Abstract

1. Introduction
We have been facing with the problem of global warming. In order to deal with the threat to the environment, the government should adopt environmental policies to reduce emissions of SO 2 or CO 2 and not to increase greenhouse gas concentrations. In this paper, we consider a problem in environmental policy design within the framework of optimal timing problems. The purpose of this paper is to derive optimal timings at which the government should adopt environmental policies to deal with increases in greenhouse gas concentrations and to reduce emissions of SO 2 or CO 2 under the continuous-time Knightian uncertainty. Furthermore, we analyze the effects of increases in Knightian uncertainty on optimal environmental policies and the reservation value. Usually, such a problem is analyzed based on the cost-benefit analysis. However, this standard approach does not consider three significant characteristics of environmental problems, that is, uncertainty, irreversibility, and the flexibility in deciding the timing of adopting envi- ronmental policies. We briefly explain these notions before we go into details. First, there exist economic uncertainty over future costs and benefits of adopting environmental policies and eco- logical uncertainty over the evolution of ecological systems.  We do not know exactly the effect of adopting environmental policies nor know the economic damages caused by increases in average temperature. Second, there exist, at least two irreversibilities to be considered in environmental problems. The one is the irreversibility with respect to environmental damage. For instance, emissions of CO 2 will increase greenhouse gas concentrations, which is considered to lead to grobal warming and damage the ecosystems. The other is the irreversibility with respect to economic damage. For example, the installation of "scrubbers" by truck companies will be sunk costs on society. Third, the adoption of environmental policies is not now-or-never decisions. That is, the government has the option to postpone the adoption of policies and can wait for the arrival of new information, for example, some data about global warming, or some innovation about scrubbers, which will enable us to put off adopting environmental policies and to avoid imposing sunk costs on society. As we have already mentioned, we have to consider two kinds of uncertainties that play important parts in environmental policy design. The one is economic uncertainty , that is, the uncertainty over future costs and benefits of adopting environmental policies to reduce emissions of SO 2 or CO 2 . When it comes to global warming, we would not grasp the resulting cost to society, or we would not exactly predict how the increase in temperature would affect agricultural outputs, or ecological systems, even if how large average temperature is expected to rise. The other is ecological uncertainty , that is, the uncertainty over the evolution of ecological systems. We could not exactly predict how further increase in greenhouse gas concentrations would affect average temperature in the future, or how much today's level of greenhouse gas concentrations could be reduced by government's policies. We also have to take care of at least two different kinds of irreversibilities that have the opposite characteristics. First, environmental policies that aim at reducing environmental damage impose sunk cost on society. For example, consider the situations in which companies owning coal-burning utilities might be forced to install scrubbers, or the situations in which they might have to scrap existing facilities and invest in more efficient machines. These sunk costs generate an opportunity cost of adopting policies immediately, rather than waiting for the arrival of new information about environmental damage and their economic concequences, for example, the information about innovations of new technologies in the near future that might enable us to remove sulfur more cheaply and efficiently. Traditional cost and benefit analysis based on the net present value approach ignores this opportunity cost, which should be considered carefully. The second irreversibility is the one with respect to environmental damage. For example, environmental pollutants are not easily removable from the atmosphere; even if the government adopts severe policies to reduce greenhouse gas emissions, it would take many years to reduce greenhouse gas concentrations in the atmosphere. The damage to environmental systems caused by higher global warming cannot