Governmental incentives for green bonds investment

Abstract

Motivated by the recent studies on the green bond market, we build a Principal–Agent model in which an investor trades on a portfolio of green and conventional bonds, both issued by the same governmental entity. The government provides incentives to the bondholder in order to increase the amount invested in green bonds. These incentives are, optimally, indexed on the prices of the bonds, their quadratic variation and covariation. We show numerically on a set of French governmental bonds that our methodology outperforms the current tax-incentives systems in terms of green investments. Moreover, it is robust to model specification for bond prices and can be applied to a large portfolio of bonds using classical optimisation methods.

Appendices

Appendix

Weak formulation of the problem

We work on the canonical space $\mathcal {Q}$ of continuous functions on [0, T] with Borel algebra $\mathcal {F}$. The $(d^g+d^c+2)$-dimensional canonical process is

$$\begin{aligned} \mathcal {B:}= \begin{pmatrix} X \\ W^g \\ W^c \\ W^I \end{pmatrix} \end{aligned}$$

and $\mathbb {F}=(\mathcal {F}_t)_{t\in [0,T]}$ is its natural filtration. We define $\mathbb {P}_0$ as the $(d^g+d^c+1)$-dimensional Wiener measure on $\mathcal {Q}$. Thus, $\mathcal {B}$ is a $(d^g+d^c+2)$-dimensional Brownian motion where $(W^g,W^c,W^I)$ has a correlation matrix $\Sigma$ under $\mathbb {P}_0$. We also define $\mathcal {M}(\Omega )$ as the set of probability measures on $(\mathcal {Q},\mathcal {F}_T)$ and

$$\begin{aligned} \mathbb {H}^2(\mathbb {P}_0)&:=\bigg \{(\pi _t)_{t\in [0,T]}:B\text { -valued}, \mathbb {F}\text {-predictable processes such that }\\&\qquad \mathbb {E}^{\mathbb {P}}\bigg [\int _0^T\Vert \pi _t\Vert _2^2 \mathrm {d}t \bigg ] <+\infty \bigg \}. \end{aligned}$$

We consider the following family of processes, indexed by $\pi \in \mathbb {H}^2(\mathbb {P}_0)$

and define the set $\mathcal {P}_m$ as the set of probability measures $\mathbb {P}^\pi \in \mathcal {M}(\mathcal {Q})$ of the form

$$\begin{aligned} \mathbb {P}^\pi = \mathbb {P}_0 \circ (\mathcal {X}^\pi )^{-1}, \; \text {for all } \pi \in \mathbb {H}^2(\mathbb {P}_0). \end{aligned}$$

Thanks to Bichteler [10], we can define a pathwise version of the quadratic variation process $\langle \mathcal {B}\rangle$ and of its density process with respect to the Lebesgue measure $\hat{\alpha }_t := \frac{d\langle \mathcal {B}\rangle _t}{\mathrm {d}t}$. As the processes $\pi \in \mathcal {A}\subset \mathbb {H}^2(\mathbb {P}_0)$ have all their coordinates strictly positive, the volatility of $\mathcal {B}$ is invertible, which implies in particular that the process $W_t = \int _0^t \hat{\alpha }_s^{-\frac{1}{2}}\mathrm {d}\mathcal {B}_s$ is an $\mathbb {R}^{d^g+d^c+2}$-valued, $\mathbb {P}$–Brownian motion with correlation matrix $\Sigma$ for every $\mathbb {P}\in \mathcal {P}_m$. According to Soner, Touzi, and Zhang [46], there exists an $\mathbb {F}^{\mathcal {B}}$–progressively measurable mapping $\beta _\pi : [0,T]\times \mathcal {Q} \longrightarrow \mathbb {R}^{d^g+d^c+2}$ such that

$$\begin{aligned}&\mathcal {B} = \beta _\pi (\mathcal {X}^\pi ),\; \mathbb {P}_0\mathrm{--a.s}, \; W = \beta _\pi (\mathcal {B}),\; \mathbb {P}^\pi \mathrm{--a.s}, \\&\hat{\alpha }(\mathcal {B})= \pi \big (\beta _\pi (\mathcal {B})\big ), \; \mathrm {d}t \otimes \mathrm {d}\mathbb {P}^\pi \mathrm{--a.e.} \end{aligned}$$

