Zhou YANG Manni LV Haisheng YANG

Abstract In this paper,the authors investigate the optimal conversion rate at which land use is irreversibly converted from biodiversity conservation to agricultural production.This problem is formulated as a stochastic control model,then transformed into a HJB equation involving free boundary.Since the state equation has singularity,it is difficult to directly derive the boundary value condition for the HJB equation.They provide a new method to overcome the difficulty via constructing another auxiliary stochastic control problem,and impose a proper boundary value condition.Moreover,they establish the existence and uniqueness of the viscosity solution of the HJB equation.Finally,they propose a stable numerical method for the HJB equation involving free boundary,and show some numerical results.

Keywords Optimal stochastic control,HJB equation,Free boundary,Land use

1 Introduction

Land is one of the most important nature resources,which has at least two common uses:Agricultural production and biodiversity conservation.It is a popular and important topic that how to allot the land to production and conservation appropriately,which is focused on in this paper.

There is a vast literature on this subject.In[1,7],the authors discover the optimal conversion rules via two-period stochastic models.The authors extend the similar arguments by means of continuous time models in[2-3].But there are no direct feedback expressions of the optimal conversion strategy in the above papers.In[10],the authors formulate this problem into a stochastic control model,and explore the properties of the optimal feedback between the conversion decision and the conservation benefit.They first transform the optimal stochastic control model into its associated Hamilton-Jacobi-Bellman(HJB for short)equation,and obtain some numerical results via the numerical method for partial differential equation(PDE for short).Unfortunately,there remains some uncertainty about the boundary value condition for the HJB equation and room for model improvement,which will be presented later.

In this paper,we suppose that land has only mutually exclusive uses: Agricultural production use and biodiversity conservation use.Land use can be converted from conservation to production,but the inverse transformation is inadmissible.Moreover,we assume that the transformation rate has an upper bound,and our aim is to obtain maximum benefit through adjusting the transformation rate.This problem can be formulated into a stochastic control model.In this model,the states are taken as the land area used as biodiversity reserveR(suppose that the total area is one unit)and the biodiversity conservation benefit per unitY,the control variable is just the transformation ratev,and the objective functional F is the expectation of the total benefit from agriculture and conservation minus the converting cost.The specific formulation of this model will be presented in Section 2.

In summary,the state equations in the optimal stochastic control problem are as follows:

As we all know,the boundary value condition is important to a PDE problem.PDE with improper boundary value condition might have multiple solutions.So,the proper boundary value condition is important to ensure that the viscosity solution of PDE(1.6)is really the value function of Problem(1.5).In the PDE(1.6),there are four boundariesy=0,r=0,y→+∞andr=1,on which the appropriate boundary values are necessary1In fact,we will show that it is sufficient to impose the boundary value condition when r=0 in Section 3..Particularly,on the boundaryr=0,since the state equation(1.2)makes no sense,it is difficult to impose the valid boundary value condition.

In fact,Leroux,et al.consider a similar model in[10],where they use the biodiversity reserveRand the biodiversity conservation benefitB=YRas the state processes.From Itˆo’s formula,Bis governed by the following stochastic differential equation(SDE for short):

Our contributions can be summarized as follows.The first contribution is that we deduce a more feasible boundary value condition for the HJB equation(1.6),which is provided in Section 3 by means of some stochastic control methods.The main difficulty in imposing the proper boundary value conditions comes from the singularity atr=0,which means that the state equation(1.2)makes no sense ifr=0.So,it is impossible to directly achieve the value ofFatr=0.In order to overcome the difficulty,we first obtain the upper bound of the upper limit ofF(y,r)asr→0+via estimating the upper bound ofF,and then obtain the lower bound of the lower limit ofF(y,r)asr→0+via another auxiliary stochastic control problem.Since we prove that the upper bound is equal to the lower bound,they are just the limit ofF(y,r)asr→0+.Thus,the value functionF(y,r)is continuous with respect toratr=0 if we let its value atr=0 be equal to the limit.Moreover,we expect that the approach proposed in this paper has wider applicability,and can be used to impose the boundary value condition for the kind of HJB equation,which is arisen from the stochastic control problem with the state equation involving singularity.

The second contribution is to establish the existence and uniqueness of the viscosity solution of the HJB equation(1.6).We can not directly apply the classical result for the existence and uniqueness of the viscosity solution of the HJB equation(1.6),due toin the coefficient functions in(1.6)blowing up whenrtends to zero.In order to obtain the existence and uniqueness of the viscosity solution,we firstly define the viscosity solution of the HJB equation(1.6)according to Definition 4.1,where the solution must have linear-growth with respect toyand proper asymptotic property asrtends to zero,which is different from the standard definition in[5].Then we construct a sequence of classical stochastic control problems(4.1)without blowing up characteristic to approximate to the original problem(1.5),and show the existence and uniqueness of the viscosity solution of the HJB equation(1.6)via establishing some proper estimates on the value functions of the stochastic control problems(4.1).

