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Portfolio management with stochastic interest rates and inflation ambiguity

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Abstract

We solve, in closed form, a stock-bond-cash portfolio problem of a risk- and ambiguity-averse investor when interest rates and the inflation rate are stochastic. The expected inflation rate is unobservable, but the investor can learn about it from observing realized inflation and stock and bond prices. The investor is ambiguous about the inflation model and prefers a portfolio strategy which is robust to model misspecification. Ambiguity about the inflation dynamics is shown to affect the optimal portfolio fundamentally different than ambiguity about the price dynamics of traded assets, for example the optimal portfolio weights can be increasing in the degree of ambiguity aversion. In a numerical example, the optimal portfolio is significantly affected by the learning about expected inflation and somewhat affected by ambiguity aversion. The welfare loss from ignoring learning or ambiguity can be considerable.

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Notes

  1. For simplicity of notation we suppress the dependence of \(\sigma _P\) on \(\bar{T}-t\).

  2. As Scheinkman and Xiong (2003), Dumas et al. (2009), and Branger et al. (2013), we assume that learning was long enough so that the variance of the estimation error has reached a steady state.

  3. To simplify the notation, we write \({\varvec{\Pi }}\) instead of \(\{(\varPi _s^S, \varPi _s^P)\}_{s \in [t,T]}\) and \(e\) instead of \(\{e_s\}_{s \in [t,T]}\). The expectation operator with respect to the probability measure \(\mathbb P ^e\) is defined as \(E_{t,\mathbf{y}}^\mathbb{P ^e}[\cdot ] \triangleq E^\mathbb{P ^e}[\cdot |X_t=x,Z_t=z,r_t=r,\hat{\beta }_t=\hat{\beta }]\).

  4. For a critique of this approach, see Pathak (2002).

  5. The hedge against the inflation risk is also zero if \(\theta =1-\gamma \), but since empirical studies support \(\gamma >1\) and \(\theta \) has to be positive, this is unlikely to be the case.

  6. It can be shown that the HJB equation for the case with observable \(\beta \)—so that the dynamics in (5) is used instead of (9)—is similar to the HJB equation (19), but some coefficients to second-order derivatives involving \(\beta \) are different.

  7. Note that the ambiguity aversion parameter depends on the precise model set-up and source of ambiguity and therefore has to be estimated on a case-by-case basis.

  8. Since Brennan and Xia (2002) estimate the parameters for real interest rates, we have adjusted their parameters to be applicable in our model.

  9. Recall that we assume a correlation of 0.3 between realized and expected inflation, \(\rho _{Z\beta }\). If, in the case with an observable expected inflation rate, we use \(\rho _{Z\beta }=-0.3\), the term \(\pi _{\beta }^S\) will in fact be slightly negative, so that the hedge against expected inflation and thus the entire portfolio weight of the stock will be decreasing with the horizon—in contrast with typical investment advice. Using \(\rho _{Z\beta }=-0.3\) in our general model with unobserved expected inflation, the portfolio weight of the stock would still be increasing with the horizon.

  10. In settings with multiple risky assets, the effect of ambiguity about a traded asset price on the optimal portfolio is not clear-cut. Flor and Larsen (2013) report that the speculative component of the bond (stock) demand increases (decreases) in \(\theta \) if the investor is uncertain about the stock price process only.

  11. It is easy to check that the assumptions of the theorem are satisfied because entries in all matrices are constant.

  12. This is the same equation as Eq. (25) but without the supremum.

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Correspondence to Alexey Rubtsov.

Appendices

Appendix A: Optimal filtering

To keep the same notation (but in boldface letters) as in Liptser and Shiryaev (2001), we rewrite the Eqs. (3), (4), and (5) in the following form

$$\begin{aligned} \left[ \begin{array}{l} \frac{dZ_t}{Z_t}\\ \frac{dS_t}{S_t}\\ dr_t \end{array}\right]&= \left[ \begin{array}{ll} \underbrace{\left( \begin{array}{c} 0\\ r_t+\alpha \\ \kappa (\bar{r}-r_t) \end{array}\right) }_{\mathbf{A}_0}+ \underbrace{\left( \begin{array}{l} 1\\ 0\\ 0 \end{array}\right) }_{\mathbf{A}_1} \beta _t \end{array} \right] dt \\&\quad + \underbrace{\left[ \begin{array}{l} 0\\ 0\\ 0 \end{array}\right] }_{\mathbf{B}_1}dW_t^{\beta }+ \underbrace{\left[ \begin{array}{c@{\quad }c@{\quad }c} \sigma _Z &{} 0 &{} 0 \\ \sigma _S \rho _{ZS} &{} \sigma _S \hat{\rho }_S &{} 0 \\ -\sigma _r \rho _{ZP} &{} -\sigma _r \hat{\rho }_{SP} &{} -\sigma _r \hat{\rho }_P \end{array} \right] }_{\mathbf{B}_2} \left[ \begin{array}{l} dW_t^Z\\ dW_t^S\\ dW_t^P \end{array} \right] \end{aligned}$$

and

$$\begin{aligned} d\beta _t=(\underbrace{\lambda \bar{\beta }}_{\mathbf{a}_0} +\underbrace{(-\lambda )}_{\mathbf{a}_1}\beta _t)dt +\underbrace{\sigma _{\beta }\hat{\rho }_{\beta }}_{\mathbf{b}_1}dW_t^{\beta }+ \underbrace{[\sigma _{\beta }\rho _{Z\beta } \ \ \ \ \sigma _{\beta }\hat{\rho }_{S\beta } \ \ \ \ \sigma _{\beta } \hat{\rho }_{P\beta }]}_{\mathbf{b}_2} \left[ \begin{array}{l} dW_t^Z\\ dW_t^S\\ dW_t^P \end{array}\right] , \end{aligned}$$

where \((W_t^Z, W_t^S, W_t^P, W_t^{\beta })^\mathsf{T }\) is a standard Brownian motion relative to the filtration \(\mathcal F _t\) and

$$\begin{aligned} \left[ \begin{array}{l} dB_t^Z\\ dB_t^S\\ dB_t^P\\ dB_t^{\beta } \end{array} \right] = \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1 &{} 0 &{} 0 &{} 0\\ \rho _{ZS} &{} \hat{\rho }_S &{} 0 &{} 0\\ \rho _{ZP} &{} \hat{\rho }_{SP} &{} \hat{\rho }_P &{} 0\\ \rho _{Z\beta } &{} \hat{\rho }_{S\beta } &{} \hat{\rho }_{P\beta } &{} \hat{\rho }_{\beta } \end{array} \right] \left[ \begin{array}{c} dW_t^Z\\ dW_t^S\\ dW_t^P\\ dW_t^{\beta } \end{array} \right] \end{aligned}$$

with

$$\begin{aligned} \hat{\rho }_S&= \sqrt{1-\rho _{ZS}^2}, \ \hat{\rho }_{SP}=\frac{\rho _{SP} -\rho _{ZS}\rho _{ZP}}{\sqrt{1-\rho _{ZS}^2}}, \ \hat{\rho }_P=\sqrt{1-\rho _{ZP}^2-\hat{\rho }_{SP}^2}, \\ \hat{\rho }_{S\beta }&= \frac{\rho _{S\beta } -\rho _{ZS}\rho _{Z\beta }}{\sqrt{1-\rho _{ZS}^2}}, \ \hat{\rho }_{P\beta }=\frac{\rho _{P\beta } -\rho _{ZP}\rho _{Z\beta } -\hat{\rho }_{SP}\hat{\rho }_{S\beta }}{\hat{\rho }_P}, \\ \hat{\rho }_{\beta }&= \sqrt{1-\rho _{Z\beta }^2 -\hat{\rho }_{S\beta }^2-\hat{\rho }_{P\beta }^2}. \end{aligned}$$