be easily reversible. Contrary to the first irreversibility, this implies that adopting environmental policies right now rather than waiting has sunk benefit , a negative opportunity cost. Again, the standard cost-benefit analysis based on the net present value approach does not consider opportunity benefit. In the literature of decision making under non-deterministic situation, decision maker's beliefs are captured by a single probability measure over the state of the world, which is cate- gorized as risk. Recently, new approaches have been proposed and axiomatized by Gilboa and Schmeidler (1989) in which decision maker's beliefs are captured by not a single but multiple probability measures over the state of the world, which is categorized as Knightian uncertainty or uncertainty in the literature.1 This paper adapts a model of the continuous-time Knightian uncertainty proposed by Chen and Epstein (2002), which is a counterpart of the static frame- work developed by Gilboa and Schmeidler (1989). We show that while an increase in risk does not affect the value of adopting environmental policies, an increase in Knightian uncertainty in- duces a decrease in the value of adopting environmental policies. This result implies that if the government is uncertainty averse, then she underestimates the value of adopting environmental policies since she makes decision on the worst case scenario. Our paper is related to Pindyck (2000, 2002). He shows that an increase in risk induces an increase in the reservation value above which environmental policies are immediately adopted and below which environmental policies are never adopted. This paper shows that an increase in Knightian uncertainty induces a decrease in the reservation value under some condition, which is a stark contrast to Pindyck (2000, 2002). The organization of this paper is as follows. Section 2 provides continuous-time models under risk and Knightian uncertainty. In order to analyze the value of adopting environmental policies under the continuous-time Knightian uncertainty in the later sections, mathematical def- initions and results are provided. Section 3 derives the value of adopting environmental policies under the continuous-time Knightian uncertainty. Section 4 provides the further characteriza- tion of the value of adopting policies, and derives the value of optimal environmental policies under the continuous-time Knightian uncertainty.  Section 5 provides sensitivity analyses, which are main results of this paper.
2. The Value of Environmental Policies under the Continuous-Time Knightian
Uncertainty
In this section, we provide continuous-time models under risk and Knightian uncertainty. At first, definitions and mathematical notions are in order.
2.1 A General Continuous-Time Model under Risk
Let (Ω, F T , P ) be a probability space, and let ( B t )o< t<T be a standard Brownian motion with ¹For example, see Knight (1921), Gilboa (1987), Schmeidler (1989) or Gilboa and Schmeidler (1989). respect to P. 2  We consider the standard filtration ( Ft )0<t<T for a standard Brownian motion ( Bt )o<t<T.3  Let ( Mt )o<t<T be a stochastic process of the stock of an environmental pollu- tant (for example, CO 2 concentrations in the atmosphere) and let E t be the evolution of the rate of emission of the pollutant at time t. We assume that the evolution of the stock of the environmental pollutant ( Mt )o≤t≤T follows a controlled arithmetic Brownian motion,
where β ∈ (0, 1] denotes the absorption rate of the environmental pollutant, δ ∈ [0, 1] denotes the natural decay rate of the stock of the environmental pollutant over time, and σ M is a constant real number. In other words, a fraction β of the emission E t goes into the atmosphere, and a fraction δ of the environmental pollutant M t on the atmosphere diffuses into the ocean, and the forests. In this paper, we ignore the stochastic fluctuation of M t for simplicity. Thus, equation (1) reduces to the following,
We assume that E t stays at the constant initial level E 0 , until an environmental policy is adopted. We assume that economic uncertainty follows a geometric Brownian motion,
where α and σ are constant real numbers. This stochastic process ( X t )o≤t≤T is assumed to capture economic uncertainty over the future costs and benefits of policy adoptions. Changes in X t over time might reflect changes in technologies. For instance, if M is SO 2 concentrations, then changes in X t might reflect the innovation of technologies that would drastically reduce the social cost of M , or population increase that would raise the social cost. Without loss of generality, it is assumed that σ > 0. We assume that the flow of social cost associated with the stock variable M t, C(M t, X t) is linear in Mt,4 that is
