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.
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Notes
When \(-p_{T-1}-\delta {\mathcal {R}}_{|p_{T-1}}[-p_T]=0\) the optimum of (7) is reached for any \(c\in [0,CA{\mathbb {X}}_{T-1}]\), hence, for this particular value we may adopt the convention \(c^*_{T-1}=CA{\mathbb {X}}_{T-1}\).
Details available from the authors upon request.
Even though the arithmetic O–U can lead to negative values, the process is frequently used to model the evolution of prices (see, for example, Alvarez and Koskela 2005; Gjolberg and Guttormsen 2002). The discussion of which process best represents timber prices is far from being settled. We refer the reader to Dixit and Pindyck (1994), Insley and Rollins (2005) and the references therein for more information.
Details available from the authors upon request.
The calculations are done in the “Appendix” of this manuscript.
We assume that at every step from \(t+1\) onwards, we either harvest nothing at all or everything available. Due to the linearity of the forestry model, this assumption is equivalent to requiring that the coefficient of c in (7) is never zero, but having a zero coefficient is an event with zero probability.
In the particular case that \(1-\delta a=0\) we observe that \(\varDelta ^m_j(p_t)\) does not depend of \(p_t\) and that Condition (28) can be verified a priori. The study of this particular case is straightforward and we omit it.
References
Alvarez, L., & Koskela, E. (2005). Wicksellian theory of forest rotation under interest rate variability. Journal of Economic Dynamics and Control, 29(3), 529–545.
Alvarez, L., & Koskela, E. (2006). Does risk aversion accelerate optimal forest rotation under uncertainty? Journal of Forest Economics, 12(3), 171–184.
Alvarez, L., & Koskela, E. (2007). Taxation and rotation age under stochastic forest stand value. Journal of Environmental Economics and Management, 54, 113–127.
Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.
Blomvall, J., & Shapiro, A. (2006). Solving multistage asset investment problems by the sample average approximation method. Mathematical Programming, 108(2–3), 571–595.
Boychuk, D., & Martell, D. (1996). A multistage stochastic programming model for sustainable forest-level timber supply under risk of fire. Forest Science, 42(1), 10–26.
Brazee, R., & Mendelsohn, R. (1988). Timber harvesting with fluctuating prices. Forest Science, 34(2), 359–372.
Carmel, Y., Paz, S., Jahashan, F., & Shoshany, M. (2009). Assessing fire risk using monte carlo simulations of fire spread. Forest Ecology and Management, 257(1), 370–377.
Clarke, J., & Harry, R. (1989). The tree-cutting problem in a stochastic environment: The case of age-dependent growth. Journal of Economic Dynamics and Control, 13(4), 569–595.
Dixit, A., & Pindyck, R. (1994). Investment under uncertainty. Princeton: Princeton University Press.
Gjolberg, O., & Guttormsen, A. (2002). Real options in the forest: What if prices are mean-reverting? Forest Policy and Economics, 4(1), 13–20.
Gong, P. (1998). Risk preferences and adaptive harvest policies for even-aged stand management. Forest Science, 44(4), 496–506.
Gong, P., & Löfgren, K. (2003). Risk-aversion and the short-run supply of timber. Forest Science, 49(5), 647–656.
Gong, P., & Löfgren, K. (2008). Impact of risk aversion on the optimal rotation with stochastic price. Natural Resource Modeling, 21(3), 385–415.
Haneveld, W. K. K., Streutker, M. H., & van der Vlerk, M. H. (2010). An alm model for pension funds using integrated chance constraints. Annals of Operations Research, 177(1), 47–62.
Hildebrandt, P., Kirchlechner, P., Hahn, A., Knoke, T., & Mujica, R. (2010). Mixed species plantations in Southern Chile and the risk of timber price fluctuation. European Journal of Forest Research, 129(5), 935–946.
Insley, M., & Rollins, K. (2005). On solving the multirotational timber harvesting problem with stochastic prices: A linear complementarity formulation. American Journal of Agricultural Economics, 87, 735–755.
Kim, Y. H., Bettinger, P., & Finney, M. (2009). Spatial optimization of the pattern of fuel management activities and subsequent effects on simulated wildfires. European Journal of Operational Research, 197(1), 253–265.
Knoke, T., Stimm, B., Ammer, C., & Moog, M. (2005). Mixed forests reconsidered: A forest economics contribution on an ecological concept. Forest Ecology and Management, 213(1), 102–116.
Lembersky, M., & Johnson, K. (1975). Optimal policies for managed stands: An infinite horizon Markov decision process approach. Forest Science, 21(2), 109–122.
Lönnstedt, L., & Svensson, J. (2000). Non-industrial private forest owners’ risk preferences. Scandinavian Journal of Forest Research, 15(6), 651–660.
Maller, R., Müller, G., & Szimayer, A. (2009). Ornstein–Uhlenbeck processes and extensions. Handbook of financial time series (pp. 421–438). Berlin, Heidelberg: Springer.
Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77–91.
