The optimal harvesting problem under price uncertainty: the risk averse case

Abstract

We study the exploitation of a one species, multiple stand forest plantation when timber price is governed by a stochastic process. Our model is a stochastic dynamic program with a weighted mean-risk objective function, and our main risk measure is the Conditional Value-at-Risk. We consider two stochastic processes, geometric Brownian motion and Ornstein–Uhlenbeck: in the first case, we completely characterize the optimal policy for all possible choices of the parameters while in the second, we provide sufficient conditions assuring that harvesting everything available is optimal. In both cases we solve the problem theoretically for every initial condition. We compare our results with the risk neutral framework and generalize our findings to any coherent risk measure that is affine on the current price.

Appendices

Appendix

1.1 Preliminaries

Before getting into the proof of the lemmas and theorems stated in the main body of the paper, we present some definitions that will be necessary throughout the proofs.

In (7), the optimal control $c_t$ in each period depends only on the current state of the forest and price through the decision function $c_t=\pi _t({\mathbb {X}}_t,p_t)$. A sequence $\varPi =\{\pi _t\}_{t\in {\mathcal {T}}}$ is called a policy. Of course, a policy is feasible if ${\mathbb {X}}_t$ and $c_t=\pi _t({\mathbb {X}}_t,p_t)$ satisfy (2) and (3) for every possible value of ${\mathbb {X}}_t$ and $p_t$ at every instant $t\in {\mathcal {T}}$. Observe that the non-negativity of the state variables $\bar{x}_t$ and $x_{a,t}$ for $a=1,\ldots ,n$ is assured by (2) and (3).

The expected benefit of a given policy $\varPi$ for an initial state ${\mathbb {X}}_{0}$ and an initial price $p_{0}$ is

$$\begin{aligned} Q^\varPi _0({\mathbb {X}}_{0},p_{0})= - p_{0}\pi _0({\mathbb {X}}_0,p_0) + \delta {\mathcal {R}}_{|p_{0}}\left[ -p_{1}\pi _1({\mathbb {X}}_1,p_1) + \delta {\mathcal {R}}_{|p_1}\left[ \cdots \right] \right] , \end{aligned}$$

(20)

if the time horizon is infinite. Correspondingly, we denote $Q^\varPi _{0,T}({\mathbb {X}}_{0},p_{0})$ the expected benefit of policy $\varPi$ whenever ${\mathcal {T}}=[1,\ldots ,T]$.

Problem (6) can be stated as the problem of finding a feasible policy that minimizes (20),

$$\begin{aligned} V_0({\mathbb {X}}_{0},p_{0}) =\left\{ \begin{array}{ll} \min _\varPi &{} Q^\varPi _0({\mathbb {X}}_{0},p_{0}) \\ \text{ s.t. } &{} \varPi \text{ is } \text{ a } \text{ feasible } \text{ policy }, \end{array}\right. \end{aligned}$$

(21)

or analogously, $V_{0,T}({\mathbb {X}}_{0},p_{0}) =\min _\varPi Q^\varPi _{0,T}({\mathbb {X}}_{0},p_{0})$ s. t. $\varPi$ is a feasible policy.

In the sequel, we will also use the expected discounted benefit from an intermediate step

$$\begin{aligned} Q^{\varPi }_{t}({\mathbb {X}}_t,p_t)= - p_{t}c_t+ \delta {\mathcal {R}}_{|p_{t}}\left[ -p_{t+1}c_{t+1} + \delta {\mathcal {R}}_{|p_{t+1}}\left[ \cdots \right] \right] \end{aligned}$$

and the corresponding value function

$$\begin{aligned} V_{t}({\mathbb {X}}_t,p_t) =\left\{ \begin{array}{ll} \min _\varPi &{}\quad Q^\varPi _{t}({\mathbb {X}}_t,p_t) \\ \text{ s.t. } &{}\quad \varPi \text{ is } \text{ a } \text{ feasible } \text{ policy }, \end{array}\right. \end{aligned}$$

as well as the analogous definitions in the finite case.