The value at t of adopting environmental policies with T an expiration time is
where r > 0 is the discount rate, and E P [ - | F t ] is the expectation with respect to P conditioned on Ft. The government's problem is formulated to decide the time when she will incur the social cost of adopting policies. The optimal time can be considered to be the solution to the optimal stopping problem of finding an ( F t )-stopping time, t ′ ∈ [0 , T ] that maximizes the value of adopting policies at period 0
It is also assumed that r > α.5 The value at t of optimal environmental policies V t , is defined by
We can show that 6
The value of adopting policies W(Xt, Mt, t) equals
as T goes to ∞. Since this model is stationary, the value of optimal environmental policies V t is shown to depend only on X t and M t , not to depend directly on t , and satisfies the following Hamilton-Jacobi-Bellman equation,
It can be shown that the optimal strategy is to stop and adopt the environmental policy right now if X t ≥ X* and to continue if X t < X*, where X* is the reservation value. From the Hamilton-Jacobi-Bellman equation above, it follows that in the continuation region,
Note that the left-hand side of this equation is the social cost associated with the stock of the environmental pollutant plus the government's expected gain of having the rights to implement environmental policies, and the right-hand side is the opportunity cost measured in terms of government's discount rate.  By applying Ito's lemma to V t and some calculations, it follows that
Combining two equations implies that
Thus, in the continuation region,
We solve this differential equation under three boundary conditions,
6
where X* is the critical value of X at or above which environmental policies should be adopted. Condition (13) reflects the fact that if X t is always zero, then the flow of the social cost associated with the stock variable C ( X t , M t ) is zero. Thus the value of optimal policies will remain to be zero. Condition (14) that follows from (9) is called the value matching condition ; when X t = X* and the government exercises her option to adopt policies, it incurs a sunk cost K and obtains the net payoff. Condition (15) is called the smooth pasting condition ; if adopting policies at X* is critical, then the derivative of the value function must be continuous.7 We guess the solution to this equation as follows:
where A, B and D are some constants. Then
The value A remains to be determined. By the boundary conditions, the negative part of the solution to (1/2)σ²γ(γ − 1) + αγ − r = 0 is ruled out. 8  Note that
Then
Thus
term is the value of the option to adopt environmental policies. The second term is the present value of the flow of social cost from the current stock of the pollutant. The third is the present value of the flow of social cost from the emission E 0 . The value function in the stopping region consists of two terms: the value of adopting environmental policies defined by (8) plus the direct cost resulting from adopting environmental policies.
2.2 Continuous-Time Knightian Uncertainty
In this subsection, we provide a model of continuous-time Knightian uncertainty, which is first proposed by Chen and Epstein (2002). 2.2.1 Densitiy Generators, Girsanov's Theorem Let L be the set of real-valued, measurable, 9 and ( F t )-adapted¹⁰ stochastic process on (Ω, F T , P ) with an index set [0,T] and let L 2 be a subset of L that is defined by
Given θ = (θ t ) ∈ L 2 , define a stochastic process ( z t )o≤t≤τ by11
A stochastic process (θ t ) ∈ L 2 is called a density generator if ( z f ) is ( F t )-martingale. Novikov's condition is one of the sufficient conditions for ( z ) to be ( F t )-martingale: 12
Let θ be a density generator and define Q θ by
Since ( z ℓ ) is ( F i )-martingale, Q θ (Ω) = E P ( z Γ ) = z δ = 1. Thus, Q θ is a probability measure, 13 and it is absolutely continuous with respect to P . 14 Furthermore, since zf is strictly positive, P is also absolutely continuous with respect to Q e . Thus, Q e is equivalent to P .15 Conversely, any probability measures equivalent to P can be obained by a density generator in this way.16 Let Θ be a set of density generators. For such a set Θ, define the set of probability measures, PO on (Ω, FT), generated by Θ, by