In particular, the canonical process $\mathcal {B}$ admits the following dynamics for all $\pi \in \mathcal {A}$

The first coordinate of the canonical process is the desired output process, the $d^g$ next coordinates are the contractible sources of risk, that is the $d^g$ green bonds and the index of conventional bond, and the last $d^c$ coordinates are the non-contractible sources of risk. Then, we can introduce easily the drift of the output process by the means of Girsanov theorem. Denote

$$\begin{aligned} \frac{\mathrm {d}\mathbb {Q}}{\mathrm {d}\mathbb {P}^\pi } := \mathcal {E}\bigg (\int _0^\cdot {\tilde{\Sigma }}(s) \mathrm {d}W_s \bigg )_T, \end{aligned}$$

a change of measure independent of the control process $\pi$, where $\tilde{\Sigma }: [0,T]\longrightarrow \mathcal {M}_{d^g+d^c+2}(\mathbb {R})$ is such that

$$\begin{aligned} \tilde{\Sigma }(t) := \begin{pmatrix} \big (\frac{r^g(t)+\eta ^g(t)\circ \sigma ^g(t)}{\sigma ^g(t)}\big )^{\top } &{} \big (\frac{r^c(t)+\eta ^c(t)\circ \sigma ^c(t)}{\sigma ^c(t)}\big )^{\top } &{} \frac{\mu ^I(t)}{\sigma ^I(t)} \\ \mathbf {0}_{d^g+d^c+1,d^g} &{} \mathbf {0}_{d^g+d^c+1,d^c} &{}\mathbf {0}_{d^g+d^c+1,1} \end{pmatrix}. \end{aligned}$$

We finally obtain the desired dynamics for the output process and the $d^g+d^c+1$ sources of risk.

We can define the functions Proof of Theorem 3.2

We can define the functions $\sigma :[0,T]\times K \longrightarrow \mathcal {M}_{d^g+d^c+2,d^g+d^c+1}(\mathbb {R}), \lambda :[0,T]\longrightarrow \mathbb {R}^{d^g+d^c+1}$ such that the set of contractible variables $(B_t)_{t\in [0,T]}$ can be rewritten for all $\pi \in \mathcal {A}$ as

$$\begin{aligned} \mathrm {d}B_t = \sigma (t,\pi _t) \big ( \lambda (t) \mathrm {d}t + \mathrm {d}W_t\big ), \end{aligned}$$

(B.1)

where for all $(t,p)\in [0,T]\times K$,

$$\begin{aligned}&\sigma (t,p) := \begin{pmatrix} \big (p^g \sigma ^g(t)\big )^\top &{} \big (p^c \sigma ^c(t)\big )^{\top } &{} p^I \sigma ^I(t) \\ \text {diag}(\sigma ^g(t)) &{} \mathbf {0}_{d^g,d^c} &{} \mathbf {0}_{d^g,1} \\ \mathbf {0}_{1,d^g}&{} \mathbf {0}_{1,d^c} &{} \sigma ^I(t) \\ \mathbf {0}_{d^c,d^g} &{} \text {diag}(\sigma ^c(t)) &{} \mathbf {0}_{d^c,1} \end{pmatrix},\; \\&\quad \lambda (t) := \begin{pmatrix} \big (\frac{r^g(t)+\eta ^g(t)\circ \sigma ^g(t)}{\sigma ^g(t)}\big )^{\top }&\big (\frac{r^c(t)+\eta ^c(t)\circ \sigma ^c(t)}{\sigma ^c(t)}\big )^{\top }&\frac{\mu ^I(t)}{\sigma ^I(t)} \end{pmatrix}^\top , \end{aligned}$$

