}}{\sup }{\mathcal {L}}^h V(t,w,x,i) \end{aligned}$$

(45)

for every $i\in \{1,\dots ,K \}$, subject to the terminal condition $V(T,w,x,i)=\log (w)$, for all $(w,x)\in {\mathbf {R}}_{+}\times {\mathbf {R}}$ and $i\in \{1,\dots ,K \}$. It follows from the form of the utility function that for all $i\in \{1,\dots ,K \}$ the value function can be rewritten as $V(t,w,x,i)=\log (w)+\nu (t,x,i)$, for some function $\nu (t,x,i)$ such that $\nu (T,x,i)=0$. Inserting the ansatz for the value function in Eq. (45) and taking first order conditions leads to

$$\begin{aligned} 0&=\frac{\mu _m}{\sigma _m^2}-h^{(1)}\beta _1-h^{(2)}\beta _2-h^{(m)},\\ 0&= \beta _1\mu _m- \lambda _1^i(x -\alpha _1^i) -\beta _1\sigma _m^2\left( h^{(m)}+ h^{(1)}\beta _1 + h^{(2)}\beta _2 \right) -\sigma ^2(h^{(1)}+h^{(2)})-h^{(1)}b_1^2 ,\\ 0&=\beta _2\mu _m+\lambda _2^i(x-\alpha _2^i)-\beta _2\sigma _m^2\left( h^{(m)}+h^{(1)}\beta _1+h^{(2)}\beta _2\right) -\sigma ^2(h^{(1)}+h^{(2)})-h^{(2)}b_2^2. \qquad {} \end{aligned}$$

Second order conditions imply that portfolio weights given in (10)–(12) are candidates to be optimal strategies. Next, we insert the optimal portfolio weights in the HJB equation. This yields the following PDE:

$$\begin{aligned} 0&=\nu _t(t,x,i)+\varTheta _1^ix^2-\varTheta _2^ix+\varTheta _3^i+\sum _{j=1}^K \nu (t,x,j)q^{ij}+ \frac{1}{2}\varGamma _2 \nu _{xx}(t,x,i)\nonumber \\&+\left( \varGamma _1+\lambda _1^i\alpha _1^i+\lambda _2^i\alpha _2^i-(\lambda _1^i+\lambda _2^i)x \right) \nu _x(t,x,i). \end{aligned}$$

(46)

We conjecture that $\nu (t,x,i)=m(t,i)x^2+n(t,i)x+u(t,i)$. Substituting this ansatz in (46) results in a quadratic equation for x. Setting the coefficients of the terms $x^2$, x and the independent term to zero yields that the functions m, n and u solve the system of ODEs given in (14)–(16) see, e.g., (2012, Theorem 3.9).

Verification. In the sequel we verify martingale conditions that ensure that V in (13) is indeed the value function. To this, let v(t, w, x, i) be a solution of the HJB equation (45) and $h\in {\mathcal {A}}$ and admissible control. By Itô’s formula we get

$$\begin{aligned}&v(T, W_T^h, X_T, Y_T)=v(t,w,x,i)+\int _t^T {\mathcal {L}}{\mathcal {v}}(r, W^h_r, X_r, Y_r)\, \mathrm dr\\&\qquad +\int _t^T \sigma _m v_w(r, W^h_r, X_r, Y_r) W^h_r \left( h^{(m)}_r+h^{(1)}_r\beta _1+h^{(2)}_r\beta _2\right) \mathrm dB^{(m)}_r\\&\qquad +\int _t^T \sigma _m+ v_x(r, W^h_r, X_r, Y_r) \left( \beta _1-\beta _2\right) \mathrm dB^{(m)}_r\\&\qquad +\int _t^T v_w(r, W^h_r, X_r, Y_r) W^h_r \sigma \left( h^{(1)}_r+h^{(2)}_r \right) \mathrm dB^{(0)}_r\\&\qquad +\int _t^T \left( v_w(r, W^h_r, X_r, Y_r) W^h_r b_1h^{(1)}_r + v_x(r, W^h_r, X_r, Y_r) b_1\right) \mathrm dB^{(1)}_r\\&\qquad +\int _t^T \left( v_w(r, W^h_r, X_r, Y_r) W^h_r b_2h^{(2)}_r - v_x(r, W^h_r, X_r, Y_r) b_2\right) \mathrm dB^{(2)}_r\\&\qquad +\int _t^T\sum _{j=1}^Kv(r, W^h_r, X_r, j)-v(r, W^h_r, X_r, Y_{r^-}) (m-\nu )(\mathrm dr \times \{j\}). \end{aligned}$$

