A representative agent model based on risk-neutral prices
aa r X i v : . [ q -f i n . M F ] J a n A representative agent model based on risk-neutral prices
Hyungbin Park ∗ Department of Mathematical Sciences,Seoul National University,1 Gwanak-ro, Gwanak-gu, Seoul, Republic of Korea
January 30, 2018
Abstract
In this paper, we determine a representative agent model based on risk-neutral informa-tion. The main idea is that the pricing kernel is transition independent, which is supportedby the well-known capital asset pricing theory. Determining the representative agent model isclosely related to the eigenpair problem of a second-order differential operator. The purposeof this paper is to find all such eigenpairs which are financially or economically meaningful.We provide a necessary and sufficient condition for the existence of such pairs, and provethat that all the possible eignepairs can be expressed by a one-parameter family. Finally, wefind a representative agent model derived from the eigenpairs.
A representative agent model is an important concept in finance, and many authors used autility function of the representative agent to solve many finance problems. However, thechoice of the utility function has been a result of coincidence rather than of consequence.Many authors have used the power functions, logarithm functions, exponential functions andtheir linear combinations as a prototype of utility function. The reason why they choose oneof these is the analytic tractability rather than inevitable economic foundation. It would begreat if one can find an inevitable reason why we should choose a specific function, and thispaper partially answers this.Our story begins with the pricing kernel. In finance, many authors made efforts toilluminate the relation between the return and risk of assets. This relation is reflected in apricing kernel, which is determined by the interaction of risk preference of market agents.The representative agent theorem says that the interaction can be understood by the utilityfunction of the market representative agent. The main purpose is to express the utilityfunction of the representative agent. This paper shares the same idea with the Ross recoverytheorem (Carr and Yu (2012), Park (2016), Qin and Linetsky (2014), Ross (2015), Walden(2017)). They utilize the risk-neutral information which can be obtained from derivative ∗ [email protected], [email protected] rices to find the pricing kernels. This result is attractive because they overturned thecommon belief that the objective measure cannot be inferred from the derivative prices.Later, we will conclude that the utility function can be determined from the risk-neutralinformation as a one-parameter family.This paper gives an interesting result. Later, in Section 5.1, we will see that if the stockprice follows the standard Black-Scholes model, then the utility function is determined asthe power function. It is an amazing result that the Black-Scholes model induces the powerutility function because two theories have been developed independently.The macroeconomic foundation of determining utility functions relies on the continuous-time consumption-based capital asset pricing model (CAPM). The standard argument of theCAPM says that the reciprocal of the pricing kernel can be expressed as L t = e βt U ′ ( X ) U ′ ( X t ) , t ≥ U ( · ) is the representative agent, β > X t ) t ≥ isthe aggregate consumption (or income) process. The reader can find out that this expressionis a continuous-time version of Eq.(9) in Ross (2015). By defining φ ( x ) := U ′ ( X ) U ′ ( x ) , (1.1)we obtain the transition independent form L t = e βt φ ( X t ) , t ≥ φ are inherited from the utility function U in the CAPM.The usual conditions on utility functions are stated in Eq.( ?? ). Therefore, from the relationin Eq.(1.1), the function φ satisfies: (i) φ > , (ii) φ ′ > , (iii) lim x → φ ( x ) = 0 , (iv)lim x →∞ φ ( x ) = ∞ . These four conditions on φ will be also called by the usual conditions .This issue will be discussed in Section 3.5.One of the most important observations is that the problem of finding the pair ( β, φ ) inthe transition independent form is transformed into an eigenvalue/eigenfunction problem ofan operator. Motivated by the paper of Carr and Yu (2012), we will see that ( β, φ ) satisfiesa second-order differential equation12 σ ( x ) φ ′′ ( x ) + k ( x ) φ ′ ( x ) − r ( x ) φ ( x ) = − β φ ( x ) . (1.2)This differential equation is a continuous version of the Perron-Frobenius type equation givenat Eq.(15) in Ross (2015).One of the main purposes of the present paper is to find the solution pairs of Eq.(1.2)with φ satisfying the usual conditions. The set of all such pairs is called the usual set andis denoted by U . The usual set is closely related to the behavior of the underlying process X, especially the Feller boundary classification of X. In Section 3.5, the usual set U will becharacterized in the terms of the boundary classifications of X. There are two main contributions in the paper. First, we provide a necessary and sufficientcondition for the existence of such pairs in the terms of the boundary classifications of X. Moreover, we will offer a representation of all such pairs ( β, φ ) . It will be proven that thepossible set can be expressed by a one-parameter family with upper bounded parameter.Second, our result gives an interesting implication on calibrating the objective dynamics. Bynow, we have used non-parametric or heuristic parametric models to calibrate the dynamics f X t under objective measure. The results of this paper tells us that one can use parametricmodels, which are economically derived, to calibrate the objective dynamics.The rest of this paper is structured as follows. Section 2 describes the underlying marketmodel of this paper and define objective measures. Section 3 gives the main result of thispaper. We transform the problem of determining the utility functions as eigenpair problems ofa second-order differential operator. And then, financially meaningful eigenpairs are chosen.Section 4 applies the main result to find a representative agent model. Section 5 presentseveral examples and the last section summarizes this paper. All proofs are in appendices. A risk-neutral financial market is defined as a probability space (Ω , ( F t ) t ≥ , Q ) having aone-dimensional Brownian motion W = ( W t ) t ≥ with the usual completed filtration ( F t ) t ≥ generated by W. All the processes in this article are assumed to be adapted to this filtration( F t ) t ≥ . We fix a positive process G = ( G t ) t ≥ , which is called a numeraire . A process S issaid to be an asset if the discounted process S/G is a martingale under Q . It is customarythat this measure Q is referred to as a risk-neutral measure when G is a money marketaccount. Assumption 1.