Assuming that for \(a \in \mathbb R \) the conditional distribution \(P(\beta _0 \le a | Z_0, S_0, r_0)\) is Gaussian \(\mathbb P \)-a.s. with mean \(\hat{\beta }_0=E[\beta _0|Z_0,S_0,r_0]\), and variance \(m_0=E[(\beta _0-\hat{\beta }_0)^2|Z_0,S_0,r_0]\) (equivalently, the distribution of \(\beta _0\) is conditionally Gaussian), we have from Theorem 12.6 in Liptser and Shiryaev (2001) that the conditional distribution \(P(\beta _t \le a | \mathcal F _t^{S,Z,r})\) is also Gaussian \(\mathbb P \)-a.s.Footnote 11

Therefore, applying Theorem 12.7 in Liptser and Shiryaev (2001) we have that the observed expected inflation rate \(\hat{\beta }_t=E[\beta _t | \mathcal F _t^{S,Z,r}]\) satisfies

$$\begin{aligned} d\hat{\beta }_t&= (\underbrace{\lambda \bar{\beta }}_{\mathbf{a}_0} +\underbrace{(-\lambda )}_{\mathbf{a}_1}\hat{\beta }_t)dt\\&\quad +\left\{ \underbrace{\left[ \sigma _Z\sigma _{\beta }\rho _{Z\beta },\ \sigma _S\sigma _{\beta }\rho _{S\beta }, \ -\sigma _r\sigma _{\beta }\rho _{P\beta }\right] }_{\mathbf{b} \circ \mathbf{B}} +\, m_t \underbrace{[1, \ 0, \ 0]}_{\mathbf{A}_1^{\mathsf{T }}} \right\} \\&\quad \times \underbrace{\left[ \begin{array}{c@{\quad }c@{\quad }c} \sigma _Z &{} \sigma _S \rho _{ZS} &{}-\sigma _r \rho _{ZP} \\ 0 &{} \sigma _S \hat{\rho }_S &{} -\sigma _r \hat{\rho }_{SP}\\ 0 &{} 0 &{} -\sigma _r \hat{\rho }_P \end{array} \right] ^{-1} \left[ \begin{array}{c@{\quad }c@{\quad }c} \sigma _Z &{} 0 &{} 0 \\ \sigma _S \rho _{ZS} &{} \sigma _S \hat{\rho }_S &{} 0 \\ -\sigma _r \rho _{ZP} &{} -\sigma _r \hat{\rho }_{SP} &{} -\sigma _r \hat{\rho }_P \end{array} \right] ^{-1}}_{(\mathbf{B} \circ \mathbf{B})^{-1}} \\&\quad \times \left\{ \left[ \begin{array}{c} \frac{dZ_t}{Z_t}\\ \frac{dS_t}{S_t}\\ dr_t \end{array}\right] - \left[ \underbrace{\left( \begin{array}{c} 0\\ r_t+\alpha \\ \kappa (\bar{r}-r_t) \end{array}\right) }_{\mathbf{A}_0}+ \underbrace{\left( \begin{array}{c} 1\\ 0\\ 0 \end{array}\right) }_{\mathbf{A}_1}\hat{\beta }_t\right] dt\right\} , \end{aligned}$$

where \(m_t=E[(\beta _t-\hat{\beta }_t)^2 | \mathcal F _t^{S,Z,r}]\) is deterministic and

$$\begin{aligned} \mathbf{b} \circ \mathbf{B}&= \mathbf{b}_1\mathbf{B}_1^\mathsf{T }+\mathbf{b}_2\mathbf{B}_2^\mathsf{T }=[0, \ 0, \ 0]\\&\quad + [\sigma _{\beta }\rho _{Z\beta }, \ \sigma _{\beta }\hat{\rho }_{S\beta }, \ \sigma _{\beta } \hat{\rho }_{P\beta }] \left[ \begin{array}{c@{\quad }c@{\quad }c} \sigma _Z &{} \sigma _S \rho _{ZS} &{} -\sigma _r \rho _{ZP}\\ 0 &{} \sigma _S \hat{\rho }_S &{} -\sigma _r \hat{\rho }_{SP}\\ 0 &{} 0 &{} -\sigma _r \hat{\rho }_P \end{array} \right] \\&= \left[ \sigma _Z\sigma _{\beta }\rho _{Z\beta }, \ \sigma _S\sigma _{\beta }\rho _{S\beta }, \ -\sigma _r\sigma _{\beta }\rho _{P\beta }\right] , \\ (\mathbf{B} \circ \mathbf{B})^{-1}&= (\mathbf{B}_1\mathbf{B}_1^\mathsf{T }+\mathbf{B}_2\mathbf{B}_2^\mathsf{T })^{-1} =(\mathbf{B}_2\mathbf{B}_2^\mathsf{T })^{-1}=(\mathbf{B}_2^\mathsf{T })^{-1}(\mathbf{B}_2)^{-1} \end{aligned}$$

where

$$\begin{aligned} (\mathbf{B}_2^\mathsf{T })^{-1}= \left[ \begin{array}{c@{\quad }c@{\quad }c} \frac{1}{\sigma _Z} &{} -\frac{\rho _{ZS}}{\sigma _Z \hat{\rho }_S} &{} \frac{\rho _{ZS}\hat{\rho }_{SP}-\hat{\rho }_S\rho _{ZP}}{\sigma _Z \hat{\rho }_S \hat{\rho }_P}\\ 0 &{} \frac{1}{\sigma _S \hat{\rho }_S} &{} -\frac{\hat{\rho }_{SP}}{\sigma _S \hat{\rho }_S \hat{\rho }_P}\\ 0 &{} 0 &{} -\frac{1}{\sigma _r \hat{\rho }_P} \end{array} \right] . \end{aligned}$$

We first evaluate \(\mathbf{b} \circ \mathbf{B} + m_t \mathbf{A}_1^\mathsf{T }\) which yields

$$\begin{aligned} \mathbf{b} \circ \mathbf{B} + m_t \mathbf{A}_1^\mathsf{T }&=\left[ \sigma _Z\sigma _{\beta }\rho _{Z\beta }+m_t, \ \sigma _S\sigma _{\beta }\rho _{S\beta }, \ -\sigma _r\sigma _{\beta }\rho _{P\beta }\right] \end{aligned}$$

Now we multiply vector \(\mathbf{b} \circ \mathbf{B} +m_t \mathbf{A}_1^\mathsf{T }\) by matrix \((\mathbf{B}_2^\mathsf{T })^{-1}\) to obtain

$$\begin{aligned}&\!\!\!\left( \mathbf{b} \circ \mathbf{B} \!+\! m_t \mathbf{A}_1^\mathsf{T }\right) (\mathbf{B}_2^\mathsf{T })^{-1} \\&\quad \!=\!\left[ \sigma _{\beta }\rho _{Z\beta }+\frac{m_t}{\sigma _Z}, \ \frac{\!-\!m_t\rho _{ZS} \!+\! \sigma _Z\sigma _{\beta }\hat{\rho }_S\hat{\rho }_{S\beta }}{\sigma _Z\hat{\rho }_S}, \ \frac{m_t(\rho _{ZS}\hat{\rho }_{SP}\!-\!\hat{\rho }_S\rho _{ZP}) \!+\!\sigma _Z\sigma _{\beta }\hat{\rho }_S\hat{\rho }_P\hat{\rho }_{P\beta }}{\sigma _Z \hat{\rho }_S\hat{\rho }_P}\right] \\&\quad \!=\![\sigma _Z A_Z, \ \sigma _S A_S, \ \sigma _P A_P], \end{aligned}$$

where

$$\begin{aligned} A_P&= \frac{m_t(\rho _{ZS}\hat{\rho }_{SP}-\hat{\rho }_S\rho _{ZP}) +\sigma _Z\sigma _{\beta }\hat{\rho }_S\hat{\rho }_P \hat{\rho }_{P\beta }}{\sigma _P\sigma _Z\hat{\rho }_S\hat{\rho }_P}, A_S=\frac{-\rho _{ZS}m_t+\sigma _Z\sigma _{\beta }\hat{\rho }_S \hat{\rho }_{S\beta }}{\sigma _S\sigma _Z \hat{\rho }_S}, \\ A_Z&= \frac{\sigma _Z\sigma _{\beta }\rho _{Z\beta }+m_t}{\sigma _Z^2}. \end{aligned}$$