The last term in the expression above corresponds to the compensated integral with respect to the jump measure of Y, that is

$$\begin{aligned}&\int _t^T\sum _{j=1}^Kv(r, W^h_r, X_r, j) -v(r, W^h_r, X_r, Y_{r^-}) (m-\nu )(\mathrm dr \times \{j\})\\&\quad =\sum _{t\le r\le T}\varDelta v(r, W^h_r, X_r, Y_r)- \int _t^T\sum _{j=1}^Kv(r, W^h_r, X_r, j)-v(r, W^h_r, X_r, Y_{r^-}) q^{Y_{r^-} j}\, \mathrm dr. \end{aligned}$$

where $\varDelta v(t, W^h_t, X_t, Y_t)=v(t, W^h_t, X_t, Y_t)-v(t, W^h_t, X_t, Y_{t^-})$ for every $t \in [0,T]$,

$$\begin{aligned} m([0,t]\times \{j\}):=\sum _{n \ge 1} {{\mathbf {1}}}_{\{Y_{T_n}=j\}}{{\mathbf {1}}}_{\{T_n\le t\}},\quad j\in \{1,\dots ,K\},\,\,t\in [0,T], \end{aligned}$$

is the jump measure of Markov chain Y with the compensator

$$\begin{aligned} \nu ([0,t]\times \{j\})=\int _0^t\sum _{i \ne j} q^{ij}{{\mathbf {1}}}_{\{Y_{r^-}=i\}}\,\mathrm dr,\quad j\in \{1,\dots ,K\},\,\,t\in [0,T]. \end{aligned}$$

and $\{T_n\}_{n \in {\mathbb {N}}}$ is the sequence of jump times of Y. Since v satisfies equation (45) we get

$$\begin{aligned}&v(T, W_T^h, X_T, Y_T) \\&\quad \le v(t,w,x,i)+ \int _t^T \sigma _m v_w(r, W^h_r, X_r, Y_r) W^h_r\left( h^{(m)}_r+h^{(1)}_r\beta _1+h^{(2)}_r\beta _2\right) \mathrm dB^{(m)}_r\\&\qquad +\int _t^T \sigma _m+ v_x(r, W^h_r, X_r, Y_r) \left( \beta _1-\beta _2\right) \mathrm dB^{(m)}_r\\&\qquad +\int _t^T v_w(r, W^h_r, X_r, Y_r) W^h_r \sigma \left( h^{(1)}_r+h^{(2)}_r \right) \mathrm dB^{(0)}_r\\&\qquad +\int _t^T \left( v_w(r, W^h_r, X_r, Y_r) W^h_r b_1h^{(1)}_r + v_x(r, W^h_r, X_r, Y_r) b_1\right) \mathrm dB^{(1)}_r\\&\qquad +\int _t^T \left( v_w(r, W^h_r, X_r, Y_r) W^h_r b_2h^{(2)}_r- v_x(r, W^h_r, X_r, Y_r) b_2\right) \mathrm dB^{(2)}_r\\&\qquad +\int _t^T\sum _{j=1}^Kv(r, W^h_r, X_r, j)-v(r, W^h_r, X_r, Y_{r^-}) (m-\nu )(\mathrm dr \times \{j\}). \end{aligned}$$

