Optimal risk management problem of natural resources: application to oil drilling

The aim of this paper is to determine the optimal balance between extraction and storage of a natural resource (in particular crude oil) over time under a large array of environmental, operational and financial constraints for an infinite maturity time. We consider a manager that owns an oil field from which he can extract oil and decides to sell or store it. This operational strategy has to be carried out in continuous time and has to satisfy physical, operational, environmental and financial constraints such as storage capacity, crude oil spot price volatility, amount available for possible extraction or maximum amount that could be invested at time t for the extraction choice. The costs of storage and extraction are also taken into account to better fit the real market scenario. We solve the optimization problem of the manager’s profit under this large array of constraints and provide an optimal strategy. We then examine different numerical scenarios to check the robustness and the corresponding optimal strategies given by our model, which is obtained by a numerical approach, with respect to different possible events related to the market, environmental policies or ecological constraints.

faster pace than it can regenerate (Behrens et al. 2007;Butchart et al. 2010). How the access to resources will be organized while economic growth is an equally important issue nowadays. Understanding the consequences of economic growth on the environment and ecosystemÕs vulnerability remains topical for scientific research.
The study of non-renewable resources extraction started with the work of Hotelling (1931), who studied general equilibrium models including the uncertainty in technology and the size of the resources. The decision function extraction of the stochastic price process has a shorter history in the non-renewable resource literature, starting with the work of Tourinho (1979).
Regarding the problem of managing an oil resource, the producer has to address his optimal choice of quantities between extraction and storage of crude oil over time. Indeed, an oil producer should decide both on the proportion of extracted oil to be sold and the one to be stored with respect to financial, environmental and practical constraints. An oil producer should determine the optimal extraction by computing the expected value of delaying the extraction of a barrel. Following the analysis of Arrow et al., the problem was recognized and discussed as an abstract optimal stopping problem by Snell. The first application of the optimal stopping problem to finance appeared in the work of Bensoussan (1984). Nevertheless, numerous analytical approximations and numerical methods have been recommended in the literature. For most option pricing problems, three numerical methods are available: finite difference, lattices, and the Monte Carlo process. Using lower and upper bound estimates, the algorithm of Glasserman (2003) addresses the high-dimensional American options, but the computational effort still grows exponentially with the number of possible exercise dates (see Jaillet et al. 2004;Carmona and Touzi 2008). To satisfy the processing limitations and the optimization issue of an exploitation of oil or gas fields, Huseby and Haavardsson (2009) claimed that the production has to be choked. This framework can find the optimal production strategy with respect to various types of objective functions. Consequently, at any specified point in time, the production is scaled down by a suitable choke factor and should not exceed the processing capacity. Huseby and Haavardsson (2009) extended this approach to the cases where the production is uncertain. Alexandrov et al. (2012) proposed a Monte Carlo real option approach as a solution to the optimization problem of a price-taker oil producer. Their method enables to analyze the impact of interest rates, size of reserves, risk aversion, expectations of oil prices, and oil reserves. The results showed that while uncertainty in the size of the reserves justifies the conclusion that full extraction is optimal, for mature producers, this uncertainty has been shrinking over time. It is possible that this result was obtained because the borrowing conditions of countries and companies are worsened by pro-cyclical extraction policies.
The aim of this paper is to address the optimal choice between the extraction and storage of crude oil over time. We examine the optimal choice between extraction and storage of crude oil over time. An oil producer should decide both on the proportion of extracted oil to be sold and the one stored. This optimal operational strategy should be conducted on a daily basis while taking into consideration physical, operational, environmental and financial constraints such as the storage capacity, crude oil spot price, total quantity available for possible extraction or maximum amount that could be invested at time t for the extraction choice. In this paper, we propose to extend massively to a much more general case the model initiated in Abid et al. (2018) in the case where there are both costs of storage and extraction, cost penalty to sell stored oil and to deal with robustness tests regarding economic, environmental and operational scenarios. We will solve this optimization problem and find the optimal strategy. Our results will show that in the case of increasing prices, the storage cost has no impact on profit and the extraction cost has a very limited impact on future income. The storage capacity, the penalty for storage before selling and the volatility of the market strongly affect the expected profit of a resource producer. We will see through robustness tests regarding economic, environmental and operational scenarios, that the storage capacity, the extraction cost and a change in the volatility of the spot oil market strongly affect the expected optimal profit of a resource producer. It is therefore a way towards efficient policy to preserve the ecosystem and environment and to not only reduce CO2 emissions but also reach the critical objectives of environmental international agreements.