Mosquera, J., Henig, M., & Weintraub, A. (2011). Design of insurance contracts using stochastic programming in forestry planning. Annals of Operations Research, 190(1), 117–130.
Ollikainen, M. (1993). A mean-variance approach to short-term timber selling and forest taxation under multiple sources of uncertainty. Canadian Journal of Forest Research, 23(4), 573–581.
Philpott, A., de Matos, V., & Finardi, E. (2013). On solving multistage stochastic programs with coherent risk measures. Operations Research, 61(4), 957–970.
Piazza, A., & Pagnoncelli, B. (2014). The optimal harvesting problem with price uncertainty. Annals of Operations Research, 217(1), 425–445.
Piazza, A., & Pagnoncelli, B. K. (2015). The stochastic mitra-wan forestry model: Risk neutral and risk averse cases. Journal of Economics, 115, 175–194.
Plantinga, A. (1998). The optimal timber rotation: An option value approach. Forest Science, 44(2), 192–202.
Rapaport, A., Sraidi, S., & Terreaux, J. (2003). Optimality of greedy and sustainable policies in the management of renewable resources. Optimal Control Applications and Methods, 24(1), 23–44.
Reeves, L., & Haight, R. (2000). Timber harvest scheduling with price uncertainty using markowitz portfolio optimization. Annals of Operations Research, 95(1–4), 229–250.
Rockafellar, R., & Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk, 2, 21–42.
Roessiger, J., Griess, V., & Knoke, T. (2011). May risk aversion lead to near-natural forestry? A simulation study. Forestry, 84(5), 527–537.
Ruszczynski, A., & Shapiro, A. (2005). Conditional risk mappings. Mathematics of Operations Research, 31, 544–561.
Shapiro, A. (2009). On a time consistency concept in risk averse multistage stochastic programming. Operations Research Letters, 37(3), 143–147.
Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on stochastic programming: Modeling and theory (Vol. 9). Philadelphia: SIAM.
Tahvonen, O., & Kallio, M. (2006). Optimal harvesting of forest age classes under price uncertainty and risk aversion. Natural Resource Modeling, 19(4), 557–585.
Thomson, T. (1992). Optimal forest rotation when stumpage prices follow a diffusion process. Land Economics, 68(3), 329–342.
Valladão, D. M., Veiga, Á., & Veiga, G. (2014). A multistage linear stochastic programming model for optimal corporate debt management. European Journal of Operational Research, 237(1), 303–311.
Yoshimoto, A., & Shoji, I. (1998). Searching for an optimal rotation age for forest stand management under stochastic log prices. European Journal of Operational Research, 105(1), 100–112.
Zheng, H. (2009). Efficient frontier of utility and cvar. Mathematical Methods of Operations Research, 70(1), 129–148.
Zhong-wei, W., & Yan, P. (2009) Measurement of forest fire risk based on var. In International conference on management science and engineering, 2009. ICMSE 2009. IEEE.
Acknowledgments
This research was partially supported by Programa Basal PFB 03, CMM. B.K. Pagnoncelli acknowledges the financial support of FONDECYT under projects 11130056 and 1120244. A. Piazza acknowledges the financial support of FONDECYT under Project 11090254 and of CONICYT Anillo ACT1106 and CONICYT REDES 140183.
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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
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),
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
and the corresponding value function
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)
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,
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 \(Q^\varPi _0\),
As the bound on \(Q^\varPi _0\) does not depend on the policy \(\varPi \), we conclude
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,
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
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
The coefficient affecting c in \(\varPhi (c)\) is
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
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
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
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
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
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
Inserting again \(V = Q^{AP}\) into the right-hand side of the Bellman’s equation (7), the argument of the min operator is
The coefficient of c is negative, and thus, the minimum is attained when \(c=\bar{x}+x_n\). So we have,
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
Using that
we have that (26) is equivalent to
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:
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
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.
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 \(\varGamma ^m(p_t)=-\delta ^m[ p_ta^m +b\sum _{l=0}^{m-1}a^l]\).Footnote 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
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:
Hence, the first term of the coefficient of c is of the form
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
Using that \(1\le a\) and \(c_t\le S\) for all t we get
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
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 \(\varDelta ^m_j(p_t)\),
The parameters’ semi-plane is divided in five regions as shown in Fig. 3 and Table 2.Footnote 7
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), \(p_t\ge b\delta /(1-a\delta )\).Footnote 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.,
In the following, we prove the properties summarized in Fig. 3. We observe in the first place that
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.
If \({{\mathrm{sign}}}(b){{\mathrm{sign}}}(a-1)>0\), i.e., in regions (ii) or (iii), (30) is equivalent to
-
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.
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.
In regions (i) and (ii) we are ready to give a sufficient condition assuring Condition (28):
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:
In order to calculate the stochastic integral we apply Itô’s isometry:
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
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Pagnoncelli, B.K., Piazza, A. The optimal harvesting problem under price uncertainty: the risk averse case. Ann Oper Res 258, 479–502 (2017). https://doi.org/10.1007/s10479-015-1963-9
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DOI: https://doi.org/10.1007/s10479-015-1963-9