The form of v and integrability condition (7) ensure that integrals with respects to Brownian motions $B^{(m)}, B^{(0)},B^{(1)},B^{(2)}$ and the compensated jump measure $m-\nu$ are true $({\mathbb {G}}, {\mathbf {P}})$-martingales. Then, taking expectations we get that

$$\begin{aligned} V(t,w,x,i)\le v(t,w,x,i), \end{aligned}$$

and the equality holds if h is a maximizer of Eq. (45). $\square$

Proof of Proposition 1

In the following we use the notation $\widehat{g(Y_t)}={\mathbf {E}}\left[ g(Y_t)|{\mathcal {F}}_t\right]$, $t\in [0,T]$. Consider the semimartingale decomposition of f(Y) given by

$$\begin{aligned} f(Y_t)= f(Y_0)+\int _0^t \langle Q{\mathbf {f}}, Y_{u^-}\rangle \, \mathrm du + M^{(1)}_t,\quad t \in [0,T], \end{aligned}$$

where $M^{(1)}$ is a $({\mathbb {G}}, {\mathbf {P}})$-martingale. Now, projecting over ${\mathbb {F}}$ leads to

$$\begin{aligned} \widehat{f(Y_t)}- \widehat{f(Y_0)}-\int _0^t \langle Q{\mathbf {f}}, {\widehat{Y}}_{u^-}\rangle \, \mathrm du = M^{(2)}_t,\quad t \in [0,T], \end{aligned}$$

where $M^{(2)}$ is an $({\mathbb {F}}, {\mathbf {P}})$-martingale. Using the martingale representation in (29) we get

$$\begin{aligned} \widehat{f(Y_t)}- \widehat{f(Y_0)}-\int _0^t \langle Q{\mathbf {f}}, {\widehat{Y}}_{u^-}\rangle \, \mathrm du = \int _0^t\gamma _u\, \mathrm dI_u,\quad t \in [0,T]. \end{aligned}$$

Let $m_t=I_t+\int _0^tX_u \varSigma ^{-1} \widehat{A(X_u,Y_u)}\,\mathrm du$, for every $t \in [0,T]$. Computing the product $f(Y)\cdot m$ and projecting on ${\mathbb {F}}$, we obtain

$$\begin{aligned} \widehat{f(Y_t)\cdot m_t}= \int _0^t m_u \langle Q{\mathbf {f}}, {\widehat{Y}}_u\rangle \, \mathrm du + \int _0^t X_u \varSigma ^{-1}\left( \widehat{f(Y_u) A(X_u,Y_u)}\right) \,\mathrm du + M^{(3)}_t,\,\, \quad {}\end{aligned}$$

(47)

for every $t \in [0,T]$ and for some $({\mathbb {F}}, {\mathbf {P}})$-martingale $M^{(3)}$. The hat in the second integrand of Eq. (47) stands for $\widehat{f(Y_u) A(X_u,Y_u)}={\mathbf {E}}\left[ f(Y_u) A(X_u,Y_u)|{\mathcal {F}}_u\right]$.

We now compute the product $\widehat{f(Y)}\cdot m$ as

$$\begin{aligned} \widehat{f(Y_t)} \cdot m_t= \int _0^t m_u \langle Q\mathbf{f}, {\widehat{Y}}_u\rangle \,\mathrm du + \int _0^t X_u \varSigma ^{-1}\widehat{f(Y_u)} \widehat{A(X_u,Y_u)}\, \mathrm du + \int _0^t \gamma _u \,\mathrm du + M^{(4)}_t. \end{aligned}$$