The following vector defines a Brownian motion (see Liptser and Shiryaev 2001, Vol.2, p.35) relative to filtration \(\mathcal F _t^{S,Z,r}\)

$$\begin{aligned}&\left[ \begin{array}{c@{\quad }c@{\quad }c} \sigma _Z &{} 0 &{} 0 \\ \sigma _S \rho _{ZS} &{} \sigma _S \hat{\rho }_S &{} 0 \\ -\sigma _r \rho _{ZP} &{} -\sigma _r \hat{\rho }_{SP} &{} -\sigma _r \hat{\rho }_P \end{array} \right] ^{-1} \left\{ \left[ \begin{array}{c} \frac{dZ_t}{Z_t}\\ \frac{dS_t}{S_t}\\ dr_t \end{array}\right] - \left[ \left( \begin{array}{c} 0\\ r_t+\alpha \\ \kappa (\bar{r}-r_t) \end{array}\right) + \left( \begin{array}{c} 1\\ 0\\ 0 \end{array}\right) \hat{\beta }_t\right] dt\right\} \nonumber \\&\quad =\left[ \begin{array}{c@{\quad }c@{\quad }c} \frac{1}{\sigma _Z} &{} 0 &{} 0 \\ -\frac{\rho _{ZS}}{\sigma _Z \hat{\rho }_S} &{} \frac{1}{\sigma _S \hat{\rho }_S} &{} 0 \\ \frac{\rho _{ZS}\hat{\rho }_{SP}-\hat{\rho }_S\rho _{ZP}}{\sigma _Z \hat{\rho }_S\hat{\rho }_P} &{} -\frac{\hat{\rho }_{SP}}{\sigma _S \hat{\rho }_S\hat{\rho }_P} &{} -\frac{1}{\sigma _r \hat{\rho }_P} \end{array} \right] \left[ \begin{array}{c} \frac{dZ_t}{Z_t}-\hat{\beta }_t dt\\ \sigma _S dB_t^S\\ -\sigma _r dB_t^P \end{array}\right] \nonumber \\&\quad =\left[ \begin{array}{c} \frac{1}{\sigma _Z}\left( \frac{dZ_t}{Z_t}-\hat{\beta }_t dt\right) \\ -\frac{\rho _{ZS}}{\sigma _Z \hat{\rho }_S} \left( \frac{dZ_t}{Z_t}-\hat{\beta }_tdt\right) +\frac{1}{\hat{\rho }_S}dB_t^S\\ \frac{\rho _{ZS}\hat{\rho }_{SP}-\hat{\rho }_S\rho _{ZP}}{\sigma _Z \hat{\rho }_S\hat{\rho }_P}\left( \frac{dZ_t}{Z_t}-\hat{\beta }_tdt\right) -\frac{\hat{\rho }_{SP}}{\hat{\rho }_S\hat{\rho }_P}dB_t^S +\frac{1}{\hat{\rho }_P}dB_t^P \end{array}\right] =\left[ \begin{array}{c} d\hat{B}_t^Z\\ d\hat{B}_t^S\\ d\hat{B}_t^P \end{array}\right] \end{aligned}$$
(22)

where \((\hat{B}_t^Z, \hat{B}_t^S, \hat{B}_t^P)^\mathsf{T }\) is the Brownian motion. From this equality we easily obtain that \(dB_t^S=\hat{\rho }_Sd\hat{B}_t^S+\rho _{ZS}d\hat{B}_t^Z,\,dB_t^P =\hat{\rho }_Pd\hat{B}_t^P+\hat{\rho }_{SP}d\hat{B}_t^S +\rho _{ZP}d\hat{B}_t^Z\).

Thus, we have

$$\begin{aligned} d\hat{\beta }_t=\lambda (\bar{\beta }-\hat{\beta }_t)dt+A_Z\sigma _Zd \hat{B}_t^Z+A_S\sigma _Sd\hat{B}_t^S+A_P\sigma _Pd\hat{B}_t^P. \end{aligned}$$

Therefore, the filtered equations can be written in terms of the Brownian motion (22) as

$$\begin{aligned} dZ_t&= Z_t\left( \hat{\beta }_tdt+\sigma _Zd\hat{B}_t^Z\right) ,\\ dS_t&= S_t\left( (r_t+\alpha ) dt+\sigma _S(\rho _{ZS}d\hat{B}_t^Z+\hat{\rho }_Sd\hat{B}_t^S)\right) ,\\ dr_t&= \kappa (\bar{r}-r_t)dt-\sigma _r\left( \hat{\rho }_P d\hat{B}_t^P+\hat{\rho }_{SP} d\hat{B}_t^S+\rho _{ZP}d\hat{B}_t^Z\right) . \end{aligned}$$

Instead of using time-dependent variance \(m_t\) in the definitions of \(A_S, A_P, A_Z\) we assume—as Scheinkman and Xiong (2003), Dumas et al. (2009), and Branger et al. (2013)—that learning was long enough and take the variance to be equal to \(m\). Here \(m\) is the limit value (as \(t \rightarrow \infty \)) of the deterministic variance \(m_t\) and it can be shown that

$$\begin{aligned} m=\frac{-\bar{K}_2+\sqrt{\bar{K}_2^2-4\bar{K}_1\bar{K}_3}}{2\bar{K}_1}, \end{aligned}$$
(23)

where

$$\begin{aligned} \bar{K}_1&= \frac{(\hat{\rho }_S\hat{\rho }_P)^2 +(\rho _{ZS}\hat{\rho }_P)^2+(\rho _{ZS}\hat{\rho }_{SP} -\hat{\rho }_S\rho _{ZP})^2}{(\sigma _Z\hat{\rho }_S\hat{\rho }_P)^2},\\ \bar{K}_2&= 2\lambda +\frac{2\sigma _Z\sigma _{\beta } \left( \rho _{Z\beta }(\hat{\rho }_S\hat{\rho }_P)^2-\rho _{ZS}\hat{\rho }_S \hat{\rho }_P^2\hat{\rho }_{S\beta }+(\rho _{ZS}\hat{\rho }_{SP} -\hat{\rho }_S\rho _{ZP})\hat{\rho }_S \hat{\rho }_P \hat{\rho }_{P\beta }\right) }{(\sigma _Z\hat{\rho }_S\hat{\rho }_P)^2},\\ \bar{K}_3&= \sigma _{\beta }^2(\rho _{Z\beta }^2+\hat{\rho }_{S\beta }^2 +\hat{\rho }_{P\beta }^2)-\sigma _{\beta }^2. \end{aligned}$$

Appendix B: Robust HJB equation

We rewrite equations for wealth \(X_t\), price level process \(Z_t\), short-term interest rate \(r_t\), and drift \(\hat{\beta }_t\) in matrix form

$$\begin{aligned} \left[ \begin{array}{c} dZ_t\\ dX_t\\ dr_t\\ d\hat{\beta }_t \end{array}\right] \!=\! \left[ \begin{array}{c} Z_t\hat{\beta }_t\\ X_t(r_t\!+\!\alpha \varPi _t^S\!+\!q\varPi _t^P)\\ \kappa (\bar{r}\!-\!r_t)\\ \lambda (\bar{\beta }\!-\!\hat{\beta }_t) \end{array}\right] dt\!+\! \left[ \begin{array}{c@{\quad }c@{\quad }c} Z_t\sigma _Z &{} 0 &{} 0\\ K_1 &{} K_2 &{} K_3 \\ -\sigma _r \rho _{ZP} &{} -\sigma _r \hat{\rho }_{SP} &{} -\sigma _r \hat{\rho }_P\\ A_Z\sigma _Z &{} A_S \sigma _S &{} A_P\sigma _P \end{array}\right] \left[ \begin{array}{c} d\hat{B}_t^Z\\ d\hat{B}_t^S\\ d\hat{B}_t^P \end{array}\right] \end{aligned}$$

where \(K_1=\sigma _S\varPi _t^S X_t \rho _{ZS} +\sigma _P \varPi _t^P X_t \rho _{ZP}, K_2=\sigma _S\varPi _t^S X_t \hat{\rho }_S+\sigma _P\varPi _t^PX_t\hat{\rho }_{SP}\), and \(K_3=\sigma _P\varPi _t^P X_t \hat{\rho }_P\).