where Q b is derived from P according to (18). In this paper, decision maker's beliefs are captured by not a single probability measure, but the set of probability measures equivalent to a probability measure P . For any θ ∈ Θ, a stochastic process ( B f )o<t<T defined by17
is a standard Brownian motion with respect to ( F t ) on (Ω, F T , Ω β ).  This follows from Girsanov's Theorem.18 2.2.2 A Set of Stochastic Differential Equations By Girsanov's theorem, the stochastic differential equation capturing economic uncertainty turns out to be
for any θ ∈ Θ. Note that the change of measure formula by way of Girsanov's Theorem does not affect the volatility term. By (20), and by applying Ito's lemma to ln X t by considering Q e as the true probability measure,
2.2.3 Rectangularity, Strong Rectangularity, i.i.d. Uncertainty, and κ-ignorance Now we define three classes of density generators for the later analyses in this paper. A set of density generators, Θ κ τ (ω) , is rectangular if there exists a set-valued stochastic process ( K t )o<t<T such that
and, there exists a compact subset K of R such that for each t, K t : Ω → K is compact-valued and convex-valued, the correspondence ( t,ω ) → K ℓ (ω), when restricted to [0, s] × Ω, is B ([0, s]) S F s - measurable for any 0 < s ≤ T,19 and 0 ∈ K t (ω) (m ⊗ P)-a.s., where m is the Lebesgue measure restricted on B ([0, T]). A set of density generators Θ K t is strongly rectangular if there exist a nonempty compact K such that 0 ∈ K t and
where m is the Lebesgue measure restricted on B ([0, T]). Any element in Θ K t satisfies Novikov's condition, which follows since K t is a subset of a compact subset K of R for all t. Note that the set K t is independent of a state ω , that is, non-stochastic, contrary to the set K t ( ω ) in (22). Next, we consider a special case of strongly rectangular sets in which K t is independent of time t. The uncertainty characterized by Θ K is i.i.d. uncertainty if there exists a compact subset K of R such that 0 ∈ K and
Note that the set OK is independent of the state and time. Finally, we consider a special case of the i.i.d.  uncertainty Θ κ , where the set K is specified as
where κ > 0. This type of uncertainty is called the κ -ignorance. The real number κ is considered to be a degree of Knightian uncertainty because the larger κ is, the larger the set of probability measures is. In order to prove Proposition 1, we adapt the theory of support functions along with Chen and Epstein (2002). Define
The strict positivity of σ for the definition of the support function is assumed in order to define θ ξ and (θ t )* below. Since K t (ω) is convex-valued, we can adapt the theory of support functions. See Rockafeller (1970) in details. In this paper, further restriction is imposed on the support function.  Define
Recall that K t is convex-valued. For this support function, define
The equality holds since σ > 0 and K t is compact-valued. Note that ( θ τ ) is a degenetate measurable process, and (θ ξ ) ∈ L. Thus (θ ξ ) ∈ Θ K t . Furthermore, if the κ -ignorance is assumed, then θ t = κ since θ t = max[ − κ, κ ]. In this case, the support function turns out to be
where K = [− κ, κ ]. Note that θ ∗ and e(σ) are independent of the state and time in case of the κ -ignorance. For the analyses in the later section, we define (θ t ) ∗ by
This is a counterpart of Equation (25).  Note that (θ t )* = − κ under the κ -ignorance since (θ t )* = min[−κ, κ].
3. The Value of Adopting Policies under Strong Rectangularity
In this section, we derive the value of adopting environmental policies W t by assuming the strong rectangularity. We assume that the government is uncertainty averse. In other words, her beliefs are captured by the set of probability measures P θ , (19), and she maximizes the infimum of expected returns over Pθ . Furthermore, we impose the strong rectangularity on Θ , whi implies that Θ is equal to Θ K ؛ . Thus, the value at t of adopting environmental policies with T an expiration time is
where C ( X t , M t ) is defined by (4) Proposition 1. Suppose that the government is uncertainty averse, and her beliefs are charac- terized by Θ K t , where Θ K t is a strongly rectangular set of density generators defined by (23) for some (K t ). Then the value of adopting environmental policies is provided by
12 where (θt)* is defined by
Proof. See Appendix.口
4. The Optimal Environmental Policy under Strong Rectangularity
In this section, we derive the values of adopting environmental policies and optimal environmen- tal policies under the i.i.d. uncertainty and the infinite time horizon in addition to the strong rectangularity.