Management strategies for oil exploitation
We consider in this paper the point of view of a manager who owns an oil extraction station and a place to store the oil, and we assume that he can exploit this one in continuous time. We denote the initial quantity of oil available in the station for possible extraction by Q D ∈ R + and we denote the maximal storage capacity by Q S ∈ R + which are two given positive constants. The first decision that the manager can have is the quantity of oil that he extracts at time t, then the part of this quantity that he sells on the market, and the part that he stores, lastly the quantity of the stock that the manager decides to sell. Thus, for any time t, we must consider a quadruple (q e t , q s t , q v t , q v,s t ) which represents the strategy of the manager q e t the amount of extracted oil per unit of time, q s t the amount of extracted oil stored per unit of time, q v t the amount of extracted oil per unit of time immediately sold on the market, q v,s t the amount of stored oil sold per unit of time.
At any time t, the quantity extracted q e t is equal to the amount of the quantity sold q v t and the quantity stored q s t . So it is enough to know only two of these quantities to know the third. Thus we now only consider the quantity stored and the quantity of fresh oil sold at time t. In Goutte et al. (2018), the authors consider the case when the extracted quantity q e t is a positive constant, that means the manager always extracts the same quantity. In this paper, we consider the more realistic cases when the extracted quantity is not assumed to be constant but just upper bounded by a given positive constant K 0 q e t = q v t + q s t ≤ K 0 . In the sequel of this paper, we consider a complete filtered probability space ( , F , F, P) and a one-dimensional F-Brownian motion B which corresponds to the uncertainty of the oil price. We now consider the set A of admissible strategies which consists in the triplet of nonnegative F-adapted processes (q s t , q v t , q v,s t ) t≥0 satisfying the following constraints. (C1) The total stored quantity must be nonnegative and we suppose this one is upper bounded by Q S which corresponds to the limit of the infrastructure 0 ≤ Q s t ≤ Q S , t ≥ 0, with the total stored amount of oil Q s is given by where Q s 0 represents the initial quantity of the stored oil. (C2) The manager cannot sell on the market a quantity stored greater than an operational bound q S per unit of time

Remark 1.1
The bound q S in condition (C2) means that even if we have an important number of barrels in our stock, we cannot sell more than the fixed quantity q S per unit of time from our stock, since we have some technical constraints which limit this quantity.
We also introduce the random variable E t which represents the total amount of oil extracted between the initial time and time t > 0 and is defined by (1.1) We now define two different cost functions: the extraction cost c e and the storage cost c stock . Both are increasing functions. The storage cost is increasing with the volume of stored barrels. This assumption states the standard economic modelling of a storage cost by implying an increasing function since it is related to operational, infrastructure, management and security costs caused by the storage of oil. While the extraction cost is increasing with the quantity that we have already extracted E, since we have to drill increasingly deeper in order to extract fresh oil then the cost is more expensive.
We assume that the manager of the station sells at time t the fresh oil to the market price P t which is given by where μ and σ are given constants with σ > 0. We also assume that the manager sells the stored oil at time t to the price (1 − ε)P t , with ε ∈ [0, 1], since the quality of stored oil can be lower quality.
We can now define fo any admissible strategy q := (q s , q v , q v,s ) ∈ A the wealth of the manager W q t at time t by where the term q v u P u is the gain of selling the quantity q v u of fresh oil at price P u on the market, q v,s u (1 − ε)P u is the gain of selling the quantity q v,s u of oil previously stored at price (1 − ε)P u , (q v u + q s u )c e (E u ) is the loss dues to the extraction cost of the global quantity q v u + q s u and c stock (S u ) the storage cost associated to the stock S u .