Let b and σ be two continuously differentiable functions on (0 , ∞ ) . Assumethat the stochastic differential equation (SDE) dX t = b ( X t ) dt + σ ( X t ) dW t , X = ξ has a unique strong solution and that the solution is non-explosive on (0 , ∞ ) . This process X is referred to as a state variable in the market. It is well-known that the process X is a univariate time-homogeneous Markov diffusion. Thenon-explosiveness means that both boundaries 0 and ∞ are inaccessible (Definition 3.3).This boundary condition is plausible because the aggregate consumption in the CAPM doesnot reach 0 and ∞ in finite time. Assumption 2.
The state variable X determines the dynamics of the numeraire G. Moreprecisely, assume that there are continuously differentiable functions r and v on (0 , ∞ ) suchthat G t = e R t ( r ( X s )+ v ( X s )) ds + R t v ( X s ) dW s . In the SDE form, the process G follows dG t G t = ( r ( X t ) + v ( X t )) dt + v ( X t ) dW t , G = 1 . (2.1) Assume that a local martingale ( e − R t v ( X s ) ds − R t v ( X s ) dW s ) t ≥ is a martingale under Q . The drift term in Eq.(2.1) may look unnatural, but it does not (Park (2016)). .2 Objective measures Given a risk-neutral market (Ω , ( F t ) t ≥ , Q ) satisfying A1 - 2, we want to find possible objec-tive measures satisfying A3 - 5 stated below. Assumption 3.
There are a real number β and a twice continuously differentiable function φ with φ ( ξ ) = 1 such that e βt φ ( X t ) /G t , t ≥ is a martingale under Q . Definition 2.1.
Let ( β, φ ) be a pair satisfying A3. A measure P φ on each F t defined by d P φ d Q (cid:12)(cid:12)(cid:12) F t = e βt φ ( X t ) /G t is called the transformed measure or the objective measure with respect to φ. This measure P φ is well-defined for all T ≥ E Q ( I A M t ) = E Q ( I A M T ) holds for A ∈ F t and 0 ≤ t ≤ T where M t := e βt φ ( X t ) /G t . Remark 2.1.
In this definition, the pair ( β, φ ) determines the objective measure P φ . Thereader may wonder why we use the superscript P φ instead of P ( β,φ ) . Later we will see that infact the number β is automatically determined by φ, thus the notation P ( β,φ ) is abundant. Remark 2.2.
Assumption 3 means that the pair ( β, φ ) can be understood as an eigenpair ofan operator. Define a pricing operator P T by P T f ( x ) := E Q X = x ( f ( X T ) /G T ) , then P T φ ( x ) = E Q X = x ( φ ( X T ) /G T ) = e − βT E Q X = x ( e βT φ ( X T ) /G T ) = e − βT φ ( x ) . The last equality holds because ( e βt φ ( X t ) /G t ) ≤ t ≤ T is a martingale and its time- value is φ ( x ) . Assumption 4.
Under an objective measure P φ , the process X = ( X t ) t ≥ approaches toinfinity as t → ∞ with probability . This assumption may be debatable, but typical empirical data says the aggregative consump-tion grows as time goes. Thus, it is reasonable to assume that the aggregate consumptionprocess is not recurrent nor converge to zero as time goes under objective measures. It iswell known that any transient time-homogeneous Markov diffusion process which does notconverge to the left boundary always approaches to the right boundary as t → ∞ . In thisrespect, this assumption is plausible.We now demonstrate reasonable assumptions on φ. The usual conditions on utility func-tion U ( x ) are as follows: (i) U ′ > U ′′ < x → U ′ ( x ) = ∞ (iv) lim x →∞ U ′ ( x ) =0 . From φ ( x ) = U ′ ( X ) /U ′ ( x ) , we obtain the following assumptions by direct calculation. Assumption 5.