(48)

for every $t \in [0,T]$, where $M^{(4)}$ is an $({\mathbb {F}}, {\mathbf {P}})$-martingale. Comparing the finite variation terms in (47) and (48), we get

$$\begin{aligned} \gamma ^{(1)}_t&=\frac{\widehat{f(Y_t) \mu _1(X_t,Y_t)}-\widehat{f(Y_t)}\widehat{\mu _1(X_t,Y_t)}}{\sigma _1},\\ \gamma ^{(2)}_t&=\frac{\sigma _1 (\widehat{f(Y_t)\mu _2(X_t,Y_t)}-\widehat{f(Y_t)}\widehat{\mu _2(X_t,Y_t)})-\sigma _2\rho (\widehat{f(Y_t)\mu _1(X_t,Y_t)}-\widehat{f(Y_t)}\widehat{\mu _1(X_tY_t)})}{\sigma _1\sigma _2\sqrt{1-\rho ^2}}, \end{aligned}$$

for every $t \in [0,T]$. By taking $f(Y_t)={{\mathbf {1}}}_{\{Y_t=e_i\}}$, we obtain the result. Finally, since the drift and diffusion coefficients in (30) are continuous, bounded and locally Lipschitz, we get that $\varvec{\pi }=(\pi ^1, \dots , \pi ^K)$ is the unique strong solution of the system (30) . $\square$

Proof of Theorem 3

Existence For notational ease we set $\sigma _1=\sqrt{\sigma ^2+b_1^2}$ and $\sigma _2=\sqrt{\sigma ^2+b_2^2}$. Assume first that function $V(t,w,x,{\mathbf {p}})$ is regular. Then it satisfies the following HJB equation

$$\begin{aligned} 0= \underset{h\in {\mathcal {A}}^{{\mathbb {F}}}}{\sup }{\mathcal {L}}_{\mathbb {F}}^h V(t,w,x,{\mathbf {p}}) \end{aligned}$$

(49)

subject to the terminal condition $V(T,w,x,{\mathbf {p}})=\log (w)$, for all $w>0$, $x\in {\mathbb {R}}$ and for every ${\mathbf {p}}\in \varDelta _K$, where ${\mathcal {L}}_{\mathbb {F}}^h$ is given by

$$\begin{aligned}&{\mathcal {L}}_{\mathbb {F}}^h f(t,w,x,{\mathbf {p}})=\left\{ f_t + f_x\left( \varGamma _1+\varvec{\mu }_1(x)^\top {\mathbf {p}}-\varvec{\mu }_2(x)^\top {\mathbf {p}}\right) + \sum _{i,j=1}^K f_{p^i} q^{ji}p^j\right. \nonumber \\&\qquad \left. + \left( r+\left( h^{(m)}+ h^{(1)}\beta _1+h^{(2)}\beta _2 \right) \mu _m +h^{(1)}\varvec{\mu }_1(x)^\top {\mathbf {p}}+ h^{(2)}\varvec{\mu }_2(x)^\top {\mathbf {p}}\right) w f_w(t,w,x,{\mathbf {p}})\right. \nonumber \\&\qquad \left. + \frac{1}{2} f_{xx}\varGamma _2 + \frac{1}{2} \sum _{i,j=1}^Kf_{p^ip^j} (H^{(i),1}({\mathbf {p}}) H^{(j),1}({\mathbf {p}})+H^{(i),2}({\mathbf {p}}) H^{(j),2}({\mathbf {p}}))\right. \nonumber \\&\qquad \left. +\frac{1}{2} f_{ww}w^2 \left( (h^{(m)}+ h^{(1)}\beta _1+h^{(2)}\beta _2 )^2\sigma _m^2 +(\sigma _1 h^{(1)}+\rho \sigma _2h^{(2)})^2 + \sigma _2^2 (1-\rho ^2){h^{(2)}}^2\right) \right. \nonumber \\&\qquad \left. + f_{wx}w \left( \sigma _m^2(\beta _1-\beta _2)(h^{(m)}+ h^{(1)}\beta _1+h^{(2)}\beta _2 )+\sigma _1^2 h^{(1)}{-} \sigma _2^2 h^{(2)}{-}\rho \sigma _1\sigma _2 (h^{(1)}-h^{(2)})\right) \right. \nonumber \\&\qquad \left. + \sum _{i=1}^K f_{wp^i}w \left( H^{(i),1}({\mathbf {p}}) (\sigma _1 h^{(1)}+\rho \sigma _2h^{(2)})+ H^{(i),2}({\mathbf {p}})\sqrt{1-\rho ^2}h^{(2)}\sigma _2\right) \right. \nonumber \\&\qquad \left. + \sum _{i=1}^K f_{xp^i} \left( (\sigma _1-\rho \sigma _2) H^{(i),1}({\mathbf {p}}) - H^{(i),2}({\mathbf {p}})\sigma _2\sqrt{1-\rho ^2}\right) \right\} \end{aligned}$$