Optimization problem
The expected utility theory has a long and prominent history in the development of decisionmaking under uncertainty. It is assumed that investor preferences can be represented by a utility function U , which embed individual risk aversion. The main classes of utility functions (see for example Gollier 2001; De Palma and Prigent 2008) are quadratic functions, which are defined only by the first two moments, the constant relative risk aversion utility functions (i.e., the power functions) and the constant absolute risk aversion utility functions (CARA), which correspond to negative exponential utility functions of the following form where γ is a positive constant which represents the risk aversion. In the following, we consider that the manager has an exponential utility function over an infinite horizon.

Value function
We recall that the set A of admissible strategies is defined by the triplet q = (q v t , q s t , q v,s t ) t≥0 such that For any initial state x = ( p, e, s) ∈ D, with D := R * + × [0, Q D ] × R + and any admissible strategy q ∈ A, we define the state process X x,q corresponding to (P, E, S), where P refers to the oil price, E is the total quantity of oil extracted and S is the amount stored, with P 0 = p, E 0 = e and S 0 = s. We know that the dynamic of X x,q is given by For any initial state x ∈ D, we only consider the admissible strategies q ∈ A such that where ρ is a positive constant which corresponds to a discount factor. We denote by A(x) the set of these admissible strategies.
For any x ∈ D and q ∈ A(x), we define the gain function J (x, q) by We can rewrite (2.3) as The objective is to maximize the gain function J over the admissible strategies A(x), and we introduce the associated value function

PDE characterization
To characterize the value function v we use the classical approach which consists on the HJB equation satisfied by v and a verification theorem. The HJB equation linked with the value function v is given by where L q is the operator associated with the diffusion (2.2) for the constant control q and is defined by Theorem 2.1 Let w ∈ C 2 (D) satisfying a quadratic growth condition, i.e., there exists a positive constant C such that admits a unique solution, denoted byX x,q t , given an initial conditionX whereq is an optimal Markovian control.

Remark 2.2
If we assume that ρ > 2μ + σ 2 , then the conditions (2.6) and (2.8) hold owing to the quadratic growth of w.
Proof (i) Let w ∈ C 2 (D) and q ∈ A(x). By Itô's formula applied to e −ρt w(X x,q t ), we obtain for any stopping time τ n e −ρT ∧τ n w(X We consider the sequence of stopping times (τ n ) n≥1 defined by Using this sequence of stopping times to take the expectation, we obtain Since w satisfies (2.5), we have Using the quadratic growth condition on w and the integrability condition on X x,q , we may apply the dominated convergence theorem and send n to infinity (2.9) By sending T to infinity and using the dominated convergence theorem, we obtain for any strategy q ∈ A(x) for any x ∈ D. (ii) By repeating the above arguments and observing that the controlq achieves equality (2.9), we have By sending T to infinity, from (2.8), we obtain where the left-hand-side term is by definition J (x,q); thus w(x) = J (x,q).

Optimal strategy
If we know the value function v, we can find the optimal strategy. To find the optimal strategy, we must find the argmax of L q w(x) + f (x, q) over D(x), which can be rewritten arg max which is equivalent to find arg max We remark that U is concave and that the inside term of U is linear in is concave. This property allows us to find the optimal strategy because it is a problem of concave maximization on a convex set. Several methods for finding the maximum value are available, such as Kuhn-Tucker conditions, subgradient projection, and Lagrange multipliers.

Algorithm to find the optimal control strategy
We introduce the following sets, which are useful for the decomposition of the domain D(x): For any x ∈ D, we can use the following method to find the optimal strategy.
Step 1: Since U is invertible, we can solve We denote by (q (k) ) k≥1 the different solutions of this equation.
Step 2: We compare all values ϕ(q (k) ) with k ≥ 1, the optimal solution is given by the sup, and the optimal strategy is given by q * ∈ {q 1 , . . .} such that ϕ(q * ) ≥ ϕ(q (k) ) for any k ≥ 1.