Appendix 1: Proof of Lemma 1

For given initial state and price ${\mathbb {X}}_0$, $p_0$ we consider the cost resulting of the application of policy $\varPi$ (not necessarily optimal) up to T: $Q^\varPi _{0,T}({\mathbb {X}}_0,p_0)$. To lighten the notation we will use $Q^\varPi _{0,T}$ instead of $Q^\varPi _{0,T}({\mathbb {X}}_0,p_0)$ as ${\mathbb {X}}_0$ and $p_0$ remain constant throughout the proof. The expression of $Q^\varPi _{0,T}$ is a slight modification of (20)

$$\begin{aligned} Q^\varPi _{0,T}= - p_{0}c_{0} + \delta {\mathcal {R}}_{|p_{0}}[ -p_{1}c_{1} + \cdots + \delta {\mathcal {R}}_{|p_{T-1}} [-p_{T}\,c_{T}]], \end{aligned}$$

Due to the fact that $p_{t}c_{t}\ge 0$ for all t (when prices follow a GBM) and the monotonicity of any coherent risk measure we know that $Q^\varPi _{0,T}\ge Q^\varPi _{0,T+1}$, hence the sequence $Q^\varPi _{0,T}$ either converges to the limit or diverges to $-\infty$ when $T\rightarrow \infty$. We now prove that $Q^\varPi _{0,T}$ is bounded below for all T,

$$\begin{aligned} Q^\varPi _{0,T}= - p_{0}c_{0} - \delta e^\mu \chi p_{0}c_{1} - \cdots - (\delta e^\mu \chi )^{T}p_{0}c_{T}\ge -p_0S \frac{1- (\delta e^\mu \chi )^{T+1}}{1- \delta e^\mu \chi }, \end{aligned}$$

where S represents the total surface of the forest and $\chi =\lambda +\kappa (1-\lambda )$.

If $\delta e^\mu \chi <1$ we get $Q^\varPi _{0,T}>-p_0S \frac{1}{1- \delta e^\mu \chi }>-\infty$ for all T. This implies that the sequence $Q^\varPi _{0,T}$ converges when T goes to infinity. This limit is the value associated to the policy $\varPi$ denoted as $Q^\varPi _0$,

$$\begin{aligned} Q^\varPi _0=-p_0\sum _{t=0}^\infty (\delta e^\mu \chi )^tc_t>-\infty \end{aligned}$$

(22)

As the bound on $Q^\varPi _0$ does not depend on the policy $\varPi$, we conclude

$$\begin{aligned} V_0({\mathbb {X}}_0,p_0)= \mathop {\mathrm{Min}}_{\varPi } Q^\varPi _0>-\infty . \end{aligned}$$

Appendix 2: Proof of Theorem 1

To prove that the greedy policy is optimal, we check that the benefit associated with it, $Q^{\textit{GP}}$, satisfies the Bellman equation (7) from any initial condition. The formula of $Q^{\textit{GP}}$ is obtained using (22), where $\chi =\lambda +\kappa (1-\lambda )$.

We consider first the infinite time horizon case. If the initial state is ${\mathbb {X}}_t=(\bar{x},x_n,\ldots , x_2,x_1)$, it is easy to see that the harvests associated to the GP are $c_{t+in}=\bar{x}+x_{n}$ for all $i\in {\mathbb {N}}$ and $c_{t+in+j}=x_{n-j}$ for $j=1,\ldots ,n-1$ and $i\in {\mathbb {N}}$, and hence,

$$\begin{aligned} Q_t^{\textit{GP}}({\mathbb {X}}_t,p_t)=- p_t \sum _{i=0}^{\infty }\left( (\delta e^\mu \chi )^{in} \bar{x}+ \sum _{j=0}^{n-1}(\delta e^\mu \chi )^{in+j} x_{n-j} \right) . \end{aligned}$$

(23)

Given $c_t=c\in [0,CA{\mathbb {X}}_t]$, the state at $t+1$ is ${\mathbb {X}}_{t+1}=(\bar{x}+x_n-c,x_{n-1},\ldots ,x_1,c)$ and the value associated to the GP is

$$\begin{aligned} Q_{t+1}^{\textit{GP}}({\mathbb {X}}_{t+1},p_{t+1})&=- p_{t+1} \sum _{i=0}^{\infty }\Big ((\delta e^\mu \chi )^{in} (\bar{x}+x_n-c)\\&\quad + \sum _{j=0}^{n-2}(\delta e^\mu \chi )^{in+j} x_{n-j-1}+(\delta e^{\mu }\chi )^{in+n-1}c \Big ). \end{aligned}$$