The third contribution is that we provide a stable numerical method for this kind of HJB equation associated with some free boundaries,which is first applied to compute the viscosity solution of HJB equation in[8-9],then extended to some financial PDE problems involving some free boundaries(e.g.in[4]).In fact,the authors apply another numerical method to the problem in[10],which looks like arising from“practical meaning”rather than from rigorous mathematical method.They assume that there exists a curve(i.e.,the free boundary)splitting the overall solution domain into two parts: The conservation region,where the optimal transformation ratev∗=0,and the conversion region withand the free boundary is just the intersection between these two regions.Moreover,they assume that the value functionis smooth in these two regions,and its first order derivative continuously crosses the free boundary.Though the assumptions may be true in this practical problem,no existing mathematical result ensures that these assumptions are right.In fact,there are several free boundaries rather than only one free boundary in some other practical problems,such as the example given in[11].In order to avoid these assumptions,we apply the finite difference method for the viscosity solution of HJB equation in[4,8-9]to the HJB equation(1.6),and provide some numerical results.

The forth contribution is to build a model which is more reasonable than the model introduced in[10](Model L for short),where the authors use the biodiversity reserveRand the biodiversity conservation benefitB=YRas the state processes.We will show that the value functionin Model L always achieves its maximum value atr=0 by means of Theorem A.1.It means that the value of land is maximum when all land is used for agriculture production,which is puzzling.The key of the puzzle comes from the fact that the biodiversity conservation benefit per unit is infinite whenb>0 andr→0+,which is impossible.So we change the stateBbyY,which represents the biodiversity conservation benefit per unit.We will show that the benefit achieves its maximum at a proper biodiversity reserve by means of the numerical result in Table 2.

This paper is organized as follows: We formulate the model,deduce the HJB equation in Section 2,and derive the boundary value condition in Section 3.We prove that the value function is a viscosity solution of the HJB equation(1.6),and establish the existence and uniqueness of the viscosity solution of the HJB equation(1.6)in Section 4.In Section 5,we apply the finite difference method for HJB equations to Problem(1.6),and obtain some numerical results.Finally,we apply the methods in Section 3 and Section 5 to Model L in Appendix A,and show a more reasonable numerical result.

2 Optimal Conversion of Land Use Model

In this section,we first formulate the model as a stochastic control problem,then transform it into its associated HJB equation.

2.1 Assumptions and model

Consider an area of land,normalized to unity.The land use can be converted from biodiversity conservation to agricultural production,and the conversion is irreversible.LetAtandRtbe the area of the land used for production and conservation at timet,respectively.Hence,At+Rt=1.

Suppose that the area of the land for biodiversity reserveRr;vsatisfies the ordinary differential equation(ODE for short)(1.1),wherevis the conversion rate from reservation land to agricultural land,and the superscriptr;vmeans thatRdepends on its initial staterand the conversion strategyv.It is clear thatAtandRtare nonnegative.Moreover,it is natural to assume that the initial conservation landr/=0 as the conversion is only from conservation land to agriculture land.So,we suppose that 0

The biodiversity conservation benefitBis governed by the SDE(1.7),where parametersα,m,σEandσCare positive constants,and represent the increasing rate of the species value in conservation land,the elasticity coefficient of conservation benefit,the volatility of the species value and the volatility of the ecosystem-specific species density,respectively.For more details,refer to[10].Zis a one-dimensional standard Brownian motion on a filtered probability spacesatisfying the usual condition.

Since the biodiversity conservation benefitBdepends on the area of conservation land,we would rather adopt the biodiversity conservation benefit per unitYas the state variable.Applying Itˆo’s formula,we deduce thatYis governed by the SDE(1.2),which is clearly positive.

Since the conversion is irreversible,and its rate has a maximum value,we letand the admissible strategy set takes the form of(1.3).

The decision-maker aims to find an admissible strategyv∗to maximize the objective functional defined as(1.4),in which the constantsφ,γ,kandρsatisfyingφ >0,γ∈(0,1),k≥0 andρ > α,represent the return per unit area,the elasticity coefficient of the utility function,the marginal cost of conversion and the discount rate,respectively.For more details,we refer to[10].In(1.4),the first term in the braces represents the flow benefit from the land for production,the second term represents the total value of biodiversity from the land for conservation,and the last term means the total cost of land conversion.

2.2 Associated HJB equation

In summary,the model can be formulated as a stochastic control problem.The decisionmaker aims to findv∗from the admissible strategy set(1.3)to maximize the objective functional(1.4)subject to(1.1)-(1.2).

From the stochastic control theory in[8,13],we know that the value functionFis a viscosity solution of the HJB equation(1.6).

3 Boundary Value Condition

In this section,we derive the boundary value condition for the HJB equation(1.6)via the original stochastic control model and the theory on PDE.