We introduce perturbations to this system by adding a drift \(\int _0^t e_sds(1, k_S, k_P)^\mathsf{T }\) to the Brownian motion \((\hat{B}_t^Z, \hat{B}_t^S, \hat{B}_t^P)^\mathsf{T }\). The resulting vector \((\tilde{B}_t^Z, \tilde{B}_t^S, \tilde{B}_t^P)^\mathsf{T }\) is a Brownian motion under probability measure \(\mathbb P ^e\). The perturbed system of equations is

$$\begin{aligned} \left[ \begin{array}{c} dZ_t\\ dX_t\\ dr_t\\ d\hat{\beta }_t \end{array}\right]&= \underbrace{\left[ \begin{array}{c} Z_t(\hat{\beta }_t-\sigma _Ze_t)\\ X_t\left( r_t+\alpha \varPi _t^S+q\varPi _t^P -\frac{K_1+k_SK_2+k_PK_3}{X_t}e_t\right) \\ \kappa (\bar{r}-r_t)+\sigma _r(\rho _{ZP}+k_S\hat{\rho }_{SP} +k_P\hat{\rho }_P)e_t\\ \lambda (\bar{\beta }-\hat{\beta }_t)-(A_Z\sigma _Z+k_SA_S\sigma _S +k_PA_P\sigma _P)e_t \end{array}\right] }_{\mathbf{M}}dt \\&\quad + \underbrace{\left[ \begin{array}{c@{\quad }c@{\quad }c} Z_t\sigma _Z &{} 0 &{} 0\\ K_1 &{} K_2 &{} K_3 \\ -\sigma _r \rho _{ZP} &{} -\sigma _r \hat{\rho }_{SP} &{} -\sigma _r \hat{\rho }_P\\ A_Z\sigma _Z &{} A_S \sigma _S &{} A_P\sigma _P \end{array}\right] }_{{\varvec{\Lambda }}} \left[ \begin{array}{c} d\tilde{B}_t^Z\\ d\tilde{B}_t^S\\ d\tilde{B}_t^P \end{array}\right] . \end{aligned}$$

According to Anderson et al. (2003), we evaluate the symmetric matrix

$$\begin{aligned} {\varvec{\Sigma }}={\varvec{\Lambda }}{\varvec{\Lambda }}^\mathsf{T }=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \sigma _{11} &{} \sigma _{12} &{} \sigma _{13} &{} \sigma _{14} \\ \sigma _{21} &{} \sigma _{22} &{} \sigma _{23} &{} \sigma _{24}\\ \sigma _{31} &{} \sigma _{32} &{} \sigma _{33} &{} \sigma _{34}\\ \sigma _{41} &{} \sigma _{42} &{} \sigma _{43} &{} \sigma _{44} \end{array}\right] \end{aligned}$$

where we defined \(\sigma _{11}\!=\!(Z_t\sigma _Z)^2\), \(\sigma _{22}\!=\!X_t^2\left( (\sigma _S\varPi _t^S)^2\!+\!(\sigma _P\varPi _t^P)^2\!+\!2\sigma _S\sigma _P\varPi _t^S\varPi _t^P\rho _{SP}\right) \), \(\sigma _{33}\!=\!\sigma _r^2, \sigma _{44}\!=\!(\sigma _Z A_Z)^2\!+\!(\sigma _S A_S)^2\!+\!(\sigma _P A_P)^2, \sigma _{12}=\sigma _Z X_t Z_t\left( \sigma _S\varPi _t^S\rho _{ZS}+\sigma _P\varPi _t^P\rho _{ZP}\right) \), \(\sigma _{13}=-\sigma _Z\sigma _rZ_t\rho _{ZP}\), \(\sigma _{14}=\sigma _Z^2A_ZZ_t\), \(\sigma _{34}=-\sigma _r\sigma _{\beta }\rho _{P\beta }\), \(\sigma _{23}=-\sigma _rX_t\left( \sigma _P\varPi _t^P +\sigma _S\varPi _t^S\rho _{SP}\right) \), \(\sigma _{24}=\sigma _{\beta }X_t\left( \sigma _S\varPi _t^S\rho _{S\beta } +\sigma _P\varPi _t^P\rho _{P\beta }\right) \).

We denote the Hessian and the gradient of the value function \(v\) with respect to state variables \(z,x,r\) and \(\hat{\beta }\), respectively as

$$\begin{aligned} \frac{\partial ^2 v}{\partial x_i \partial x_j} \triangleq \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} v_{zz} &{} v_{zx} &{} v_{zr} &{} v_{z\hat{\beta }}\\ v_{zx} &{} v_{xx} &{} v_{xr} &{} v_{x\hat{\beta }}\\ v_{zr} &{} v_{xr} &{} v_{rr} &{} v_{r\hat{\beta }}\\ v_{z\hat{\beta }} &{} v_{x\hat{\beta }} &{} v_{r\hat{\beta }} &{} v_{\hat{\beta }\hat{\beta }} \end{array}\right] , \quad \left( \frac{\partial v}{\partial x_i}\right) \triangleq \left[ \begin{array}{c} v_z\\ v_x\\ v_r\\ v_{\hat{\beta }} \end{array}\right] . \end{aligned}$$

Let \(\varvec{\pi }=(\pi ^S, \pi ^P)\) be the vector of fractions of wealth invested at time \(t \in [0,T]\) in the stock (\(\pi ^S\)) and the bond (\(\pi ^P\)), then according to Anderson et al. (2003), the robust HJB equation is

$$\begin{aligned} v_t+\sup \limits _{\varvec{\pi } \in \mathbb R ^2}\inf \limits _{e \in \mathbb R } \left( \mathbf{M}^\mathsf{T }\left( \frac{\partial v}{\partial x_i}\right) +\frac{1}{2}\mathrm{trace }\left( {\varvec{\Sigma }} \frac{\partial ^2 v}{\partial x_i \partial x_j}\right) +\frac{e^2}{2\varPsi }\right) =0. \end{aligned}$$