Inserting $V_{t+1}=Q_{t+1}^{\textit{GP}}$ into the rhs of the Bellman equation (7), the argument of the $\mathop {\mathrm{Min}}$ operator is

$$\begin{aligned} \varPhi (c)= & {} -p_t c+\delta {\mathcal {R}}_{|p_t} \left[ -p_{t+1} \sum _{i=0}^{\infty }\left( (\delta e^\mu \chi )^{in} (\bar{x}+x_n-c) \right. \right. \nonumber \\&+\, \left. \left. \sum _{j=0}^{n-2}(\delta e^\mu \chi )^{in+j} x_{n-j-1}+(\delta e^{\mu }\chi )^{in+n-1}c \right) \right] \nonumber \\= & {} -p_tc - p_t \sum _{i=0}^{\infty }\Big ((\delta e^\mu \chi )^{in+1} (\bar{x}+x_n-c) \nonumber \\&+\,\sum _{j=0}^{n-2}(\delta e^\mu \chi )^{in+j+1} x_{n-j-1}+(\delta e^{\mu }\chi )^{in+n}c \Big ) \end{aligned}$$

(24)

The coefficient affecting c in $\varPhi (c)$ is

$$\begin{aligned} \text{ coeff }(c) \!=\!-p_t \left( \!1-\!\sum _{i=0}^\infty (\delta e^\mu \chi )^{in+1}\!+\! \sum _{i=0}^\infty (\delta e^\mu \chi )^{in+n} \!\right) \!=-p_t(1\!-\delta e^\mu \chi ) \sum _{i=0}^\infty (\delta e^\mu \chi )^{in}\!>\!0 \end{aligned}$$

As $\text{ coeff }(c)<0$, the minimum in (7) is attained when $c=\bar{x}+x_n$. Inserting this value of c in (24) yields

$$\begin{aligned} \varPhi (\bar{x}+x_n)= & {} -p_t(\bar{x}+x_n) - p_t \sum _{i=0}^{\infty }\left( \sum _{j=0}^{n-2}(\delta e^\mu \chi )^{in+j+1} x_{n-j-1}+(\delta e^{\mu }\chi )^{in+n}(\bar{x}+x_n) \right) \\= & {} - p_t \sum _{i=0}^{\infty }\left( (\delta e^{\mu }\chi )^{in}(\bar{x}+x_n)+ \sum _{j=1}^{n-1}(\delta e^\mu \chi )^{in+j} x_{n-j} \right) =Q^{\textit{GP}}({\mathbb {X}}_t,p_t), \end{aligned}$$

showing that (7) holds and that the GP is optimal.

The proof for the finite horizon case is similar but more involved. Indeed, let t be expressed as $T-(kn+j)$, where $k=\lfloor {\frac{T-t}{n}}\rfloor$ and $j\in \{0,\ldots ,n-1\}$ is the remainder of the integer division of $(T-t)$ by n. This way of expressing t puts in evidence that after completing k cycles there will be still j time steps to go until reaching the end of the horizon. Thus, the expected benefit associated to the GP is

$$\begin{aligned} Q^{\textit{GP}}_t({\mathbb {X}}_t,p_t)&=-p_t\left[ \sum _{i=0}^{k-1}(\delta e^{\mu }\chi )^{in}\left( \bar{x}+\sum _{l=0}^{n-1}(\delta e^{\mu }\chi )^l x_{n-l}\right) \right. \\&\quad \left. +\, (\delta e^{\mu }\chi )^{kn}\left( \bar{x}+\sum _{l=0}^j (\delta e^{\mu }\chi )^l x_{n-l} \right) \right] . \end{aligned}$$

The first term of the rhs represents the expected benefit of the k completed cycles, while the second term corresponds to the last j steps. Again, we need to check that $Q^{\textit{GP}}$ satisfies (7) and that the minimum is attained for $c=CA{\mathbb {X}}_t$. We leave the details to the reader.

Appendix 3: Proof of Theorem 2

To prove that the accumulating policy is optimal, we check that the benefit associated with it ($Q^{AP}$) satisfies the dynamic programming equation (7). Let t be expressed as $T-(kn+j)$, where $k=\lfloor {\frac{T-t}{n}}\rfloor$ and $j\in \{0,\ldots ,n-1\}$ is the remainder of the integer division of $(T-t)$ by n. After some computations we can prove that

$$\begin{aligned} Q^{AP}_t({\mathbb {X}}_t,p_t) = -p_t\Big [(\delta e^\mu \chi )^k (\bar{x}+\sum _{l=0}^j x_{n-l}) +\sum _{i=1}^k (\delta e^\mu \chi )^{in+j}S \Big ] \end{aligned}$$

(25)

where S represents the total surface of the forest and $\chi =\lambda +\kappa (1-\lambda )$. We point out that for $k=0$, we follow the convention $\sum _{1}^0 (\cdot )=0$.

For the rest of the proof we divide the study into two cases depending on the value of j: (i) $j>0$ and (ii) $j=0$.