Data
The data are taken from the daily WTI oil prices in the time period from 01/01/2005 to 31/12/2016. Figure 1 shows the historical evolution of the crude oil price, revealing that this one has fluctuated considerably over our sample period, especially during the periods of global crises  Figure 2 shows an exceptional volatility of the crude oil price return during the financial world crisis (2008)(2009) and during the last two years, which has led to an increase in the uncertainty in the world economy as well as in the financial markets. In fact, the fluctuation of the WTI crude oil price is the result of the report published by the US agency AP (Associated Press), which indicates that the American tanks used to store oil have been practically full for several weeks. That is, the United States was able to sell all of the crude oil that it produced or imported. Moreover, over the past seven weeks, the United States has produced and imported an average of one million barrels more oil per time unit than it consumes. Stored mainly at Cushing in Oklahoma, the amount of available crude oil has reached a level not seen in 80 years (US Department of Energy). Thus, the U.S. government even fears the "tank tops", the limit where another drop of oil cannot be stored. As a result, oil prices in the markets fell sharply and continued to decline during the subsequent months. Therefore, these points confirm the interest of an optimization strategy that balances the storage and sale of crude oil.
Returns are calculated from the price data by taking the natural logarithm of the ratio of two successive prices. The statistical properties reveal that the average oil return is − 0.0002 over our sample period. The skewness and kurtosis coefficients are, respectively, − 0.0706 and 7.306, indicating that the oil return distribution is skewed towards the left and revealing the leptokurtic behaviour of the return distributions with fat tails. Additionally, the oil return distribution deviates significantly from normality, as shown by the Jarque-Bera test results.