(50)

for every function $f:[0,T]\times {\mathbb {R}}^+ \times {\mathbb {R}}\times \varDelta _K \rightarrow {\mathbb {R}}$, which is bounded, differentiable with respect to time and twice differentiable with respect to $(w,x,{\mathbf {p}})$ with bounded derivatives. By the form of the utility function we have that the value function has the form $V(t,w,x,\pi )=\log (w)+v(t,x, \pi )$, for some function $v(t,x,\pi )$, such that $v(T,x,{\mathbf {p}})=0$ for all $(x,{\mathbf {p}})\in ({\mathbb {R}}\times \varDelta _K)$. By inserting the first ansatz in Eq. (50) and considering the first order condition we get that the candidate for an optimal strategy is given by (37), (38),(39). Since $V(t,w,x,{\mathbf {p}})$ is concave and increasing in w, the second order condition implies that (37),(38) and (39) is the maximizer and the optimal portfolio strategy. Here, we choose v of the form $v(t,x,{\mathbf {p}})={{\bar{m}}}({\mathbf {p}})x^2+{{\bar{n}}} (t,{\mathbf {p}})x+{{\bar{u}}}(t,{\mathbf {p}})$. Inserting this ansatz in Eq. (50) leads to the system of linear partial differential equations in (41), (42), (43).

Verification. To conclude that V is the value function, we show a verification result. Let ${\widetilde{V}}(t,w,x,{\mathbf {p}})$ be a solution of (49) with the boundary condition ${\widetilde{V}}(T,w,x,{\mathbf {p}})=\log (w)$. Let $h\in {\mathcal {A}}^{{\mathbb {F}}}$ be an ${\mathbb {F}}$-admissible control, let $W^h$ the solution to Eq. (34). By applying Itô’s formula we get

$$\begin{aligned}&{\widetilde{V}}(T, W_T^h, X_T,\varvec{\pi }_T)= {\widetilde{V}}(t,w,x,{\mathbf {p}})+\int _t^T {\mathcal {L}}_{\mathbb {F}}^h {\widetilde{V}}(u, W^h_u, X_u, \varvec{\pi }_u)\, \mathrm du \\&\qquad + \int _t^T \left( {\widetilde{V}}_w(u, W^h_u, X_u, \varvec{\pi }_u) W^h_u (h^{(m)}_u+ h^{(1)}_u\beta _1+ h^{(2)}_u \beta _2 )\right. \\&\qquad \left. + {\widetilde{V}}_x(u, W^h_u, X_u, \varvec{\pi }_u) (\beta _i-\beta _2) \right) \sigma _m \mathrm dB^{(m)}_u\\&\qquad +\int _t^T \left( {\widetilde{V}}_w(u, W^h_u, X_u, \varvec{\pi }_u) W^h_u (\sigma _1h^{(1)}{+}\rho \sigma _2 h^{(2)}) {+} (\sigma _1-\rho \sigma _2) {\widetilde{V}}_x(u, W^h_u, X_u, \varvec{\pi }_u)\right) \mathrm dI^{(1)}_u \\&\qquad +\int _t^T\sum _{i=1}^K{\widetilde{V}}_{p^i}(u, W^h_u, X_u, \varvec{\pi }_u) {{\bar{H}}}^{(i),1}(\varvec{\pi }_t) \mathrm dI^{(1)}_u\\&\qquad + \int _t^T \left( \sigma _2 \sqrt{1-\rho ^2}( {\widetilde{V}}_w(u, W^h_u, X_u, \varvec{\pi }_u) W^h_u h^{(2)}_u -{\widetilde{V}}_x(u, W^h_u, X_u, \varvec{\pi }_u)) \right) \mathrm dI^{(2)}_u\\&\qquad + \int _t^T \sum _{i=1}^K{\widetilde{V}}_{p^i}(u, W^h_u, X_u, \varvec{\pi }_u) H^{(i), 2} (\pi _t) \mathrm dI^{(2)}_u. \end{aligned}$$