4.1 The Value of Optimal Environmental Policies under the Strong Rectangularity
The optimal time is the solution to the optimal stopping problem of finding an ( F t )-stopping time, t' ∈ [0, T ] that maximizes the value of adopting policies at period 0 Thus, the value at t of optimal environmental policies V t , is defined by
where C(Xt, Mt) is defined by (4). Appendix shows that V t is a solution to the following Hamilton-Jacobi-Bellman equation,
where W t is defined below by (32). Further assumptions enable us to solve this type of the We discuss this Hamilton-Jacobi-Bellman equation, otherwise difficult to solve analytically. topic in the next two subsections.
4.2 The Value of Adopting Policies under the i.i.d. Uncertainty and the
Infinite-Time Horizon
In this subsection, we derive the value of adopting policies by assuming the i.i.d. uncertainty, the infinite-time horizon and no-existence of expiration date. Under the assumption of the i.i.d. uncertainty, the support function (24) reduces to
and
Furthermore, θ ∗ = argmin { σx | x ∈ K } = min K . Note that θ ∗ and θ ∗ are independent of time and the state. Under the i.i.d. uncertainty, (26) reduces to
where λ ≡ r + δ − α + σθ* Thus, by assuming that there exists no expiration date, together with the i.i.d. un- certainty and the infinite-time horizon, the value of adopting environmental policies reduces to
as T goes to ∞ , since r > α , σ > 0, δ > 0, and 0 ∈ K , which implies the positivity of λ .
4.3 The Value of Optimal Environmental Policies under i.i.d. uncertainty and
Infinite Horizon
In this subsection, we derive the value of optimal environmental policies V t under the i.i.d. uncertainty and the infinite-time horizon. In order to solve the HJB equation (28) analytically, we assume that the continuous-time Knightian uncertainty is independent of time, in other words, it is characterized by the i.i.d. uncertainty, the planning horizon is infinite, and there exists no expiration date,22 which implies that V t depends on X t and M t , not on time t directly. Thus we can write V t = V(X t , M t ) for some function V : R × R + → R . Then the Hamilton- Jacobi-Bellman equation (28) turns out to be
where V : R ± × R ± → R. From the Hamilton-Jacobi-Bellman equation (33), it follows that
in the continuation region. The left-hand side of this equation is the social cost assosiated with the stocks of environmental pollutants plus the government's expected minimum gain of having the rights to carry out environmental policies, and the right-hand side is the opportunity cost measured in terms of government's discount rate. If P θ is singleton, then this equation reduces to (10). In the continuation region, it is shown that23
Thus in the continuation region, it follows that
We solve this differential equation with the following boundary conditions,
where X* is the critical value of X at or above which environmental policies should be adopted. The three conditions have been already explained within the framework of risk. Condition (34) reflects the fact that if X is always zero, then the flow of the social cost associated with the stock variable C ( X t , M t ) is zero. Thus the value of optimal policies will remain to be zero. Condition (35) is the value matching condition, and condition (36) is the smooth pasting condition. By solving the differential equation with the three boundary conditions, we obtain the following optimal strategy,
It can be shown that γ > 1 (see Appendix). The value function in the continuation region consists of three components. The first term is the value of the option to adopt the environmental policy. The second term is the present value of the flow of social cost from the current stock of pollutants. The third is the present value of the flow of social cost from the emission E 0 . The value function in the stopping region consists of two terms: the value of adopting environmental policies defined by (32) plus the direct cost resulting from adopting the environmental policy.
5. Sensitivity Analyses
In this section, we analyze the effects of increases in risk and Knightian uncertainty on the value of adopting environmental policies and the value of optimal environmental policies. We also analyze the effects of increases in risk and Knightian uncertainty on the optimal timing of adopting environmental policies.
5.1 An Increase in Risk (the case of no-Knightian uncertainty)
In this subsection, we consider the case of no-Knightian uncertainty. We show that an increase in risk does not affect the value of adopting policies W t , and an increase in risk does induce increases in the value of optimal environmental policies V t in the stopping region and the reservation value X*. Proposition 2.  In the case of no Knightian uncertainty, an increase in risk induces no change in the value of adopting policies W t , an increase in risk induces an increase in the value of optimal environmental policies V t in the stopping region, and an increase in risk induces an increase in the reservation value X*. Proof. See Appendix.