Thanks to Assumption 2.2 and the definition of $\mathcal {A}$, the functions $\sigma$, and $\lambda$ are bounded. As the function $\sigma (t,\pi )$ is continuous in time for some constant control process $\pi \in \mathcal {A}$, there always exists a weak solution to (B.1). Thanks to the boundedness of the function $\lambda$, we can use Girsanov’s theorem which guarantees that every $\pi \in \mathcal {A}$ induces a weak solution for

$$\begin{aligned} B_t = B_0 + \int _0^t \sigma (s,\pi _s) \mathrm {d}W^{\prime }_s, \; \frac{\mathrm {d}\mathbb {P}^\prime }{\mathrm {d}\mathbb {P}}\bigg |_{\mathcal {F}_T} = \mathcal {E}\bigg (\int _0^\cdot \lambda (s)\cdot \mathrm {d}W_s \bigg )_T, \end{aligned}$$

where $W^{\prime }$ is a $\mathbb {P}^{\prime }$–Brownian motion.

The cost function $k: K\rightarrow \mathbb {R}$ is measurable and bounded by boundedness of the elements of K. We introduce the norms

$$\begin{aligned} \Vert Z^e\Vert _{\mathbb {H}^p}^p = \sup _{\pi \in \mathcal {A}}\mathbb {E}^{\pi }\bigg [\bigg ( \int _0^T\big \Vert \tilde{\sigma }(t,\pi _t) Z_t\big \Vert ^2 \mathrm {d}t\bigg )^{p/2}\bigg ], \; \Vert Y^e\Vert _{\mathbb {D}^p}^p = \sup _{\pi \in \mathcal {A}}\mathbb {E}^{\pi }\bigg [\sup _{t\in [0,T]} |Y_t|^p\bigg ], \end{aligned}$$

for any $\mathbb {F}$-predictable, $\mathbb {R}^{d^g+d^c+2}$-valued process $Z^e$ and $\mathbb {R}$-valued process $Y^e$, and for all $(t,p)\in [0,T]\times K$, $\widetilde{\sigma }:[0,T]\times K\rightarrow \mathcal {M}_{d^g+d^c+2}(\mathbb {R})$ is such that

$$\begin{aligned} \widetilde{\sigma }^2(t,\pi _t) = \sigma (t,p) \sigma ^\top (t,p). \end{aligned}$$

We also define the functions $H^e: [0,T]\times \mathbb {R}^{d^g+d^c+2}\times \mathbb {S}_{d^g+d^c+2}(\mathbb {R})\times \mathbb {R}\longrightarrow \mathbb {R}$ and $h^e: [0,T]\times \mathbb {R}^{d^g+d^c+2}\times \mathbb {S}_{d^g+d^c+2}(\mathbb {R})\times \mathbb {R}\times K \longrightarrow \mathbb {R}$ as

$$\begin{aligned}&H^e(t,z,g,y):=\sup _{p\in K}h^e(t,z,g,y,p), \\&h^e(t,z,g,y,p):= -\gamma k(p)y + z\cdot \sigma (t,p)\lambda (t) + \frac{1}{2}\mathrm {Tr}\Big [g\sigma (t,p)\Sigma (\sigma (t,p))^\top \Big ]. \end{aligned}$$

We introduce the set of so-called admissible incentives $\mathcal {ZG}^e$ as the set of $\mathbb {F}$-predictable processes $(Z^e,\Gamma ^e)$ valued in $\mathbb {R}^{d^g+d^c+2}\times \mathbb {S}_{d^g+d^c+2}(\mathbb {R})$ such that

$$\begin{aligned} \Vert Z^e\Vert _{\mathbb {H}^p}^p + \Vert Y^{e,Z^e,\Gamma ^e}\Vert _{\mathbb {D}^p}^p <+\infty , \end{aligned}$$