By Eq. (50) we get

$$\begin{aligned}&{\widetilde{V}}(T, W_T^h, X_T,\varvec{\pi }_T)\le {\widetilde{V}}(t,w,x,{\mathbf {p}}) \nonumber \\&\qquad + \int _t^T \left( {\widetilde{V}}_w(u, W^h_u, X_u, \varvec{\pi }_u) W^h_u (h^{(m)}_u+ h^{(1)}_u\beta _1+ h^{(2)}_u \beta _2 )\right. \nonumber \\&\qquad \left. + {\widetilde{V}}_x(u, W^h_u, X_u, \varvec{\pi }_u) (\beta _i-\beta _2) \right) \sigma _m \mathrm dB^{(m)}_u\nonumber \\&\qquad +\int _t^T \left( {\widetilde{V}}_w(u, W^h_u, X_u, \varvec{\pi }_u) W^h_u (\sigma _1h^{(1)}{+}\rho \sigma _2 h^{(2)}) {+} (\sigma _1-\rho \sigma _2) {\widetilde{V}}_x(u, W^h_u, X_u, \varvec{\pi }_u)\right) \mathrm dI^{(1)}_u \nonumber \\&\qquad +\int _t^T\sum _{i=1}^K{\widetilde{V}}_{p^i}(u, W^h_u, X_u, \varvec{\pi }_u) {{\bar{H}}}^{(i),1}(\varvec{\pi }_t) \mathrm dI^{(1)}_u\nonumber \\&\qquad + \int _t^T \left( \sigma _2 \sqrt{1-\rho ^2}( {\widetilde{V}}_w(u, W^h_u, X_u, \varvec{\pi }_u) W^h_u h^{(2)}_u -{\widetilde{V}}_x(u, W^h_u, X_u, \varvec{\pi }_u)) \right) \mathrm dI^{(2)}_u\nonumber \\&\qquad + \int _t^T \sum _{i=1}^K{\widetilde{V}}_{p^i}(u, W^h_u, X_u, \varvec{\pi }_u) H^{(i), 2} (\varvec{\pi }_t) \mathrm dI^{(2)}_u. \end{aligned}$$

(51)

Note that stochastic integrals with respect to $B^{(m)},I^{(1)}$ and $I^{(2)}$ are true martingales. This is a consequence of the fact that function ${\widetilde{V}}(t,w,x,{\mathbf {p}})=\log (w)+{{\bar{m}}}(t)x^2+\bar{n}(t,{\mathbf {p}})x+{{\bar{u}}}(t, {\mathbf {p}})$ solves the HJB equation, that $(h^{(m)}, h^{(1)}, h^{(2)})$ is an ${\mathbb {F}}$-admissible strategy and that functions $\bar{m}(t),{{\bar{n}}}(t,{\mathbf {p}}), {{\bar{u}}}(t,{\mathbf {p}})$ and their derivatives are bounded over the compact interval $[0,T]\times \varDelta _K$. Then taking the expectation on both sides of inequality (51) implies that $V(t,w,x,{\mathbf {p}})\le {\widetilde{V}}(t,w,x,{\mathbf {p}})$. Moreover if $({h^{(m)}}^*,{h^{(1)}}^*,{h^{(2)}}^ *)$ is a maximizer of Eq. (49), then we obtain the equality $V(t,w,x,{\mathbf {p}})= {\widetilde{V}}(t,w,x,{\mathbf {p}})$. $\square$