In particular,

$$\begin{aligned} \begin{array}{cl} &{}\sup \limits _{\varvec{\pi } \in \mathbb R ^2}\inf \limits _{e \in \mathbb{R }}\left\{ v_t\!+\!z\left( \hat{\beta }-\sigma _Ze\right) v_z \!+\!x\left( r\!+\!\alpha \pi ^S+q\pi ^P-[\sigma _S\pi ^Sa_1 +\sigma _P\pi ^Pa_2]e\right) v_x\right. \\ &{}\quad +\left( \kappa (\bar{r}-r)+\sigma _ra_2e\right) v_r+\left( \lambda (\bar{\beta } -\hat{\beta })-a_3e\right) v_{\hat{\beta }}+\frac{1}{2}(z\sigma _Z)^2v_{zz}\\ &{}\quad +\, \sigma _Zxz\left( \sigma _S\pi ^Sn_1+\sigma _P\pi ^Pn_2\right) v_{zx} -\sigma _Z\sigma _rzn_2v_{zr}+\sigma _Z^2A_Zzv_{z\hat{\beta }}\\ &{}\quad +\, \frac{1}{2}x^2\left( (\sigma _S\pi ^S)^2+(\sigma _P\pi ^P)^2 +2\sigma _S\sigma _P\pi ^S\pi ^Pn_4\right) v_{xx} \\ &{}\quad -\, \sigma _rx\left( \sigma _P\pi ^P+\sigma _S\pi ^Sn_4\right) v_{xr} +\sigma _{\beta }x\left( \sigma _S\pi ^Sn_5 +\sigma _P\pi ^Pn_6\right) v_{x\hat{\beta }} +\frac{1}{2}\sigma _r^2v_{rr}\\ &{}\quad \left. -\, \sigma _r\sigma _{\beta }n_6v_{r\hat{\beta }} +\frac{1}{2}\left( (A_Z\sigma _Z)^2+(A_S\sigma _S)^2 +(A_P\sigma _P)^2\right) v_{\hat{\beta }\hat{\beta }} +\frac{e^2}{2\varPsi }\right\} =0, \end{array} \end{aligned}$$
(24)

where \(n_1=\rho _{ZS}, n_2=\rho _{ZP}, n_3=\rho _{Z\beta },\,n_4=\rho _{SP}, n_5=\rho _{S\beta }\), and \(n_6=\rho _{P\beta }\).

To find the infimum over \(e\), we take the derivative with respect to \(e\) and set it equal to zero.

$$\begin{aligned} \frac{d}{de}\left( -z\sigma _Zev_z-x(\sigma _S\pi ^Sa_1 +\sigma _P\pi ^Pa_2)ev_x +\sigma _ra_2ev_r-a_3ev_{\hat{\beta }}+\frac{e^2}{2\varPsi }\right) =0. \end{aligned}$$

The value \(e^*\) that gives the infimum is

$$\begin{aligned} e^*=\varPsi \left( z\sigma _Zv_z+x(\sigma _S\pi ^Sa_1 +\sigma _P\pi ^Pa_2)v_x-\sigma _ra_2v_r+a_3v_{\hat{\beta }}\right) . \end{aligned}$$

To simplify the notation we use \(\phi =1-\gamma \). Let us look for a solution in the form \(v(t,z,x,r,\hat{\beta }) =\frac{1}{\phi }\left( \frac{x}{z}\right) ^{\phi }h(t,r,\hat{\beta })\) and assume that \(\varPsi =\frac{\theta }{h}\left( \frac{x}{z}\right) ^{-\phi }\). Plugging these functions into \(e^*\) we obtain

$$\begin{aligned} e^*=\theta \left( \sigma _S\pi ^Sa_1+\sigma _P\pi ^Pa_2-\sigma _Z -\frac{\sigma _ra_2}{\phi }\frac{h_r}{h}+\frac{a_3}{\phi } \frac{h_{\hat{\beta }}}{h}\right) . \end{aligned}$$

Plugging \(v(t,z,x,r,\hat{\beta }),\,e^*\) into (24) and dividing by \(\frac{1}{\phi }\left( \frac{x}{z}\right) ^{\phi }\) we have

$$\begin{aligned}&\sup \limits _{\varvec{\pi } \in \mathbb{R }^2}\left\{ h_t\!-\!\phi \hat{\beta }h\!+\!\phi \left( r\!+\!\alpha \pi ^S \!+\!q\pi ^P\right) h\!+\!\kappa \left( \bar{r}\!-\!r\right) h_r\!+\!\lambda \left( \bar{\beta } \!-\!\hat{\beta }\right) h_{\hat{\beta }}\!+\!\frac{\phi (\phi \!+\!1)\sigma _Z^2}{2}h\right. \nonumber \\&\quad -\phi ^2\sigma _Z\left( \sigma _S\pi ^Sn_1\!+\!\sigma _P\pi ^Pn_2\right) h \!+\!\phi \sigma _Z\sigma _rn_2h_r\!-\!\phi \sigma _Z^2A_Zh_{\hat{\beta }}\nonumber \\&\quad +\frac{\phi (\phi -1)}{2}\left( (\sigma _S\pi ^S)^2\!+\!(\sigma _P\pi ^P)^2 \!+\!2\sigma _S\sigma _P\pi ^S\pi ^Pn_4\right) h\!-\!\phi \sigma _r\left( \sigma _P\pi ^P \!+\!\sigma _S\pi ^Sn_4\right) h_r\nonumber \\&\quad +\phi \sigma _{\beta }\left( \sigma _S\pi ^Sn_5 \!+\!\sigma _P\pi ^Pn_6\right) h_{\hat{\beta }}\!+\!\frac{\sigma _r^2}{2}h_{rr} \!-\!\sigma _r\sigma _{\beta }n_6h_{r\hat{\beta }}\nonumber \\&\quad +\frac{1}{2}\left( (A_Z\sigma _Z)^2\!+\!(A_S\sigma _S)^2 \!+\!(A_P\sigma _P)^2\right) h_{\hat{\beta }\hat{\beta }}\nonumber \\&\quad \left. -\frac{\phi \theta }{2}h\left( \sigma _S\pi ^Sa_1\!+\!\sigma _P\pi ^Pa_2 \!-\!\sigma _Z \!-\!\frac{\sigma _ra_2}{\phi }\frac{h_r}{h} \!+\!\frac{a_3}{\phi }\frac{h_{\hat{\beta }}}{h}\right) ^2\right\} \!=\!0. \end{aligned}$$
(25)

The values of \(\pi ^S\) and \(\pi ^P\) that give the supremum in (25) are

$$\begin{aligned} \pi ^S&= A+B\frac{h_r}{h}+C\frac{h_{\hat{\beta }}}{h}, \nonumber \\ \pi ^P&= D+E\frac{h_r}{h}+F\frac{h_{\hat{\beta }}}{h}, \end{aligned}$$
(26)

where

$$\begin{aligned} A&= \frac{\alpha \sigma _P(1-\phi +\theta a_2^2)+q\sigma _S\left( (\phi -1)n_4-\theta a_1a_2\right) }{(\phi -1)\sigma _S^2\sigma _P\left( (\phi -1)(1-n_4^2) -\theta (a_1^2+a_2^2-2a_1a_2n_4)\right) }\\&\quad + \frac{\sigma _Z\sigma _S\sigma _P\left( (\phi n_1-\theta a_1)(\phi -1-\theta a_2^2)-(\phi n_2-\theta a_2)((\phi -1)n_4-\theta a_1a_2)\right) }{(\phi -1)\sigma _S^2\sigma _P\left( (\phi -1)(1-n_4^2) -\theta (a_1^2+a_2^2-2a_1a_2n_4)\right) },\\ B&= \frac{\theta \left( -(\phi -1)a_1\sigma _ra_2-\phi \sigma _ra_2^2n_4 +(\phi -1)\sigma _r a_2^2 n_4 +\phi \sigma _ra_1a_2\right) }{\phi (\phi -1) \sigma _S\left( (\phi -1)(1-n_4^2)-\theta (a_1^2+a_2^2-2a_1a_2n_4)\right) },\\ C&= \frac{\theta (\phi -1)a_3(a_1-a_2n_4)+\phi \sigma _{\beta }\left( (1 -\phi +\theta a_2^2)n_5-((1-\phi )n_4+\theta a_1a_2)n_6\right) }{\phi (\phi -1)\sigma _S\left( (\phi -1)(1-n_4^2) -\theta (a_1^2+a_2^2-2a_1a_2n_4)\right) },\\ D&= \frac{q\sigma _S(1-\phi +\theta a_1^2)+\alpha \sigma _P\left( (\phi -1)n_4-\theta a_1a_2\right) }{(\phi -1)\sigma _P^2\sigma _S\left( (\phi -1)(1-n_4^2) -\theta (a_1^2+a_2^2-2a_1a_2n_4)\right) }\\&\quad +\frac{\sigma _Z\sigma _P\sigma _S\left( (\phi n_2-\theta a_2)(\phi -1-\theta a_1^2)-(\phi n_1-\theta a_1)((\phi -1)n_4-\theta a_1a_2)\right) }{(\phi -1)\sigma _P^2\sigma _S\left( (\phi -1)(1-n_4^2) -\theta (a_1^2+a_2^2-2a_1a_2n_4)\right) },\\ E&= \frac{-\theta (\phi -1)(\sigma _ra_2^2-\sigma _r a_1a_2n_4)+\phi (\phi -1)\sigma _r(1-n_4^2)-\phi \theta \sigma _r(a_1^2 -a_1a_2n_4)}{\phi (\phi -1)\sigma _P\left( (\phi -1)(1-n_4^2) -\theta (a_1^2+a_2^2-2a_1a_2n_4)\right) },\\ F&= \frac{\theta (\phi -1)a_3(a_2-a_1n_4)+\phi \sigma _{\beta }\left( (1 -\phi )(n_6-n_4n_5)+\theta (a_1^2n_6-a_1a_2n_5)\right) }{\phi (\phi -1) \sigma _P\left( (\phi -1)(1-n_4^2)-\theta (a_1^2+a_2^2-2a_1a_2n_4)\right) }. \end{aligned}$$