(i) Here we have $t+1=T-(k'n+j')$ where $k'=k$ and $j'=j-1\in \{0,\ldots ,n-2\}$ and $Q^{AP}_{t+1}(A{\mathbb {X}}_t+Bc,p_{t+1})$ can be expressed as

$$\begin{aligned} - p_{t+1}\Big [ (\delta e^\mu \chi )^{j-1} (\bar{x}+x_n-c+\sum _{l=0}^{j-1} x_{n-l-1}) + \sum _{i=1}^k (\delta e^\mu \chi )^{in+j-1}S \Big ], \end{aligned}$$

where $\chi =\lambda +(1-\lambda )\kappa$. Inserting $V = Q^{AP}$ into the right-hand side of the dynamic programming equation (7), the argument of the min operator, $\varPhi (c)$, is

$$\begin{aligned}&-\,p_tc+\delta {\mathcal {R}}_{|p_t}\big [ -p_{t+1}\big [(\delta e^\mu \chi )^{j-1} (\bar{x}+x_n-c+\sum _{l=0}^{j-1} x_{n-l-1}) + \sum _{i=1}^k (\delta e^\mu \chi )^{in+j-1}S \big ] \big ] \\&\quad = -\,p_tc-p_t\big [(\delta e^\mu \chi )^{j} (\bar{x}+x_n-c+\sum _{l=0}^{j-1} x_{n-l-1}) + \sum _{i=1}^k (\delta e^\mu \chi )^{in+j}S \big ] \big ) \\&\quad = -\,p_tc(1-(\delta e^\mu \chi )^{j} )-p_t\big [(\delta e^\mu \chi )^{j} (\bar{x}+x_n+\sum _{l=1}^{j} x_{n-l}) + \sum _{i=1}^k (\delta e^\mu \chi )^{in+j}S \big ] \\&\quad = -\,p_tc(1-(\delta e^\mu \chi )^{j} )+Q^{AP}_t({\mathbb {X}}_t,p_t). \end{aligned}$$

As the coefficient of c is non-negative, the minimum is attained when $c=0$ and $\varPhi (0)$ is exactly $Q^{AP}_t({\mathbb {X}}_t,p_t),$ showing that equation (7) holds.

(ii) Case $t=T-kn$. In this case, we have $t+1=T-[(k-1)n+n-1)]$ and $Q_{t+1}^{AP}(A{\mathbb {X}}_t+Bc,p_{t+1})$ can be expressed as

$$\begin{aligned}&-\,p_{t+1}\Big [(\delta e^\mu \chi )^{n-1} (\bar{x}+x_n-c+\sum _{l=0}^{n-2} x_{n-l-1}+c) + \sum _{i=1}^{k-1} (\delta e^\mu \chi )^{in+n-1}S \Big ]\\&\quad = -\,p_{t+1}\left[ (\delta e^\mu \chi )^{n-1} S + \sum _{i=2}^{k} (\delta e^\mu \chi )^{in-1}S \right] = -p_{t+1} \sum _{i=1}^{k} (\delta e^\mu \chi )^{in-1}S. \end{aligned}$$

Inserting again $V = Q^{AP}$ into the right-hand side of the Bellman’s equation (7), the argument of the min operator is

$$\begin{aligned} \varPhi (c)= & {} -p_tc+\delta {\mathcal {R}}_{|p_t} \left[ -p_{t+1} \sum _{i=1}^{k} (\delta e^\mu \chi )^{in-1}S\right] . \end{aligned}$$

The coefficient of c is negative, and thus, the minimum is attained when $c=\bar{x}+x_n$. So we have,

$$\begin{aligned} \varPhi (\bar{x}+x_n)= & {} -p_t(\bar{x}+x_n) - p_t \sum _{i=1}^{k} (\delta e^\mu \chi )^{in}S. \end{aligned}$$

The right-hand side is exactly (25) when $j=0$, hence we have $\varPhi (\bar{x}+x_n)=Q^{AP}_t(\cdot ,\cdot )$ and equation (7) is satisfied.

In both cases, we have shown that $Q^{AP}_t(\cdot ,\cdot )$ satisfies equation (7), hence it is the value function and the proposed policy is optimal.