Optimization results
In this part, we present some numerical results about the solution of the HJB Eq. (2.4) related to the value function v. Indeed to the best of our knowledge, it is impossible to find an explicit solution to this HJB equation since the control set D(x), in which the optimal strategy q * lives, evolves with the time and the constraints are modified. Thus, we propose an approximation of the value function v and of the associated optimal strategy q * . For that we used a finite difference scheme which leads to the resolution of a Controlled Markov Chain problem. This category of problems is deeply studied by Kushner and Dupuis (2001) who state that the solution of a given HJB equation can be approximated by the solution of a Controlled Markov chain problem. The convergence of the solution of our numerical scheme towards the solution of the HJB equation, when the space step goes to zero, can be shown using the standard local consistency argument provided in Kushner and Dupuis (2001) (i.e. the first and the second moments of the approximating Markov chain converge to those of the continuous process X representing the state of the system). The main advantage of this probabilistic approach is that it does not require the use of any analytical properties of the real solution. We may refer in particular to Budhiraja and Ross (2007), Hindy et al. (1993), for numerical schemes involving a Controlled Markov Chain problem to approximate a control problem. We bring to the reader's attention that the computed approximated control q * is only a candidate for the optimal policy. Further investigations are needed to show that the numerical computed strategy is optimal for the control problem. This latter point could be the topic of a future work.
In this general case, we adopt the following parameters and functions: -The storage cost c stock is assumed to be given by c stock (x) = exp (ξ s x) for any x ≥ 0, where ξ s = log(10)/Q D is a constant for normalizing the units. This cost function means that the storage cost increases exponentially with the increasing quantity that must be stored. This well reflects the actual conditions with higher cost for security, pollution rules and infrastructure when the amount of stored oil is higher. -The extraction cost c e is assumed to be given by c e (x) = exp(ξ e x) for any x ≥ 0, where ξ e is a positive constant which will be given in the sequel. This cost function increases exponentially with the quantity of extracted oil. Indeed, since we have to go more deeply to extract oil, the corresponding extraction cost inceases. -The total amount available for extraction is 10,000,000 barrels. -We cannot extract more than K 0 = 10,000 barrels per time unit. -We cannot sell more than q S = 100,000 barrels that were stored previously per time unit. This means that q v,s t ≤ q S = 100,000 for any t > 0. -The market parameters are ρ = 0.05, μ = 0.01 and σ = 0.02. -The risk-aversion parameter will be equal to γ = 0.0005.
We consider two different cases for the market: a standard market with all constraints and a simple market without the extraction cost and storage cost. The results show the optimized returns for state variables at time t > 0, (P, S, E), where P denotes the current price per oil barrel, S denotes the total amount of barrels stored and E denotes the total amount of oil already extracted. Figure 3 shows the corresponding optimized value function of our investment problem for a fixed price P = 55$ for different values of S and E.  shows the corresponding optimal strategy q * := (q v, * , q s, * , q v,s, * ) for different values of S and E. Figure 3 suggests the following: -The value function is maximized at V > 18 when the storage is full (S = 10 × 10 6 ) and when there is no extraction initially (E = 0). Indeed, it corresponds to the case in which we have a full available stock and a full reserve available for extraction. -The storage increases the value function for all possible already extracted amounts. Here, storage plays the role of security against the volatility of the market price. -For a fixed stored quantity, when the total amount of already extracted oil increases, the value function decreases. This result is expected because the available reserve is less important, and therefore, the potential income decreases.
If we analyse Fig. 4, which shows the optimal strategy q * := (q v, * , q s, * , q v,s, * ) obtained, we can see the following: For the quantity extracted at time t: When the stock is full or close to full, we prefer to sell a higher fraction of the extracted oil. Indeed, as the stock is full we can not stock any new quantity extracted. So there are only two possible solutions: sell more stock to have new available stock at the next step; or reduce the quantity we extract. Therefore, when the stock is quite empty, we prefer to store a much higher fraction of the extracted oil. Indeed, this is economically reasonable because when the stock is quite empty more than 10% of the possible quantity, the best strategy is to place the oil barrel in the stock to secure our gain and make it impervious to a possible change of price. Conversely, when the stock is full, it is useless to put more oil in the stock, and the best strategy is to sell a higher fraction of the extracted oil. The case of an empty stock is interesting. In this case, the best strategy is always to sell all of the oil and never store anything. Indeed, when stock is empty, we observe that we stored nothing and we sell all extracted quantity directly. This is too an operational cost reason since the management of a natural resource has a running cost and if the producer has no stock he has to cover his running cost first and sell his production before to think about storage and financial trading optimization. For a middle scenario in which the stock is half full and the reserve available for extraction is also half full, the best strategy is to sell one part of the extracted oil and store the other part. We observe in a large vision that there are always compensatory effects linked to the fact that financial, operational and cost factors are stated and impemented in our optimization problems. We always extract the maximum possible amount of oil. This means that for any t > 0 we have q e t = q v t + q s t = K 0 = 10,000. For the quantity stored and sold at time t: The results are homogeneous with respect to the total amount already extracted. This means that the control variable q v,s t does not depend on the size of the reserve but only on the amount available in the stock. For a larger amount in the stock, we choose to sell a greater amount of oil.