(B.2)

for some $p>1$ where for $y^e_0\in \mathbb {R}$,

$$\begin{aligned} Y_t^{e, y_0^e, Z^e,\Gamma ^e} := y_0^e + \int _0^t Z^e_s \mathrm {d}B_s + \frac{1}{2}\mathrm {Tr}\Big [\Gamma ^e_s \mathrm {d}\langle B\rangle _s \Big ] - H^e\big (s,Z_s^e,\Gamma _s^e,Y_s^{e, Z^e,\Gamma ^e}\big ) \mathrm {d}s. \end{aligned}$$

Condition (B.2) guarantees that the process $(Y_t^{e, y_0^e, Z^e,\Gamma ^e})_{t\in [0,T]}$ is well defined: provided that the right-hand side integrals are well defined, and by noting that $H^e$ is Lipschitz in its last variable (since the cost function k is bounded), $(Y_t^{e, y_0^e, Z^e,\Gamma ^e})_{t\in [0,T]}$ is the unique solution of an ODE with random coefficient. Moreover, as K is a compact set and $h^e$ is continuous with respect to its last variable, the supremum with respect to p is always attained. As $(Z^e,\Gamma ^e)=(\mathbf {0}_{d^g+d^c+2},\mathbf {0}_{d^g+d^c+2,d^g+d^c+2})\in \mathcal {ZG}^e$, this set is non-empty and we are in the setting of Cvitanić, Possamaï, and Touzi [18]. Using [18, Proposition 3.3 and Theorem 3.6], we obtain that without reducing the utility of the Principal, any admissible contract admits the representation

$$\begin{aligned} U_A(\xi ) = Y_T^{e, Z^e,\Gamma ^e}, \end{aligned}$$

Define for all $t\in [0,T]$ the processes

$$\begin{aligned}&Z_t =: -\frac{Z_t^e}{\gamma Y_t^{e,y_0^e, Z^e,\Gamma ^e}}, \; \Gamma _t := -\frac{\Gamma _t^e}{\gamma Y_t^{e, y_0^e, Z^e,\Gamma ^e}},\; Y_t^{y_0,Z,\Gamma } \\&\quad = y_0 + \int _0^T Z_s \mathrm {d}B_s + \frac{1}{2}\mathrm {Tr}\big [\big (\Gamma _s + \gamma Z_s Z_s^\top \mathrm {d}\langle B\rangle _s \big ] - H\big (s,Z_s,\Gamma _s\big ) \mathrm {d}s, \end{aligned}$$

where $H:[0,T]\times \mathbb {R}^{d^g+d^c+2}\times \mathbb {S}_{d^g+d^c+2}(\mathbb {R})\longrightarrow \mathbb {R}$ is defined by $H(t,z,g) = \sup _{p\in K} h(t,z,g,p)$ and

$$\begin{aligned} \begin{aligned} \mathcal {ZG}:= \bigg \{(Z_t,\Gamma _t)_{t\in [0,T]}&: \mathbb {R}^{d^g+d^c+2}\times \mathbb {S}_{d^g+d^c+2}(\mathbb {R})\text {-valued, }\mathbb {F}\text {-predictable processes s.t}\\&\big (-\gamma Z_t U_A(Y_t^{y_0,Z,\Gamma }),-\gamma \Gamma _t U_A(Y_t^{y_0,Z,\Gamma })\big )_{t\in [0,T]}\in \mathcal {ZG}^e \bigg \}. \end{aligned} \end{aligned}$$

(B.3)

An application of Itō’s formula leads to $\xi = Y_T^{y_0,Z,\Gamma }$. Thus, we obtain the desired representation for admissible contracts and $V_A( Y_T^{y_0,Z,\Gamma })=U_A(y_0)$. The characterisation of $\mathcal {A}(Y_T^{y_0,Z,\Gamma })$ is a direct consequence of Cvitanić, Possamaï, and Touzi [18, Proposition 3.3].