The first claim in this proposition states that an increase in risk does not have any effect on the value of adopting environmental policies. The third claim implies that the more risk there exists over the future social cost of pollutants, the greater is the incentive to wait rather than to adopt environmental policies immediately. Note that while the second and third claims are presented in Pindyck (2000), the first claim is not.
5.2 An Increase in Knightian uncertainty (the case of κ ignorance)
In this subsection, we analyze the effects of Knightian uncertainty on the value of adopt- ing policies and the reservation value.  We show that an increase in Knightian uncertainty induces a decrease in the value of adopting policies, and induces a decrease in the reservation value under some condition. Recall that in the case of no-Knightian uncertainty, an increase in risk does not affect the value of adopting policies W t , and an increase in risk does induce an increase in the reservation value X*. Finally, we provide the following proposition. Proposition 3.  We assume the same conditions in Proposition 2, and assume the κ-ignorance. Then, an increase in Knightian uncertainty induces a decrease in the value of adopting policies W t , and an increase in Knightian uncertainty induces a decrease in the reservation value X* for parameters that make the absolute value of ∂γ/∂κ sufficiently small. Proof. See Appendix.
The first claim in this proposition states that an increase in Knightian uncertainty has the negative effect on the value of adopting environmental policies.  This result implies that if the government has imprecise knowledge about the state of the world, then she underestimates the value of adopting environmental policies since she makes decisions on the worst case scenario. This is a stark contrast to the first claim in Proposition 2. Contrary to the first claim in this proposition, which holds without any restrictions, the second claim in this proposition holds for sets of parameters that make the absolute value of ∂γ/∂κ sufficiently small.
Appenix 1: Derivations of Mathematical Results
Derivation of (8).
At first, we derive the value at t of adopting policies as follows:
as T → ∞, where the second equality follows from Fubini's Theorem and solving the stochastic differential equation dX t = α X t dt + σ X t dB t ,24 and the fifth equality follows from solving the ordinal differential equation dMt/dt = βE 0 − σ Mt. 25  By letting E 0 = 0, we obtain the value of adopting policies W ( X t , M t ) = − X t M t / ( r + δ − α). Thus, the stationarity is obtained. □
Derivation of HJB in case of risk
The Hamilton-Jacobi-Bellman equation follows since
where the first equality follows from the definition of V t , the third equality follows from the definition of W t , the fourth holds by the law of iterated integrals, the fifth follows from the definition of V t , the sixth holds by approximating e-rdt by (1 - rdt ), and the last equality holds by eliminating higher order terms than dt. □
Proof of Proposition 1
In order to prove Proposition 1, we have to show the next lemma. Lemma 1. For any s ≥ t and for any θ ∈ Θ K t
where θ ∗ ≡ argmin {σx|x ∈ K} = min K . Proof. Let s ≥ t and let θ ∈ Θ K t . Then
Thus
where the inequality follows from the monotonicity of conditional expectation. 26口Now we are in a position to prove Proposition 1. Proof of Proposition 1.
where the second equality holds by (19), the third equality holds by Fubini's theorem for con- ditional expectation, 27 the fourth equality holds by (21), the sixth equality holds by Lemma 1, the seventh equality follows from the fact that (θ ∗ ) is a degenerate stochastic process, the eighth equality holds by the fact that ( B f *) is a Brownian motion with respect to Q θ +, and the tenth equality follows since M s = −μ + ( M t + μ )e−δ(s−t) for all s ≥ t, where μ = −(β/δ)E 0 . Thus the口proof is completed.