Appendix 4: Proof of Lemma 2

Due to (19), we only need to show that

$$\begin{aligned} \frac{\delta b}{1-\delta a}\ge & {} \frac{b}{1-a}\Big [1-\frac{1-\delta ^j}{a^m(1-\delta ^ja^j)} \Big ]. \end{aligned}$$

(26)

Using that

$$\begin{aligned} \frac{\delta b}{1-\delta a}=\frac{b}{1-a}\Big [1-\frac{1-\delta }{1-\delta a}\Big ], \end{aligned}$$

we have that (26) is equivalent to

$$\begin{aligned} \frac{1}{1-a}\Big [\frac{1-\delta }{1-\delta a}\Big ]\le & {} \frac{1}{1-a}\Big [\frac{1-\delta ^j}{a^m(1-\delta ^ja^j)} \Big ]\\&\quad \iff \frac{1}{1-a}\Big [\frac{a^m(1-\delta ^ja^j)}{1-\delta a}\Big ] \le \frac{1}{1-a}\Big [\frac{1-\delta ^j}{1-\delta } \Big ]\\&\quad \iff \frac{1}{1-a}\Big [a^m \sum _{l=0}^{j-1}(\delta a)^l \Big ] \le \frac{1}{1-a}\Big [\sum _{l=0}^{j-1}\delta ^l\Big ]. \end{aligned}$$

Given that $a\in (0,1)$, the last inequality is always valid.

Appendix 5: Proof of Lemma 3

Given initial state and price ${\mathbb {X}}_0$, $p_0$ we denote by $Q^\varPi _{0,T}({\mathbb {X}}_0,p_0)$ the cost resulting of the application of a (not necessarily optimal) policy $\varPi$ up to T. As in Lemma 1 we denote $Q^\varPi _{0,T}=Q^\varPi _{0,T}({\mathbb {X}}_0,p_0)$. Knowing that ${\mathcal {R}}_{p_t}[-p_{t+1}]=-ap_t-b$, where a and b are defined in (17), we can write $Q^\varPi _T$ as:

$$\begin{aligned} Q^\varPi _{0,T}= & {} - p_{0}c_{0} + \delta {\mathcal {R}}_{|p_{0}}[ -p_{1}c_{1} + \delta {\mathcal {R}}_{|p_{1}}[ -p_{2}c_{2}+ \cdots \\&+\, \delta {\mathcal {R}}_{|p_{T-3}} [-p_{T-2}c_{T-2}+ \delta {\mathcal {R}}_{|p_{T-2}} [-p_{T-1}c_{T-1}+ \delta {\mathcal {R}}_{|p_{T-1}} [-p_{T}\,c_{T}]]]]]\\= & {} - p_{0}c_{0} + \delta {\mathcal {R}}_{|p_{0}}[ -p_{1}c_{1} +\delta {\mathcal {R}}_{|p_{1}}[ -p_{2}c_{2} +\cdots \\&+\, \delta {\mathcal {R}}_{|p_{T-3}} [-p_{T-2}c_{T-2}+ \delta {\mathcal {R}}_{|p_{T-2}} [-p_{T-1}(c_{T-1}+\delta a c_T)- \delta bc_T]]]]\\= & {} - p_{0}c_{0} + \delta {\mathcal {R}}_{|p_{0}}[ -p_{1}c_{1} + \delta {\mathcal {R}}_{|p_{1}}[ -p_{2}c_{2} \cdots \\&+\, \delta {\mathcal {R}}_{|p_{T-3}} [-p_{T-2}(c_{T-2}+ \delta ac_{T-1}+(\delta a)^2c_{T})-b\delta c_{T-1}- b\delta ^2(a+1)c_T]]]\\&\vdots \\= & {} -p_0\sum _{t=0}^T (\delta a)^tc_t - b\sum _{t=1}^T \delta ^t c_t \frac{1-a^t}{1-a} \end{aligned}$$

We will show that $\lim _{T\rightarrow \infty } Q^\varPi _{0,T}$ exists and define the value associated with policy $\varPi$ for the infinite time horizon as the value of that limit. To this end we compute

$$\begin{aligned} |Q^\varPi _{0,T+\tau }-Q^\varPi _{0,T}|\le & {} p_0\sum _{t=T+1}^{T+\tau }c_t(\delta a)^t + |b| \sum _{t=T+1}^{T+\tau } \delta ^t c_t \frac{1-a^t}{1-a} \nonumber \\\le & {} p_0\sum _{t=T+1}^{\infty }c_t(\delta a)^t + |b| \sum _{t=T+1}^{\infty } \delta ^t c_t \frac{1}{1-a}\nonumber \\\le & {} (\delta a)^{T+1} p_0 S \frac{1}{1-\delta a} + \delta ^{T+1} S |b| \frac{1}{1-\delta }\frac{1}{1-a}. \end{aligned}$$

(27)

As the last expression converges to 0 when $T\rightarrow \infty$ for all $\tau \in {\mathbb {N}}$, we conclude that $\lim _{T\rightarrow \infty } Q^\varPi _{0,T}$ exists.