Since the HJB equation(1.6)is degenerate aty=0 and satisfies Fichera condition in[6],the boundary value ofFaty=0 should be determined by itself.Specifically,we should first deduce the ODE forF(0,r)through takingy=0 in the HJB equation(1.6),and obtain the value ofF(0,r)by means of solving the ODE.SinceFhas a linear growth with respect toy,which will be proven in Theorem 3.1,the standard theory for Cauchy problem(refer to[11-12])implies that the boundary value ofF(y,r)aty→+∞is also unnecessary.Note that the derivative ofFwith respect torand the second-order derivative ofFwith respect toyin the HJB equation(1.6)are non-positive and non-negative,respectively.So,the HJB equation(1.6)is a forward parabolic PDE,and the boundary value atr=1 is not needed.Hence,it is sufficient to impose the boundary value ofFatr=0.

However,it is not trivial to obtain the boundary value ofFatr=0 since the SDE ofYis meaningless whenr=0.The key idea to overcome the difficulty is to consider the limit ofF(y,r)asr→0+.In fact,we should find a continuous solution of the HJB equation(1.6).So,F(y,0)is just the limit ofF(y,r)asr→0+,and it is sufficient to find the limit.

with the admissible strategy set

4 Existence and Uniqueness of the Viscosity Solution

In the following,we prove the uniqueness of the viscosity solution.Otherwise,there exist at least two viscosity solutionsF1andF2.Consider the approximation stochastic control problem(4.1),and denoteFnbywith the boundary valueFi,i=1,2,respectively.It is clear that for any fixed(y,r)∈An,

So,we deduce that

5 Numerical Method and Results

In this section,we apply the numerical method for HJB equations in[8-9]to PDE(1.6),and obtain some numerical illustrations of our model.

In order to finish the computation in finite steps,we restrict the HJB equation(1.6)in a bounded domain rather than the unbounded domainy >0,0

where

and

Through the above difference equations and the standard iteration method,we obtain some numerical illustrations.Unless otherwise stated,values of parameters used in this section are presented in Table 1,which are the same as in[10].

Table 1 The parameters.

Figure 1 shows the optimal conversion boundaryr=R(y)(the blue curve),under which is the conservation zone,and above which is the conversion zone.If(y,r)lies in the conservation zone,then the optimal strategy is no conversion,i.e.,v∗=0,keeping the reserve land for biodiversity conservation.In the case of(y,r)belonging to the conversion zone,the optimal strategy is to convert the land for conservation into that for production at the rate ofuntil the land area for biodieversity conservationrfirst hits the optimal conversion boundary.Moreover,we find that the optimal conversion boundary is increasing with respect to biodiversity conservation benefit per unity.It means that the optimal land area for conservation at highyshould be more than that at lowery,whereas,the optimal land area for agricultural production at highyshould be less than that at lowery.It is clear that the characteristics are consistent with reality.

Figure 1 The optimal conversion boundary.

Figure 2 plots land valueFas a function of biodiversity value per unit landyand reserve landr.From this figure,we find that land valueFis increasing with respect toy,but it does not have the same monotonicity with respect tor,which is clearly illustrated in Table 2.In Table 2,the bold numbers represent the maximum value ofFwith respect torfor somey.Table 2 shows thatFfirst increases and then decreases with respect tor,and the maximum value ofFachieves at somer∗∈(0,1)rather thanr∗=0.The results are reasonable and coincide with our intuition.

Figure 2 Land values F,as a function of reserve land r and biodiversity value per unit land y.

Table 2 Land values F(in billions of dollars)(in body of table),in terms of biodiversity value per unit land y(in millions of dollars)and reserve land r.

Appendix Some Results for Model L

We derive the following inequality through takingr→0+in(5.8),

Figure 3 Land values F,as a function of reserve land r and biodiversity value b.

Table 3 Land values (in billions of dollars)(in body of table),in terms of biodiversity value b(in millions of dollars)and reserve land r.

Table 3 Land values (in billions of dollars)(in body of table),in terms of biodiversity value b(in millions of dollars)and reserve land r.

br 10.90.80.70.60.50.40.30.20.10 25 4.19 4.38 4.48 4.56 4.63 4.69 4.76 4.82 4.87 4.914.95 13 3.64 3.83 3.93 4.00 4.06 4.11 4.16 4.21 4.26 4.31 4.35 12 3.60 3.79 3.89 3.96 4.02 4.07 4.12 4.16 4.21 4.26 4.30 10 3.52 3.70 3.80 3.87 3.93 3.98 4.03 4.07 4.12 4.164.21 9 3.46 3.65 3.75 3.82 3.88 3.93 3.97 4.01 4.06 4.104.15 7 3.38 3.57 3.67 3.74 3.80 3.85 3.89 3.92 3.96 4.014.05 6 3.34 3.53 3.63 3.70 3.76 3.81 3.85 3.88 3.92 3.964.00 5 3.30 3.49 3.59 3.66 3.72 3.77 3.81 3.84 3.87 3.91 3.95 4 3.26 3.45 3.55 3.62 3.68 3.73 3.77 3.80 3.83 3.863.91

AcknowledgementWe thank Prof.Ying Hu and Prof.Yan Zeng for their valuable comments and helpful suggestions,which helped to significantly improve the presentation of the paper.