Derivation of HJB in case of Knightian Uncertainty

where the first equality follows from the definition of V t , the third equality follows from (32), the fourth follows from the definition of Pθ , the fifth holds by the law of iterated integrals, the sixth follows from the strong rectangularity (41), the eighth follows from the definition of V t , the ninth holds by approximating e-rdt by (1 – rdt ), and the last equality holds by eliminating higher order terms than dt. □
Derivation of V t under Knightian uncertainty
By Ito's lemma,
Thus we can show that
where the first equality follows from the assumption of the i.i.d. uncertainty, the fourth equality holds by the negativity of ∂V t /∂X t in the continuation region, and the last equality follows from the definition of θ ⋆ . Therefore, in the continuation region, it follows that
with the following boundary conditions:
We guess the solution to this equation as follows:
where A, B and D are some constants. Then
Thus,
The value A remains to be determined. By the boundary conditions, the negative part of the solution to (1/2)σ²γ(γ − 1) + (α − σθ ∗ )γ − r = 0 is ruled out. Note that
where the inequality holds by the assumption that r > α + σκ , and the last equality holds since α, σ > 0 and θ ∗ = −κ < 0, which implies that |α − σθ ∗ + σ 2 /2| = α − σθ ∗ + σ 2 /2. Then
Thus
Therefore, the derivation is completed. □ Proof of ∂γ/∂σ² < 0 and ∂γ/∂κ < 0. Let
Note that Q 1 (1) = α − r < 0, Q 1 (0) = − r < 0, and γ 1 > 1 by (16), where γ 1 is the positive part of this quadratic equation. By differentiating this quadratic equation totally with respect to σ 2 , it follows that
∂Q1/∂σ² = (1/2)γ(γ − 1) > 0 and ∂Q1/∂γ1 > 0 at γ1 > 1 imply
Note that Q2(1) = α − r + σκ < 0,28 Q2(0) = − r < 0, and γ1 > 1 by (38), where γ1 is the positive part of this quadratic equation. By differentiating this quardratic equation totally with respect to κ , it follows that
∂Q₂/∂κ = σγ > 0 at γ₁ > 1 and ∂Q₂/∂γ₁ > 0 at γ₁ imply
Thus, the proof is completed. □
Proof of the negativity of ∂Vt/∂Xt
In order to show that ∂Vi/∂Xt < 0 in the continuation region, it suffices to show that in the continuation region,
where A is defined by
The inequality is proved as follows:
for any X t < X *, γ > 1, δ > 0, and r > α + σκ. Furthermore,
for any γ > 1, δ > 0, and r > α + σκ. Thus, the proof is completed. □
Proof of Proposition 2
By (8), it follows that ∂W t / ∂σ² = 0. Define V t ' as follows:
By taking the logarithm of both sides of the equation and differentiating both sides of the equation with respect to γ, it follows that
Recall that ∂γ/∂σ² < 0 by (39). Thus
Moreover, it follows that
Thus
which completes the proof. □
Proof of Proposition 3
It follows from (32) that ∂Wt/∂κ < 0. By differentiating X* with respect to κ , it follows that ∂X* дк
Thus, ∂X*/∂κ < 0 if
Note that the negativity of ∂γ/∂κ follows from (40). □
Appendix 2
Let 0 ≤ s ≤ t ≤ T, let x be an F T -measurable function, and let Θ be rectangular. Then,
where the first equality follows from the law of iterated integral. The second equality follows from Lemma 4 below. See Nishimura and Ozaki (2003) in details. Let θ be a density generator, let ( z e ) be defined by
and define the measure Q f by
Lemma 2. Qf is a probability measure satisfying
where Q  is defined by
Lemma 3.  Let 0 ≤ s ≤ t ≤ T and let x be an F t -measurable function.  Then E 2 ′ [x| F s] depends only on (θu)s<u<t· These lemmas together with the assumption that Θ is strongly rectangular prove the following lemma. Lemma 4. Let 0 ≤ s ≤ t ≤ T and let x be an Ft-measurable function. Also assume that Θ is strongly rectangular. Then under the assumption that the minima exist,