Substituting the values of \(\pi ^S, \pi ^P\) given in (26) into the Eq. (25) and dividing by \(h\), we obtain

$$\begin{aligned}&h_t+(\phi r-\phi \hat{\beta }+C_1)h+(-\kappa r+C_2)h_r+(-\lambda \hat{\beta }+C_3)h_{\hat{\beta }} \nonumber \\&\quad +C_4h_{rr}+C_5h_{r\hat{\beta }}+C_6h_{\hat{\beta } \hat{\beta }}+C_7\frac{h_r^2}{h}+C_8\frac{h_{\hat{\beta }}^2}{h} +C_9\frac{h_rh_{\hat{\beta }}}{h}=0, \end{aligned}$$
(27)

where

$$\begin{aligned} C_1&= \frac{\phi \sigma _Z^2}{2}\left( 1+\phi -\theta \right) +\phi \left( \alpha A+qD\right) +\frac{\phi (\phi -1)}{2}\left( \sigma _S^2A^2+\sigma _P^2D^2 +2\sigma _S\sigma _Pn_4AD\right) \\&\quad -\,\phi ^2\sigma _Z\left( \sigma _Sn_1A\!+\!\sigma _Pn_2D\right) \!-\!\frac{\phi \theta }{2}\left( \sigma _Sa_1A+\sigma _Pa_2D\right) ^2 +\phi \theta \left( \sigma _Sa_1A+\sigma _Pa_2D\right) \sigma _Z,\\ C_2&= \kappa \bar{r}+\sigma _Z\sigma _rn_2\left( \phi -\theta \right) +\phi \left( \alpha B+qE\right) -\phi \theta \left( \sigma _Sa_1A+\sigma _Pa_2D\right) \left( \sigma _Sa_1B+\sigma _Pa_2E\right) \\&\quad +\,\phi (\phi -1)\left( \sigma _S^2AB+\sigma _P^2DE+\sigma _S\sigma _Pn_4(AE+BD) \right) \\&\quad -\,\phi \sigma _r\left( \sigma _PD+\sigma _Sn_4A\right) -\phi ^2\sigma _Z\left( \sigma _Sn_1B+\sigma _Pn_2E\right) \\&\quad -\,\phi \theta \left( -\frac{\sigma _ra_2}{\phi }\left( \sigma _Sa_1A +\sigma _Pa_2D\right) -\sigma _Z\left( \sigma _Sa_1B+\sigma _Pa_2E\right) \right) ,\\ C_3&= \lambda \bar{\beta }-\left( \sigma _Z\sigma _{\beta }n_3+m\right) \left( \phi -\theta \right) +\phi \left( \alpha C+qF\right) -\phi ^2\sigma _Z\left( \sigma _Sn_1C+\sigma _Pn_2F\right) \\&\quad +\, \phi (\phi -1)\left( \sigma _S^2AC+\sigma _P^2DF+\sigma _S\sigma _Pn_4(AF+CD) \right) +\phi \sigma _{\beta }\left( \sigma _Sn_5A+\sigma _Pn_6D\right) \\&\quad -\, \phi \theta \left( \sigma _Sa_1A+\sigma _Pa_2D\right) \left( \sigma _Sa_1C +\sigma _Pa_2F\right) \\&\quad -\, \phi \theta \left( \frac{a_3}{\phi }\left( \sigma _Sa_1A+\sigma _Pa_2D\right) -\sigma _Z\left( \sigma _Sa_1C+\sigma _Pa_2F\right) \right) ,\\ C_4&= \frac{\sigma _r^2}{2},\\ C_5&= -\sigma _r\sigma _{\beta }n_6,\\ C_6&= \frac{1}{2}\left( (A_Z\sigma _Z)^2+(A_S\sigma _S)^2 +(A_P\sigma _P)^2\right) ,\\ C_7&= -\frac{\phi \theta }{2}\left( \frac{\sigma _rn_2}{\phi }\right) ^2 +\frac{\phi (\phi -1)}{2}\left( \sigma _S^2B^2+\sigma _P^2E^2 +2\sigma _S\sigma _Pn_4BE\right) \\&\quad -\phi \sigma _r\left( \sigma _Sn_4B+\sigma _PE\right) -\frac{\phi \theta }{2}\left( \sigma _Sa_1B+\sigma _Pa_2E\right) ^2 +\theta \sigma _ra_2\left( \sigma _Sa_1B+\sigma _Pa_2E\right) ,\\ C_8&= -\frac{\phi \theta }{2}\left( \frac{\sigma _Z\sigma _{\beta }n_3 +m}{\phi \sigma _Z}\right) ^2+\frac{\phi (\phi -1)}{2}\left( \sigma _S^2C^2 +\sigma _P^2F^2+2\sigma _S\sigma _Pn_4CF\right) \\&\quad +\phi \sigma _{\beta }\left( \sigma _Sn_5C+\sigma _Pn_6F\right) -\frac{\phi \theta }{2}\left( \sigma _Sa_1C+\sigma _Pa_2F\right) ^2-\theta a_3\left( \sigma _Sa_1C+\sigma _Pa_2F\right) ,\\ C_9&= \phi (\phi -1)\left( \sigma _S^2BC+\sigma _P^2EF +\sigma _S\sigma _Pn_4(BF+CE)\right) \\&\quad -\phi \sigma _r\left( \sigma _PF+\sigma _Sn_4C\right) +\phi \sigma _{\beta }\left( \sigma _Sn_5B+\sigma _Pn_6E\right) \\&\quad -\phi \theta \left( \sigma _Sa_1B+\sigma _Pa_2E\right) \left( \sigma _Sa_1C +\sigma _Pa_2F\right) \\&\quad -\theta a_3\left( \sigma _Sa_1B+\sigma _Pa_2E\right) +\theta \sigma _ra_2\left( \sigma _Sa_1C+\sigma _Pa_2F\right) . \end{aligned}$$

The function \(h(t,r,\hat{\beta })\) that solves (27) is given by

$$\begin{aligned} h(t,r,\hat{\beta })=\exp \left( \tilde{a}(t)\hat{\beta } +\tilde{b}(t)r+c(t)\right) , \end{aligned}$$

where functions \(\tilde{a}, \tilde{b}, c\) are the solution to the following system of ordinary differential equations