Green investments with stochastic interest rates

1.1 Framework

In the article, we considered a deterministic structure for the short-term rates. However, this omits some important stylised facts of the yield curve. In this section we show that at the expense of the use of stochastic control, the government can provide incentives based on short-term rates following a one factor stochastic model.

We now assume that the vectors of short rate dynamics of the green bonds are given by

$$\begin{aligned} \mathrm {d}r_t^{g} := a^{g}(t,r_t^g) \mathrm {d}t + \text {diag}(b^{g}) \mathrm {d}W_t^{g,r}, \end{aligned}$$

(C.1)

where $b^g\in \mathbb {R}^{d^g}_+$, $a^{g}:[0,T]\times \mathbb {R}^{d^g}\longrightarrow \mathbb {R}^{d^g}$ and $W^{g,r}$ is a $d^g$-dimensional Brownian motion of correlation matrix $\Sigma ^{g,r}$.

Remark C.1

For notational simplicity, we assume no dependence between the risk sources of the short-term rates and the ones of the bonds. Allowing such dependence is straightforward and does not lead to a higher dimension of the control problem.

We contract only on the portfolio process, the risk factors of the green bonds and of the stochastic short-term rate of the green bonds, and the risk factor of the index of conventional bonds. The new sets of state variables are

where the superscript S stands for stochastic, which can be written as

where for all $t\in [0,T]$, $p=(p^g,p^c,p^I)\in \mathbb {R}^{d^g}\times \mathbb {R}^{d^c}\times \mathbb {R}$, $r^g\in \mathbb {R}^{d^g}$

We now specify the new set of admissible contracts that we consider for the incentives proposed by the government.

1.2 Representation of admissible contracts

Define $\mathcal {C}^S$ as the set of admissible contracts in the case of stochastic short-term rates (the admissibility conditions are the same as for the set $\mathcal {C}$) and for any $\pi \in \mathcal {A}$ we introduce the following quantities

We define $h^S:[0,T]\times \mathbb {R}^{2d^g+d^c+2}\times \mathbb {S}_{2d^g+d^c+2}(\mathbb {R})\times \mathbb {R}^{d^g}\times K\longrightarrow \mathbb {R}$ such that

$$\begin{aligned} h^S(t,z,g,r^g,p) = -k(p) + z \cdot \mu ^S(t,p,r^g) + \frac{1}{2}\mathrm {Tr}\Big [ g\Sigma ^S(t,p)\Sigma (\Sigma ^S(t,p))^{\top } \Big ], \end{aligned}$$

and for all $(t,z,g,r^g)\in [0,T]\times \mathbb {R}^{2d^g+d^c+2}\times \mathbb {S}_{2d^g+d^c+2}(\mathbb {R})$ we define

$$\begin{aligned} \mathcal {O}^S(t,z,g,r^g):= \Big \{{\hat{p}}\in K: {\hat{p}} \in \underset{p\in K}{\text {argmax }}h^S(t,z,g,r^g,p)\Big \}, \end{aligned}$$

as the set of maximisers of $h^S$ with respect to its last variable for $(t,z,g,r^g)$ fixed. Following Schäl [44], there exists at least one Borel-measurable map ${\hat{\pi }}:[0,T]\times \mathbb {R}^{2d^g+d^c+2}\times \mathbb {S}_{2d^g+d^c+2}(\mathbb {R})\times \mathbb {R}^{d^g}\longrightarrow K$ such that for every $(t,z,g,r^g)\in [0,T]\times \mathbb {R}^{2d^g+d^c+2}\times \mathbb {S}_{2d^g+d^c+2}(\mathbb {R})\times \mathbb {R}^{d^g}$, ${\hat{\pi }}(t,z,g,r^g)\in \mathcal {O}^S(t,z,g,r^g)$. We denote by $\mathcal {O}^S$ the set of all such maps. By analogy with Theorem 3.2, the following theorem states the form of any admissible contracts in this setting.