Appendix 6: Proof of Theorem 3

We state ${\mathbb {X}}_{t+1}$ and equation (7) in terms of ${\mathbb {X}}_t$ and c as follows.

$$\begin{aligned} {\mathbb {X}}_t=\begin{pmatrix}\bar{x}_t\\ x_{n,t}\\ x_{n-1,t}\\ \vdots \\ \vdots \\ x_{1,t} \end{pmatrix} \longrightarrow {\mathbb {X}}_{t+1}=A{\mathbb {X}}_t+Bc=\begin{pmatrix}\bar{x}_t+ x_{n,t}- c\\ x_{n-1,t}\\ x_{n-2,t}\\ \vdots \\ \vdots \\ c \end{pmatrix},\\ V_t({\mathbb {X}}_t,p_t)=\min _c \big \{- p_tc + \delta {\mathcal {R}}_{|p_t}(V_{t+1}({\mathbb {X}}_{t+1},p_{t+1}))\big \}. \end{aligned}$$

The main idea of the proof is to consider the role played by c in all the possible expressions of $V_{t+1}(\cdot ,\cdot )$. This is not an easy task, but despite all the possible harvesting policies, the coefficient of c has a particular structure: it is the sum of terms of the form $\varDelta ^{m_i}_{j_i}(p_t)$ (as defined in (18)), for some values of $m_i\in {\mathbb {N}}$ and $j_i\in \{0,\ldots ,n-1\}$ plus possibly one negative term $\varGamma ^m(p_t)=-\delta ^m[ p_ta^m +b\sum _{l=0}^{m-1}a^l]$.⁶

Indeed, from $t+1$ on, two different situations can arise: (i) nothing is harvested in the next n steps or (ii) the first harvest occurs at $t=j_0$ with $1\le j_0<n$.

In case (i), the state at $t+n$ will be

$$\begin{aligned} {\mathbb {X}}_{t+n}=(S-c,c,0,\ldots ,0)^T. \end{aligned}$$

It is easy to see that the influence of c extinguishes as the constraint on the harvest is $c_{t+n}\le S$. We do not know the complete expression of $V_{t+1}(\cdot ,\cdot )$ but we do know that the coefficient of c is simply $\varGamma ^0(p_t)=-p_t$ with no $\varDelta ^m_j(p_t)$ terms.

In case (ii), the first harvest after t takes place at $t+j_0$ and (7) can be written as:

$$\begin{aligned} V_t({\mathbb {X}}_t,p_t)= & {} \min _c \big \{- p_tc + \delta {\mathcal {R}}_{|p_{t+1}}[\delta {\mathcal {R}}_{|p_{t+2}}[\ldots \delta {\mathcal {R}}_{p_{t+j_0-1}}[-p_{t+j_0}(\bar{x}_t+\cdots +x_{n-j_0,t}-c)\\&+\, \delta {\mathcal {R}}_{|p_{t+j_0}}[V_{t+j_0+1}({\mathbb {X}}_{t+j_0+1},p_{t+j_0+1})]]]]\big \}. \end{aligned}$$

Hence, the first term of the coefficient of c is of the form

$$\begin{aligned} \varDelta ^0_{j_0}(p_t)=-p_t(1-\delta ^{j_0}a^{j_0})-\frac{b}{1-a}(-\delta ^{j_0}+\delta ^{j_0}a^{j_0}). \end{aligned}$$

There might be more terms including c in the expression of $V_{t+j_0+1}(\cdot ,\cdot )$. For a complete characterization of the coefficient of c we refer the reader to Piazza and Pagnoncelli (2014) where the analogous result in the risk neutral case is presented. The construction of the coefficient of c follows the same lines, the reader only needs to substitute the operator ${\mathbb {E}}_{|p_t}$ for ${\mathcal {R}}_{|p_t}$.

The number of terms comprising the coefficient of c, may or not be finite. In the infinite case, Lemma 3 implies that the sum converges.

The proof is completed by showing that the coefficient of c is negative. But, Lemma 2 shows that $\varDelta ^m_j\le 0$ when condition (16) holds, which finishes the proof.

Appendix 7: Proof of Lemma 4

For values of $a\in (0,1)$ the proof presented for Lemma 3 is valid. For values of $a\in [1,1/\delta )$ we need to modify the proof from (27) onwards.