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \tilde{a}^{\prime }-\phi -\lambda \tilde{a}=0, &{} \tilde{a}(T)=0,\\ \tilde{b}^{\prime }+\phi -\kappa \tilde{b}=0, &{} \tilde{b}(T)=0,\\ c^{\prime }+C_1+C_2\tilde{b}+C_3\tilde{a}+(C_4+C_7)\tilde{b}^2\\ \quad +(C_6+C_8)\tilde{a}^2+(C_5+C_9)\tilde{a}\tilde{b}=0, &{} c(T)=0. \end{array}\right. \nonumber \\ \end{aligned}$$
(28)

Solving this system of equations we obtain

$$\begin{aligned} \tilde{a}(t)&= \frac{\phi }{\lambda }\left( e^{-\lambda (T-t)}-1\right) ,\\ \tilde{b}(t)&= -\frac{\phi }{\kappa }\left( e^{-\kappa (T-t)}-1\right) , \end{aligned}$$

and

$$\begin{aligned} c(t)&= C_1(T-t)+C_2\frac{\phi }{\kappa }(T-t-b_{\kappa }(T-t)) +C_3\frac{\phi }{\lambda }\left( b_{\lambda }(T-t)-T+t\right) \nonumber \\&\quad +(C_4+C_7)\left( \frac{\phi }{\kappa }\right) ^2\left( T-t-2b_{\kappa }(T-t) +b_{2\kappa }(T-t)\right) \nonumber \\&\quad +(C_6+C_8)\left( \frac{\phi }{\lambda }\right) ^2\left( b_{2\lambda }(T-t) -2b_{\lambda }(T-t)+T-t\right) \nonumber \\&\quad +(C_5+C_9)\left( \frac{\phi ^2}{\lambda \kappa }\right) \left( b_{\lambda +\kappa }(T-t)-b_{\lambda }(T-t)-b_{\kappa }(T-t)+T-t\right) ,\quad \quad \quad \end{aligned}$$
(29)

where the function \(b_{\nu }(T-t)\) is given by (2).

Appendix C: Suboptimal investment strategy

The worst-case value of \(e\) is the same as in Appendix B. Let us look for a solution in the form \(v^{{\varvec{\Pi }}}(t,z,x,r,\hat{\beta }) =\frac{1}{\phi }\left( \frac{x}{z}\right) ^{\phi }h^{{\varvec{\Pi }}}(t,r, \hat{\beta })\) and assume that \(\varPsi =\frac{\theta }{h^{{\varvec{\Pi }}}}\left( \frac{x}{z}\right) ^{-\phi }\). Then we obtain the equationFootnote 12

$$\begin{aligned}&h_t^{{\varvec{\Pi }}}-\phi \hat{\beta }h^{{\varvec{\Pi }}}+\phi \left( r +\alpha \pi ^S+q\pi ^P\right) h^{{\varvec{\Pi }}}+\kappa \left( \bar{r}-r \right) h_r^{{\varvec{\Pi }}}+\lambda \left( \bar{\beta } -\hat{\beta }\right) h_{\hat{\beta }}^{{\varvec{\Pi }}} \nonumber \\&\quad +\frac{\phi (\phi +1)\sigma _Z^2}{2}h^{{\varvec{\Pi }}}-\phi ^2\sigma _Z\left( \sigma _S\pi ^Sn_1+\sigma _P\pi ^Pn_2 \right) h^{{\varvec{\Pi }}}+\phi \sigma _Z\sigma _rn_2h_r^{{\varvec{\Pi }}} \nonumber \\&\quad -\phi \sigma _Z^2A_Zh_{\hat{\beta }}^{{\varvec{\Pi }}}+\frac{\phi (\phi -1)}{2}\left( (\sigma _S\pi ^S)^2+(\sigma _P\pi ^P)^2 +2\sigma _S\sigma _P\pi ^S\pi ^Pn_4\right) h^{{\varvec{\Pi }}}\nonumber \\&\quad -\phi \sigma _r\left( \sigma _P\pi ^P+\sigma _S\pi ^Sn_4\right) h_r^{{\varvec{\Pi }}}+\phi \sigma _{\beta }\left( \sigma _S\pi ^Sn_5+\sigma _P\pi ^Pn_6 \right) h_{\hat{\beta }}^{{\varvec{\Pi }}}+\frac{\sigma _r^2}{2}h_{rr}^{{\varvec{\Pi }}} \nonumber \\&\quad -\sigma _r\sigma _{\beta }n_6h_{r\hat{\beta }}^{{\varvec{\Pi }}}+\frac{1}{2}\left( (A_Z\sigma _Z)^2+(A_S\sigma _S)^2 +(A_P\sigma _P)^2\right) h_{\hat{\beta }\hat{\beta }}^{{\varvec{\Pi }}}\nonumber \\&\quad -\frac{\phi \theta }{2}h^{{\varvec{\Pi }}}\left( \sigma _S\pi ^Sa_1 +\sigma _P\pi ^Pa_2-\sigma _Z -\frac{\sigma _ra_2}{\phi }\frac{h_r^{{\varvec{\Pi }}}}{h^{{\varvec{\Pi }}}} +\frac{a_3}{\phi }\frac{h_{\hat{\beta }}^{{\varvec{\Pi }}}}{h^{{\varvec{\Pi }}}} \right) ^2=0. \end{aligned}$$
(30)

The Eq. (30) can be written as

$$\begin{aligned}&h_t^{{\varvec{\Pi }}}+(\phi r-\phi \hat{\beta }+A_1)h^{{\varvec{\Pi }}}+(-\kappa r+A_2)h_r^{{\varvec{\Pi }}}+(-\lambda \hat{\beta }+A_3)h_{\hat{\beta }}^{{\varvec{\Pi }}} \nonumber \\&\quad +A_4h_{rr}^{{\varvec{\Pi }}}+A_5h_{r\hat{\beta }}^{{\varvec{\Pi }}} +A_6h_{\hat{\beta }\hat{\beta }}^{{\varvec{\Pi }}} +A_7\frac{(h_r^{{\varvec{\Pi }}})^2}{h^{{\varvec{\Pi }}}}+A_8 \frac{(h_{\hat{\beta }}^{{\varvec{\Pi }}})^2}{h^{{\varvec{\Pi }}}}+A_9 \frac{h_r^{{\varvec{\Pi }}}h_{\hat{\beta }}^{{\varvec{\Pi }}}}{h^{{\varvec{\Pi }}}}=0,\quad \quad \end{aligned}$$
(31)

where

$$\begin{aligned} A_1&= \phi (\alpha \pi ^S+q\pi ^P)+\frac{\phi (\phi +1)\sigma _Z^2}{2} -\phi ^2\sigma _Z(\sigma _S\pi ^Sn_1+\sigma _P\pi ^Pn_2)\\&\quad +\frac{\phi (\phi -1)}{2}\left( (\sigma _S\pi ^S)^2+(\sigma _P\pi ^P)^2 +2\sigma _S\sigma _P\pi ^S\pi ^Pn_4\right) -\frac{\theta }{2}Q^2\phi ,\\ A_2&= \kappa \bar{r}+\phi \sigma _Z\sigma _rn_2-\phi \sigma _r(\sigma _P\pi ^P +\sigma _S\pi ^Sn_4) +Q\sigma _ra_2\theta ,\\ A_3&= \lambda \bar{\beta }-\phi \sigma _Z^2A_Z +\phi \sigma _{\beta }(\sigma _S\pi ^Sn_5+\sigma _P\pi ^Pn_6)-Qa_3\theta ,\\ A_4&= \frac{\sigma _r^2}{2},\\ A_5&= -\sigma _r\sigma _{\beta }n_6,\\ A_6&= \frac{1}{2}\left( (A_Z\sigma _Z)^2+(A_S\sigma _S)^2+(A_P\sigma _P)^2 \right) ,\\ A_7&= -\frac{\sigma _r^2a_2^2\theta }{2\phi },\\ A_8&= -\frac{a_3^2\theta }{2\phi },\\ A_9&= \frac{\sigma _ra_2a_3\theta }{\phi },\\ Q&= \sigma _S\pi ^Sa_1+\sigma _P\pi ^Pa_2-\sigma _Z. \end{aligned}$$

Assuming that a suboptimal strategy \({\varvec{\Pi }}\) does not depend on the state variables, the only difference between (27) in Appendix B and (31) is that the coefficients \(A_1, A_2, A_3\) (in contrast with \(C_1, C_2, C_3\)) can be time-dependent.