Theorem C.2

Without reducing the utility of the Principal, we can restrict the study of admissible contracts to the set $\mathcal {C}^S_1$ where any $\xi \in \mathcal {C}^S_1$ is of the form $\xi =Y_T^{y_0,Z^S,\Gamma ^S,{\hat{\pi }}}$ where for all $t\in [0,T]$,

$$\begin{aligned} Y_t^{y_0,Z^S,\Gamma ^S,{\hat{\pi }}}&:= y_0 + \int _0^t Z^S_s\cdot \mathrm {d}B_s + \frac{1}{2}\mathrm {Tr}\Big [\big (\Gamma ^S_s + \gamma Z^S_s (Z_s^S)^{\top }\big )\mathrm {d}\langle B^S\rangle _s\Big ]\nonumber \\&\quad - h^S\big (s,Z^S_s,\Gamma ^S_s,r_s^g,{\hat{\pi }}(s,Z_s,\Gamma _s,r_s^g)\big )\mathrm {d}s, \end{aligned}$$

(C.2)

where ${\hat{\pi }}\in \mathcal {O}^S$ and $(Z^S_t)_{t\in [0,T]}, (\Gamma _t^S)_{t\in [0,T]}$ are respectively $\mathbb {R}^{2d^g+2+d^c}$ and $\mathbb {S}_{2d^g+2+d^c}(\mathbb {R})$-valued, $\mathbb {F}$-predictable processes satisfying similar conditions as the elements of $\mathcal {ZG}$. We denote the set of admissible incentives as $\mathcal {ZG}^S$. Moreover in the present case of stochastic rates for green bonds

$$\begin{aligned}&V^A(Y_T^{y_0,Z^S,\Gamma ^S,{\hat{\pi }}})=U_A(y_0),\;\\&\quad \mathcal {A}\big (Y_T^{y_0,Z^S,\Gamma ^S,{\hat{\pi }}}\big ) = \Big \{\big ({\hat{\pi }}(t,Z^S_t,\Gamma ^S_t,r_t^g)\big )_{t\in [0,T]}, \hat{\pi }\in \mathcal {O}^S, (Z^S_t,\Gamma ^S_t)_{t\in [0,T]}\in \mathcal {ZG}^S\Big \}. \end{aligned}$$

We now set

where for all $t\in [0,T]$

We define $h^{\text {obs},S}:[0,T]\times \mathbb {R}^{2d^g+2}\times \mathbb {S}_{2d^g+2}(\mathbb {R})\times \mathbb {R}^{d^g}\times K\longrightarrow \mathbb {R}$ such that

$$\begin{aligned}&h^{\text {obs}}(t,z^{\text {obs},S},g^{\text {obs},S},r^g,p) = -k(p) + z^{\mathrm{obs},S} \cdot \mu ^{\text {obs},S}(t,p,r^g)\\&\quad + \frac{1}{2}\mathrm {Tr}\Big [ g^{\mathrm{obs},S}\Sigma ^{\mathrm{obs},S}(t,p)\Sigma (\Sigma ^{\mathrm{obs},S}(t,p))^{\top } \Big ], \end{aligned}$$

and for all $(t,z^{\text {obs},S},g^{\text {obs},S},r^g)\in [0,T]\times \mathbb {R}^{2d^g+2}\times \mathbb {S}_{2d^g+2}(\mathbb {R})\times \mathbb {R}^{d^g}$ we define

$$\begin{aligned} \mathcal {O}^{\text {obs},S}(t,z^{\text {obs}},g^{\text {obs}},r^g) := \Big \{{\hat{p}}\in K: {\hat{p}}\in \underset{p\in K}{\text {argmax }}h^{\text {obs},S}(t,z^{\text {obs},S},g^{\text {obs},S},r^g,p)\Big \}. \end{aligned}$$