We have that

$$\begin{aligned} |Q^\varPi _{T+\tau }-Q^\varPi _T| { \le } p_0\sum _{t=T+1}^{T+\tau }c_t(\delta a)^t + |b| \sum _{t=T+1}^{T+\tau } \delta ^t c_t \sum _{j=0}^{t-1}a^j. \end{aligned}$$

Using that $1\le a$ and $c_t\le S$ for all t we get

$$\begin{aligned} |Q^\varPi _{T+\tau }-Q^\varPi _T| { \le } (\delta a)^{T+1} p_0 S \frac{1}{1-\delta a} + S |b| \sum _{t=T+1}^{\infty } t (a\delta )^t. \end{aligned}$$

As the sum $\sum _{t=1}^\infty t (a\delta )^t$ converges whenever $a\delta <1$, its T-tail must go to zero when T goes to infinity.

Finally, we have that the right hand side of the inequality above converges to 0 when $T\rightarrow \infty$ and we conclude that $\lim _{T\rightarrow \infty } Q^\varPi _T$ exists.

Appendix 8: Proof of Theorem 4

This theorem is a generalization of Theorem 3 from $(a,b)\in (0,1)\times {\mathbb {R}}_+$ to $(a,b)\in (0,1/\delta )\times {\mathbb {R}}$. The proof of Theorem 3 consist in characterizing the coefficient of c in the Bellman equation (7). It is shown that this coefficient is the sum of infinite terms of the form $\varDelta ^m_j(p_t)$. This construction does not depend on the value of a and b but relies exclusively in the fact that the conditional risk measure is affine on $p_t$. Hence, this part of the proof extends directly to the more general setting of this theorem.

The characterization of the coefficient’ sign relies on Lemma 2 presented in Sect. 5 that gives a sufficient condition assuring that

$$\begin{aligned} \varDelta ^m_j(p_t)\le 0 \quad \text{ for } \text{ all } \; m\le T-t\quad \hbox { and for all }\; j\in \{1,\ldots ,n\}. \end{aligned}$$

(28)

Lemma 2 is valid for $(a,b)\in (0,1)\times {\mathbb {R}}_+$. In the following we study the extension of this lemma to $(a,b)\in (0,1/\delta ) \times {\mathbb {R}}$. We start by noticing that (18) is not valid for $a=1$. We will use the following representation of $\varDelta ^m_j(p_t)$,

$$\begin{aligned} \left\{ \begin{array}{ll} \delta ^{m_i}\left\{ -p_ta^{m_i}(1-\delta ^{j_i}a^{j_i})-\frac{b}{1-a}\Big [1-\delta ^{j_i}-a^{m_i}(1-\delta ^{j_i}a^{j_i})\Big ]\right\} &{}\quad \text{ if } \; a\ne 1\\ \delta ^m\{-p_t(1-\delta ^j)-b[m-\delta ^j(m+j)]\}&{}\quad \text{ if } \;a=1, \end{array}\right. \end{aligned}$$

The parameters’ semi-plane is divided in five regions as shown in Fig. 3 and Table 2.⁷

Fig. 3

Semi-plane of parameters a and b

Table 2 Parameters regions

In the following, we look for conditions implying Condition (28), as this is sufficient to prove that the coefficient of $c^*_t$ is negative and hence $c^*_t=CA{\mathbb {X}}_t$. We will see that the following conditions imply Condition (28):

• 1. In region (i), $p_t\ge b\delta /(1-a\delta )$ (this is Lemma 2).

• 2. In region (ii), $p_t\ge b\delta /(1-a\delta )$.⁸

• 3. In region (iii), no sufficient condition assuring Condition (28) is found in the infinite horizon case.

• 4. In region (iv), $p_t\ge b/(1-a)$

Let us denote by $r^m_j(a,b)$ to the rhs of (19) when $a\ne 1$ and the corresponding expression for $a=1$, i.e.,

$$\begin{aligned} r^m_j(a,b)=\left\{ \begin{array}{l@{\quad }l} \frac{b}{1-a}\Big [1-\frac{1-\delta ^j}{a^m(1-\delta ^ja^j)} \Big ] &{} \text{ if } a\ne 1\\ \frac{-b}{1-\delta ^j}[m-\delta ^j(m+j)] &{} \text{ if } a= 1 \end{array}\right. \end{aligned}$$

In the following, we prove the properties summarized in Fig. 3. We observe in the first place that

$$\begin{aligned} \varDelta ^m_j(p_t)\le 0 \iff p_t\ge r^m_j(a,b) \qquad \text{ if } 1-\delta a>0 \end{aligned}$$

(29)

We start by determining whether $r^0_1(a,b)=\frac{b\delta }{1-\delta a}$ bounds $r^m_j(a,b)$ (below or above) for all m and $j=1,\ldots ,n$. We study the case $a\ne 1$, leaving the easier particular case $a=1$ to the reader. We also make the observation that if $b=0$ then $r^m_j=0$ for all m and j.