The function \(h^{{\varvec{\Pi }}}(t,r,\hat{\beta })\) that solves (31) is given by

$$\begin{aligned} h^{{\varvec{\Pi }}}(t,r,\hat{\beta })=\exp \left( \tilde{a}(t)\hat{\beta } +\tilde{b}(t)r+c^{{\varvec{\Pi }}}(t)\right) \!, \end{aligned}$$

where the function \(c^{{\varvec{\Pi }}}\) is the solution to the following ordinary differential equation

$$\begin{aligned}&(c^{{\varvec{\Pi }}})^{\prime }+A_1+A_2\tilde{b}+A_3\tilde{a}+(A_4+A_7)\tilde{b}^2 +(A_6+A_8)\tilde{a}^2\nonumber \\&\quad +(A_5+A_9)\tilde{a}\tilde{b}=0, \ c^{{\varvec{\Pi }}}(T)=0. \end{aligned}$$
(32)

Appendix D: Detection-error probability

Define the conditional characteristic functions

$$\begin{aligned} f_1(\omega , t, N)&= E^\mathbb{P }[\exp (i\omega \xi _{1,N}) \ | \ \mathcal F _t^{S,Z,r}]=E^\mathbb{P }[\varXi _{1,N}^{i\omega } \ | \ \mathcal F _t^{S,Z,r}], \\ f_2(\omega , t, N)&= E^\mathbb{Q }[\exp (i\omega \xi _{1,N}) \ | \ \mathcal F _t^{S,Z,r}]=E^\mathbb{Q }[\varXi _{1,N}^{i\omega } \ | \ \mathcal F _t^{S,Z,r}] \\&= E^\mathbb{P }[\exp (i\omega \xi _{1,N})\exp (\xi _{1,N})|\mathcal F _t^{S,Z,r}] =E^\mathbb{P }[\varXi _{1,N}^{i\omega + 1}|\mathcal F _t^{S,Z,r}], \end{aligned}$$

where \(i=\sqrt{-1}\) and \(\xi _{1,t}=\ln \xi _t^{e^*}\).

Since the conditional characteristic functions are martingales, the Feyman-Kac theorem implies that functions \(f_1\) and \(f_2\) satisfy

$$\begin{aligned} \frac{\partial f_1}{\partial t}+\frac{1}{2} \varXi _{1,t}^2 (e_t^*)^2\frac{\partial ^2 f_1}{\partial \varXi _{1,t}^2}&= 0, \ \ \ f_1(\omega , N, N)=\varXi _{1,N}^{i\omega },\end{aligned}$$
(33)
$$\begin{aligned} \frac{\partial f_2}{\partial t}+\frac{1}{2} \varXi _{1,t}^2 (e_t^*)^2\frac{\partial ^2 f_2}{\partial \varXi _{1,t}^2}&= 0, \ \ \ f_2(\omega , N, N)=\varXi _{1,N}^{i\omega +1}. \end{aligned}$$
(34)

Let us look for a solution in form \(f_1(\omega ,t,N)=\varXi _{1,t}^{i\omega }e^{D(t)}\). Substituting the trial solution into (33) and dividing the result by \(\varXi _{1,t}^{i\omega }e^{D(t)}\) yields

$$\begin{aligned} D^{\prime }(t)+\frac{1}{2}i\omega (i\omega -1)(e_t^*)^2=0, \ D(N)=0. \end{aligned}$$

Solving this equation we obtain

$$\begin{aligned} D(t)=-\frac{1}{2}\omega ^2\int \limits _t^N (e_s^*)^2 ds -\frac{1}{2}i\omega \int \limits _t^N (e_s^*)^2 ds. \end{aligned}$$

Therefore,

$$\begin{aligned} f_1(\omega , t, N)&= \exp \left( i\omega \left[ -B_t - \int \limits _t^N \frac{(e_s^*)^2}{2} ds-\frac{1+k_S^2+k_P^2}{2}\int \limits _0^t(e_s^*)^2ds\right] \right. \\&\quad \quad \quad \,\,\left. -\, \omega ^2\int \limits _t^N \frac{(e_s^*)^2}{2} ds\right) \end{aligned}$$

where \(B_t=\int \limits _0^t\left( e_s^*d\hat{B}_s^Z+k_Se_s^*d\hat{B}_s^S+k_Pe_s^*d\hat{B}_s^P\right) \)

Similarly, we use \(f_2(\omega ,t,N)=\varXi _{1,t}^{i\omega +1}e^{E(t)}\) as a trial solution, which after substitution into (34) and division by \(\varXi _{1,t}^{i\omega +1}e^{E(t)}\) yields

$$\begin{aligned} E^{\prime }(t)+\frac{1}{2}i\omega (i\omega +1)(e_t^*)^2=0, \ E(N)=0. \end{aligned}$$

Solving this ordinary differential equation we obtain

$$\begin{aligned} E(t)=-\frac{1}{2}\omega ^2\int \limits _t^N (e_s^*)^2 ds+\frac{1}{2} i\omega \int \limits _t^N (e_s^*)^2 ds. \end{aligned}$$

Thus,

$$\begin{aligned} f_2(\omega , t, N)&= \exp \left( i\omega \left[ -B_t-\frac{1+k_S^2+k_P^2}{2}\int \limits _0^t (e_s^*)^2ds+\frac{1}{2}\int \limits _t^N (e_s^*)^2ds\right] \right. \\&\quad \quad \quad \,\,\left. -B_t-\frac{1+k_S^2+k_P^2}{2}\int \limits _0^t(e_s^*)^2ds-\frac{1}{2}\omega ^2 \int \limits _t^N (e_s^*)^2 ds\right) . \end{aligned}$$

It is obvious that

$$\begin{aligned} \mathrm{Re }\left( \frac{f_1(\omega ,0,N)}{i\omega }\right)&= -\frac{1}{\omega }\exp \left( -\frac{1}{2}\omega ^2\int \limits _0^N (e_s^*)^2 ds\right) \sin \left( \frac{1}{2}\omega \int \limits _0^N (e_s^*)^2 ds\right) , \\ \mathrm{Re }\left( \frac{f_2(\omega ,0,N)}{i\omega }\right)&= \frac{1}{\omega }\exp \left( -\frac{1}{2}\omega ^2\int \limits _0^N (e_s^*)^2 ds\right) \sin \left( \frac{1}{2}\omega \int \limits _0^N (e_s^*)^2 ds\right) . \end{aligned}$$

Therefore, the detection-error probability (see formula 861.22 in Dwight 1973) is

$$\begin{aligned} \varepsilon _N(\theta )&= \frac{1}{2}-\frac{1}{2\pi }\int \limits _0^{\infty } \left( \mathrm{Re }\left[ \frac{f_2(\omega ,0,N)}{i\omega }\right] -\mathrm{Re }\left[ \frac{f_1(\omega ,0,N)}{i\omega }\right] \right) d\omega \\&= \frac{1}{2}-\frac{1}{2} \mathrm{erf }\left( \frac{\sqrt{\tilde{K}}}{2}\right) , \end{aligned}$$

where \(\tilde{K}=\dfrac{1}{2}\int _0^N(e_s^*)^2ds\) and \(\mathrm{erf }(x)=\dfrac{2}{\sqrt{\pi }}\int _0^x e^{-t^2}dt\).

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Munk, C., Rubtsov, A. Portfolio management with stochastic interest rates and inflation ambiguity. Ann Finance 10, 419–455 (2014). https://doi.org/10.1007/s10436-013-0238-1

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