Robustness tests regarding economic, environmental and operational scenarios
We would now like to measure and test the robustness and the optimal value function obtained for different scenarios. These scenarios will reflect economic, financial, environmental or operational changes and constraints.
-Production and Planet environmental ecosystem preservation constraints, such as a decrease of the quantity available for possible extraction or of the total possible quantity that can be stored. -Cost and Penalty constraints, such as an increase of the storage cost or of the extraction cost. We will also analyse an increase of the selling penalty for stored oil. -Financial crisis events, such as a cut in the fossil fuels price or an increase of the crude volatility price or drift parameter. -Risk aversion analysis with respect to the producer.
To address these scenarios, we will consider five possible state variables at time t > 0, (P, S, E) i , with i ∈ {1, 2, . . . , 5}, where we recall that P denotes the current price of a oil barrel, S denotes the total number of barrels stored and E denotes the total amount of oil already extracted. These states are as follows:  Results are given for a state price P = 55$ Average life case: The first case will be the mean or middle life of our resource exploitation case, where for a price P, we obtain (P, S, E) 1 = (P, Q S 2 , Q D 2 ). This means that we are in a state where half of both the storage capacity and the amount of oil available for possible extraction are already achieved. Beginning life case: The second case will be the beginning life of our resource exploitation case, where for a price P, we obtain (P, S, E) 2 = (P, 0, 0). This means that no oil has already been extracted and stored. The entire possible available resource and storage capacities are available. Full capacity stored case: The third case will be the case in which the maximum storage capacity Q S is achieved but we have nothing extracted (i.e., we begin the life of our resource exploitation with a full stock). For a given price P, we obtain (P, S, E) 3 = (P, Q S , 0). Available resource exhausted without stock: The fourth case will be the case in which the total quantity of oil available in the station for possible extraction Q D is achieved. The available resource is exhausted, and we have no stock (i.e., the stock is empty). For a given price P, we obtain (P, S, E) 4 = (P, 0, Q D ). Available resource exhausted with full stock: The fifth case will be again the case in which the total quantity of oil available in the station for possible extraction Q D is achieved but we have a full available stock. For a given price P, we obtain (P, S, E) 5 = (P, Q S , Q D ).
Remark 3.1 These five cases correspond to interesting economic and operational cases. The fourth case is the worst case scenario because no more resources are available for extraction and the stock is empty. In this case, the possible future income is zero. This worst case will give always the same optimized value function equal to (q s , q v , q s,v ) is (0, 0, 0). This implies that the corresponding value function gives ϕ(q s , We clearly see in Table 1 that the third case is the best case scenario: the full capacity stored case. Indeed, for this case, we are in the situation where the maximal storage capacity Q S is achieved but we have nothing extracted. For the following results, the market parameters for the standard case are ρ = 0.05, μ = 0.01, σ = 0.02 and γ = 0.0001.

Financial crisis events
We begin with the studies of financial crisis events. Figure 5 gives the results for the average life case with respect to the observed price state. We see that as the price increases, the future possible income increases which is economically reasonable.
In Table 2, we address the scenario case of an increasing volatility of the crude oil price σ . This increase will reflect economic or financial events such as financial crises, political instability in the Middle East (OPEC countries) or war, which can affect crude oil production   Results are given for a state price P = 55$, with the gain or loss with respect to the standard case given in the first row or in Table 1 provided in parenthesis and therefore the oil price in the world economy. We observe that in all possible states of our resource, an increase of the volatility implies a decrease of the value function and therefore a financial loss. In the average life case, a 50% increase in the volatility leads to a loss of 0.65%, and a 100% increase of the volatility leads to a loss of 1.6%. The results shown in Table 2 demonstrate that an increase of the instability in crude oil price implies a real loss in our optimal natural resource investment problem.
In Table 3, we address the scenario of a variation of the crude drift price μ. We see that a decrease of the drift reduces our potential future income. For the average life case, the loss is 2.41%. On the other hand, an increase of the drift induces an increase of the value function which is in concordance with the financial intuition. For instance, for the average life case, an increase of 100% of the drift implies a gain of 7.62%.

Risk aversion
In Table 4, we address the scenario of a variation in the risk aversion parameter γ of the producer. We see that an increase in the aversion of risk implies an increase in the future income. We obtain a change in the future income from 9.1282 to 13.1014, corresponding to an increase of 53.35%. This result can be explained by the huge estimated volatility of the market σ = 2%. Indeed, a more prudent producer will be less impacted by this price volatility and will therefore obtain an increase in the future possible income.