(30)

If ${{\mathrm{sign}}}(b){{\mathrm{sign}}}(a-1)>0$, i.e., in regions (ii) or (iii), (30) is equivalent to

$$\begin{aligned}&a^m\frac{1-\delta ^j a^j}{1-\delta a} \lesseqqgtr \frac{1-\delta ^j}{1-\delta } \\&\quad \iff a^m\sum _{l=0}^{j-1}(\delta a)^j \lesseqqgtr \sum _{l=0}^{j-1}\delta ^j \end{aligned}$$

• If $a>1$ (region (ii)), the inequality above holds with “$\ge$”. Hence, $r^0_1(a,b)\ge r^m_j(a,b)$.

• If $a<1$ (region (iii)), it holds with “$\le$”. Hence, $r^0_1(a,b)\le r^m_j(a,b)$.

If ${{\mathrm{sign}}}(b){{\mathrm{sign}}}(a-1)<0$, i.e., in regions (i) or (iv), (30) is equivalent to

• If $a>1$ (region (iv)), the inequality above holds with “$\le$”. Hence, $r^0_1(a,b)\le r^m_j(a,b)$.

• If $a<1$ (region (i)), it holds with “$\ge$”. Hence, $r^0_1(a,b)\ge r^m_j(a,b)$.

Table 3 summarizes these results.

Table 3 Bounds for $r^m_j(a,b)$

Putting this information together with that of (29), we conclude that to assure Condition (28) it is sufficient to have the conditions indicated in Table 4.

Table 4 Bounds for $p_t$

In regions (i) and (ii) we are ready to give a sufficient condition assuring Condition (28):

$$\begin{aligned} p_t\ge r^0_1(a,b)=\frac{\delta b}{1-\delta a}. \end{aligned}$$

To reach some conclusion in regions (iii) and (iv) we need some extra information of $r^m_j(a,b)$.

In region (iv) it is very easy to check that $r^m_j(a,b)\le b/(1-a)$. Hence $p_t\ge b/(1-a)$ is sufficient to assure Condition (28).

In region (iii) is a bit different. We have that $\lim _{m\rightarrow \infty } r^m_j(a,b)=+\infty$, hence $p_t$ cannot be greater than $r^m_j$ for all m. Hence, no conclusion can be drawn in the infinite horizon case. However, in the finite horizon case Condition (28) only requires having $p_t\ge r^m_j$ for $m\le T-t$. Furthermore, some calculation shows that $r^{m+1}_j>r^m_j$ and $r^{m}_j>r^m_{j+1}.$ Hence, we can propose a condition depending on the value of $T-t$: $p_t\ge r^{T-t}_1=\frac{b}{1-a}\Big [1-\frac{1-\delta }{a^{T-t}(1-\delta a)}\Big ].$

Appendix 9: Mean deviation risk calculation for O–U

For the particular case of $p=2$ and a O–U process, the conditional MDR is given by:

$$\begin{aligned} {\text {MDR}}_{|p_t}(-p({t+1}))&= -p_t e^{-\eta } - \bar{p}(1-e^{-\eta }) \\&\quad +\,c\left( {\mathbb {E}}\left[ \left| \int _t^{t+1} \sigma e^{\eta (s-(t+1))}dW(s) \right| ^2 \right] \right) ^{1/2}. \end{aligned}$$

In order to calculate the stochastic integral we apply Itô’s isometry:

$$\begin{aligned} {\mathbb {E}}\left( \left| \int _0^T G(t,W_t) dW_t \right| ^2\right) = {\mathbb {E}}\left( \int _0^T \left| G(t,W_t) \right| ^2 dt \right) , \end{aligned}$$

(31)

for a stochastic process $G(t,W_t) \in {\mathbb {L}}^2(0,T)$. Using (31) and noting that in our case the process G is deterministic, we have

$$\begin{aligned} \left( {\mathbb {E}}\left[ \left| \int _t^{t+1} \sigma e^{\eta (s-(t+1))} dW(s) \right| ^2 \right] \right) ^{1/2} = \left( {\mathbb {E}}\left[ \int _t^{t+1} (\sigma e^{\eta (s-(t+1))})^2 ds \right] \right) ^{1/2} \\ = \sigma \left( {\mathbb {E}}\left[ \int _t^{t+1} e^{2\eta (s-(t+1))} ds \right] \right) ^{1/2} = \sigma \left( \frac{1}{2\eta } - \frac{e^{-2\eta }}{2\eta } \right) ^{1/2}. \end{aligned}$$