The function φ satisfies the following conditions(i) φ > (ii) φ ′ > (iii) lim x → φ ( x ) = 0 (iv) lim x →∞ φ ( x ) = ∞ . n this paper, the four conditions above are called the usual conditions on φ. It can be easily checked that a function φ satisfies the usual conditions if and only if U ( x ) := R x · /φ ( y ) dy satisfies the usual conditions on utility function. Definition 2.2.
For a real number β and a positive function φ ∈ C (0 , ∞ ) , we say ( β, φ ) isan admissible pair if the pair satisfies A3 - 5. The main purpose of this paper is to find all admissible pairs ( β, φ ) (i.e., satisfying A3 - 5) forgiven risk-neutral market (Ω , ( F t ) t ≥ , Q ) with A1 - 2. The main contribution of this articleis to investigate A5. The followings are of interest to us.(i) Find sufficient conditions on the risk-neutral market satisfying A1 - 2 such that thereexists an admissible pair.(ii) If such admissible pairs exist, express all such pairs.It will be proven that such pairs can be expressed by a one-parameter family. In this section, we observe that the problem of finding pairs ( β, φ ) satisfying A3 is transformedinto a problem of finding eigenpairs of a second-order differential operator. Define an operator L by L φ ( x ) = 12 σ ( x ) φ ′′ ( x ) + k ( x ) φ ′ ( x ) − r ( x ) φ ( x ) , (3.1)where k := b − σv. See Park (2016) for proof of the following theorem.
Theorem 3.1.
Let β be a real number and φ be a twice continuously differentiable function.If the pair ( β, φ ) satisfies Assumption 3, then ( β, φ ) is an eigenpair of the operator −L . Conversely, if ( β, φ ) is an eigenpair of the operator −L , then e βt φ ( X t ) /G t , t ≥ is a local martingale under Q . Through this paper, a generic solution pair of the differential equation L h = − λh will bedenoted by ( λ, h ) . This theorem guides the strategy of this paper. We first restrict ourattention to the solution pairs ( λ, h ) of L h = − λh with h > h ( X ) = 1 , then weexclude pairs ( λ, h ) which does not satisfy A3 - 5. Theorem 3.2. If r ( x ) ≥ , there exists a number λ ≥ . such that it has two linearlyindependent positive solutions for λ < λ, has no positive solution for λ > λ and has one ortwo linearly independent solutions for λ = λ. Define r := inf x> r ( x ) . By considering r ( x ) − r, it is obtained that λ ≥ r . (3.2)Refer to page 146 and 149 in Pinsky (1995) for proof. The condition that r ( x ) ≥ r ( X t ) is usually nonnegative in practice.We express the pair ( λ, h ) more efficiently by the following way. A solution h of a second-order differential equation is uniquely determined by the initial value h ( X ) and the initial erivative h ′ ( X ) . By normalizing, we may assume h ( X ) = 1 , then a solution is determinedby h ′ ( X ) . Thus, a solution pair ( λ, h ) can be represented by ( λ, h ′ ( X )) under the assumptionthat h ( X ) = 1 . Occasionally we use the terminology without ambiguity: the tuple ( λ, h ′ ( X ))is corresponding to the pair ( λ, h ) . The two terms tuple and pair will be used to distinguishbetween these meanings. Using the notion of tuples, we define the set of all admissible tuplesby the following way. A := (cid:8) ( λ, h ′ ( ξ )) ∈ R (cid:12)(cid:12) ( λ, h ) satisfies Assumption 3, 4 and 5 (cid:9) . Motivated by Theorem 3.1, we first consider the set of all solution pairs ( λ, h ) of L h = − λh with h > . We recall that X = ξ in Assumption 1. Definition 3.1.
We say ( λ, h ′ ( ξ )) ∈ R is a candidate tuple or we say ( λ, h ) is a candidatepair if ( λ, h ) is a solution pair of L h = − λh with h ( · ) > and h ( ξ ) = 1 . Denote the set ofthe candidate tuples by C . C := (cid:8) ( λ, h ′ ( ξ )) ∈ R (cid:12)(cid:12) L h = − λh, h > , h ( ξ ) = 1 (cid:9) . We briefly state properties of C . Theorem 3.2 guarantees that the set C is nonempty if r ( x ) ≥ . Recall that λ be the maximum value of the first coordinate of elements of C , thatis, λ = max { λ | ( λ, z ) ∈ C } . For any λ with λ ≤ λ, we set M λ := sup ( λ,z ) ∈C z , m λ := inf ( λ,z ) ∈C z . Occasionally, we use notations M ( λ ) and m ( λ ) instead of M λ and m λ , respectively, to avoiddouble subscripts such as h M λ and h m λ . Propositioin 3.1.