Production and Planet environmental ecosystem preservation constraints
In Table 5, we address the scenario of a decrease of the available amount for possible extraction. This scenario reflects the possibility of an environmental agreement or constraints that limit the extraction volume of the resource. At the current levels, the consumption of fossil energy appears strongly unsustainable. Carbon stranded assets place the necessity to embrace a transition away from a fossil-fuel driven society at the centre of contemporary discussion and analysis. Changes in the values and attitudes are to be expected as the fossil fuel companies' shareholders fully integrate environmental issues within their developmental framework. This is why we address this scenario and the next scenario in which there is a decrease of the total possible amount of oil that can be stored. The results listed in Table 5 are in agreement with our expectations because a decrease of the maximum possible extracted volume capacity reduces the value function and the expected income. The value of the expected income changes from 9.1282 to 7.9320 (loss of 7.15%) with a reduction by ten for the average life case. However, if we examine the beginning life case, which represents the future income of new resource production, the observed changes is from 1.2595 to 0.1286, corresponding to a loss of 89.78%. This means that if we have a greater reduction in the volume of extraction, a new resource plan becomes essentially useless because the future possible income vanishes (Table 6).
We now examine the results with respect to a decrease of the total possible amount of oil that we can store. This scenario reflects the infrastructure investment possibility and all investment that a resource company can make in this sector. Here, as well, we observe that a loss in the quantity that can be sold from stock per time unit implies a loss in the future possible income. We obtain a loss of 73.82% for the average life case if we decide to impose a reduction of ten times for this parameter.

Cost and Penalty constraints
In Table 7, we provide the results for the scenario of an increase in the selling penalty for stored oil. This scenario can reflect environmental penalties for extracting the oil and not    Results are given for a state price P = 55$, with the gain or loss with respect to the standard case given in the first row or in Table 1 provided in parenthesis selling it immediately or a loss in term of quality of the barrel of oil. Indeed, the choice of storage implies that the volume of extracted oil is greater than the volume demanded by the market. Of course, we observe that an increase of this penalty decreases the possible future income. For example, a loss of 6.85% is obtained for the average life case if the selling penalty is zero. We now address the two scenarios of a modification of the storage cost and extraction cost.
Examination of the results presented in Table 8 shows that the effect of the storage cost is quite small. Indeed, in the worst case (i.e., a higher value for the storage cost), the loss is only 0.05%. This means that an environmental policy aimed at reducing pollution and preserving the environment based on increasing the storage cost is useless. Therefore, this is not an effective approach for saving the planet. Results are given for a state price P = 55$, with the gain or loss with respect to the standard case given in the first row or in Table 1 provided in parenthesis. We recall that ξ s = log(10)/Q D Results are given for a state price P = 55$, with the gain or loss with respect to the standard case given in the first row or in Table 1 provided in parenthesis. We recall that ξ e = log(15)/Q D On the other hand, examination of the results for the extraction cost presented in Table 9 shows that an increase of the extraction cost has an impact on the future incomes and thus can be used to manage the environment system. We obtain a reduction in income of 2.26% in this case.

Conclusion
We examine the optimal choice between extraction and storage of crude oil over time. An oil producer should decide on the proportion of extracted oil to be sold and the proportion stored. This optimal operational strategy should be conducted on a daily basis while taking into consideration physical, operational, environmental and financial constraints such as the storage capacity, crude oil spot price, total quantity available for possible extraction or maximum amount that could be invested at time t for the extraction choice. In this paper, we solve this optimization problem and find the optimal strategy. Our results show that in the case of increasing prices, the storage cost has no impact on profit and the extraction cost has a very limited impact on future income. The storage capacity, the penalty for storage before selling and the volatility of the market strongly affect the expected profit of a resource producer. This means that an efficient policy to reduce global warming and preserve the ecosystem and environment must be related to storage capacity, infrastructure or penalties for using this production method. Indeed, we have seen in our robustness tests regarding economic, environmental and operational scenarios that the storage capacity, the extraction cost and a change in the volatility of the spot oil market strongly affect the expected optimal profit of a resource producer. It is so a way to efficient policy to preserve the ecosystem and environment and to reduce CO2 emissions and reach the expectation of some international agreements.