Using again Schäl [44], there exists one Borel-measurable map ${\hat{\pi }}:[0,T]\times \mathbb {R}^{2d^g+2}\times \mathbb {S}_{2d^g+2}(\mathbb {R})\times \mathbb {R}^{d^g}\longrightarrow ~B$ such that for every $(t,z^{\text {obs},S},g^{\text {obs},S},r^g)\in [0,T]\times \mathbb {R}^{2d^g+2}\times \mathbb {S}_{2d^g+2}(\mathbb {R})\times \mathbb {R}^{d^g}$, ${\hat{\pi }}(t,z^{\text {obs},S},g^{\text {obs},S},r^g)\in \mathcal {O}^{\text {obs},S}(t,z^{\text {obs},S},g^{\text {obs},S},r^g)$ and $\mathcal {O}^{\text {obs},S}$ denotes the set of all such maps. We consider the subset of admissible contracts

where any contract in $\mathcal {C}_2^S$ is of the form $Y_T^{y_0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }}}$ where for all $t\in [0,T]$,

$$\begin{aligned} \begin{aligned} Y_t^{y_0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }}} := y_0 + \int _0^t&Z_s^{\mathrm{obs},S}\cdot \mathrm {d}B_s^{\mathrm{obs},S} + \frac{1}{2}\mathrm {Tr}\Big [\big (\Gamma ^{\mathrm{obs},S}_s + \gamma Z_s^{\mathrm{obs},S}(Z_s^{\mathrm{obs},S})^{\top }\big )d\langle B^{\mathrm{obs},S}\rangle _s\Big ] \\&- h^{\mathrm{obs},S}\Big (s,Z_s^{\mathrm{obs}},\Gamma _s^{\mathrm{obs}},r_s^g,{\hat{\pi }}\big (s,Z_s^{\mathrm{obs},S},\Gamma _s^{\mathrm{obs},S},r_s^g\big )\Big )\mathrm {d}s, \end{aligned} \end{aligned}$$

(C.3)

where $y_0\ge 0$, ${\hat{\pi }}\in \mathcal {O}^{\text {obs},S}$ and $(Z^{\text {obs},S},\Gamma ^{\text {obs},S})\in \mathcal {ZG}^{\text {obs},S}$ with

$$\begin{aligned} \mathcal {ZG}^{\text {obs},S}&:= \Big \{(Z^{\text {obs},S},\Gamma ^{\text {obs},S}): \mathbb {R}^{2d^g+2}\times \mathbb {S}_{2d^g+2}(\mathbb {R})\text {-valued, }\\&\quad \mathbb {F}\text {-predictable s.t } Y_T^{y_0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }}}\in \mathcal {C}_2^S \Big \}. \end{aligned}$$

We can now formulate the stochastic control problem faced by the government.

1.3 The Hamilton–Jacobi–Bellman equation

Let us define the process $(Q^{y_0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }}}_t)_{t\in [0,T]}$ where for all $(t,y_0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }})\in [0,T]\times \mathbb {R}\times \mathcal {ZG}^{\text {obs},S}\times \mathcal {O}^{\text {obs},S}$

$$\begin{aligned} Q_t^{y_0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }}}&:= X_t - \int _0^t \sum _{i=1}^{d^g}\big (G_i -{\hat{\pi }}_i^{g}(s,Z_s^{\mathrm{obs},S},\Gamma _s^{\mathrm{obs},S},r_s^g)\big )^2 ds -Y_t^{y_0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }}}. \end{aligned}$$

The optimisation problem of the government that we consider here is

$$\begin{aligned} \begin{aligned} \widetilde{V}_0^P = \sup _{y_0\ge 0}\sup _{(Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }})\in \mathcal {ZG}^{\text {obs},S}\times \mathcal {O}^{\text {obs},S}}\mathbb {E}^{{\hat{\pi }}(Z^{\text {obs},S},\Gamma ^{\text {obs},S})}&\Bigg [-\exp \Big (-\nu Q_T^{y_0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }}}\Big )\Bigg ]. \end{aligned} \end{aligned}$$

(C.4)

Due to the presence of state variables in the best response of the Agent, the optimal control of the Principal will no longer be deterministic, and we have to rely on the Hamilton–Jacobi–Bellman formulation of the stochastic control problem. First, we note that the supremum over $y_0$ is attained by setting $y_0=0$. Next, the state variables of the control problem are $\big (t,B_t^{\text {obs},S},Q_t^{0,Z^{\text {obs},S},\Gamma ^{\text {obs},S},{\hat{\pi }}}\big )$ and as it is standard in control problems with CARA utility function, the last variable can be simplified. Define $P^S=\mathbb {R}^{2d^g+2}\times \mathbb {S}_{2d^g+2}(\mathbb {R})$, and the Hamiltonian $H^{{\hat{\pi }}}:[0,T]\times P^S \times \mathbb {R}\times P^S \longrightarrow \mathbb {R}$

$$\begin{aligned}&H^{\hat{\pi }}(t,z,g,u,u_b,u_{bb}) := \nu u \bigg ( z \cdot \mu ^{\mathrm{obs},S}\big (t,{\hat{\pi }}(t,z,g,r^g),r^g\big ) + \sum _{i=1}^{d^g}\big (G_i -{\hat{\pi }}_i^{g}(t,z,g,r^g)\big )^2 \\&+ \frac{1}{2}\mathrm {Tr}\bigg [(g+\gamma z z^{\top })\Sigma ^{\mathrm{obs},S}\big (t,{\hat{\pi }}(t,z,g,r^g)\big )\Big (\Sigma ^{\mathrm{obs},S}\big (t,{\hat{\pi }}(t,z,g,r^g)\big )\Big )^{\top }\bigg ] \\&\quad - h^{\mathrm{obs},S}\big (t,z,g,r^g,{\hat{\pi }}(t,z,g,r^g)\big ) \bigg ) \\&+ \frac{1}{2} \nu ^2 u \mathrm {Tr}\bigg [z z^{\top } \Sigma ^{\mathrm{obs},S}\big (t,{\hat{\pi }}(t,z,g,r^g)\big )\Big (\Sigma ^{\mathrm{obs},S}\big (t,{\hat{\pi }}(t,z,g,r^g)\big )\Big )^{\top }\bigg ] \\&\quad + u_b \cdot \mu ^{\mathrm{obs},S}\big (t,{\hat{\pi }}(t,z,g,r^g),r^g\big ) \\&+ \frac{1}{2}\mathrm {Tr}\bigg [ \Sigma ^{\mathrm{obs},S}\big (t,{\hat{\pi }}(t,z,g,r^g)\big )\Big (\Sigma ^{\mathrm{obs},S}\big (t,{\hat{\pi }}(t,z,g,r^g)\big )\Big )^{\top } u_{bb} \bigg ]. \end{aligned}$$

The value function of the control problem of the Principal is related to the following Hamilton–Jacobi–Bellman equation

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \partial _t U(t,b) + \underset{(z,g,{\hat{\pi }})\in P^S \times \mathcal {O}^{\text {obs},S}}{\sup } H^{\hat{\pi }}\big (t,z,g,b,U,U_b,U_{bb}\big ) =0,\\ U(T,b)= -1, \end{array}\right. } \end{aligned} \end{aligned}$$

(C.5)

where $U:[0,T]\times \mathbb {R}^{2d^g+2}\longrightarrow \mathbb {R}$ and for all $(i,j)\in \{1,\dots ,2d^g+2\}$, $(U_b)_i = \partial _{b_i} U, (U_{bb})_{i,j}=\partial _{b_i b_j}U$, in the sense that ${\widetilde{V}}^P_0=U(0,b_0)$ where $B_0^{\text {obs},S}=b_0$ and $y_0=0$. Thus, the incentives provided to the investor are obtained up to the resolution of a $(2d^g + 2)$-dimensional HJB equation. Although it provides greater flexibility on the modelling of short-term rates, this approach can only be applied to a small portfolio of bonds using classic numerical schemes on sparse grids.