Let λ ≤ λ. For any z with m λ ≤ z ≤ M λ , the tuple ( λ, z ) is in C . The above proposition can be easily shown by the fact that the equation L h = − λh has twolinearly independent solutions and any solution can be expressed by the linear combinationsof the two solutions. For rigorous proof, refer to Park (2016). Therefore, the λ -slice of C isa connected and compact set. For a candidate pair ( λ, h ) , Theorem 3.1 says that ( e λt h ( X t ) /G t ) t ≥ is a local martingaleunder Q . Propositioin 3.2.
Let ( λ, h ) be a candidate pair such that ( e λt h ( X t ) /G t ) t ≥ is a martingale(that is, ( λ, h ) satisfies Assumption 3). Then a process ( B ht ) t ≥ defined by dB ht = − ( σh ′ h − − v )( X t ) dt + dW t is a Brownian motion under the transformed measure P h . In this case, the P h -dynamics of X is dX t = ( b − vσ + σ h ′ h − )( X t ) dt + σ ( X t ) dB ht = ( k + σ h ′ h − )( X t ) dt + σ ( X t ) dB ht . (3.3)Occasionally, we use the notation P and ( B t ) t ≥ instead of P h and ( B ht ) t ≥ , respectively,without ambiguity. Even when ( e λt h ( X t ) /G t ) t ≥ is not a martingale, we can consider thediffusion process in Eq.(3.3). Definition 3.2.
The diffusion process ( X t ) t ≥ defined by dX t = ( k + σ h ′ /h )( X t ) dt + σ ( X t ) dB t is called the diffusion process induced by the pair ( λ, h ) or the tuple ( λ, h ′ ( ξ )) . .3 Divergence to infinity We shift our attention to Assumption 4. The following theorem specifies which candidatetuples induce transformed measures satisfying Assumption 4. For proof, see Park (2016).
Theorem 3.3.
Let λ ≤ λ. The diffusion process induced by tuple ( λ, M λ ) approaches toinfinity as t → ∞ with probability one. For z with m λ ≤ z < M λ , the diffusion processinduced by tuple ( λ, z ) approaches to zero as t → ∞ with positive probability. In conclusion, it is obtained that
A ⊆ { ( λ, M λ ) ∈ C | λ ≤ λ } . From now, we mainly focus on the tuples ( λ, M λ ) with λ ≤ λ. We now explore the martingale condition discussed in Section 3.2. For any given tuple( λ, h ′ ( ξ )), we know the process e λt h ( X t ) G − t is a local martingale. Consider the set of tuples( λ, M λ ) which induce the martingales e λt h ( X t ) G − t , that is, M := { ( λ, M λ ) | e λt h ( X t ) G − t is a martingale } . Clearly, A is a subset of M . The following theorem states that the set is connected in R . Refer to Park (2016) for proof.
Theorem 3.4.
Let δ < λ ≤ λ and let ( δ, g ) and ( λ, h ) be the candidate pairs correspondingto ( δ, M δ ) and ( λ, M λ ) , respectively. If e δt g ( X t ) G − t is a martingale, so is e λt h ( X t ) G − t . We assume that sufficiently many candidate pairs satisfy the martingale condition. Inother words, the number λ defined by λ := inf { λ | ( λ, M λ ) ∈ M } is sufficiently small. Thisassumption is to guarantee the existence of admissible pair. If the set is too small or is empty,there may not exist an admissible pair.There is an useful criteria to check the martingale condition. Let ( λ, h ) be a candidatepair of L h = − λh. This pair satisfies the martingale condition if and only if the followingtwo conditions hold: Z ξ dx h ( x ) e − R xξ k ( s ) σ s ) ds Z ξx dy h ( y ) σ ( y ) e R yξ k ( s ) σ s ) ds = ∞ , Z ∞ ξ dx h ( x ) e − R xξ k ( s ) σ s ) ds Z xξ dy h ( y ) σ ( y ) e R yξ k ( s ) σ s ) ds = ∞ . (3.4)We recall Definition 3.2. The above criteria implies that the process e λt h ( X t ) G − t is amartingale if and only if the diffusion process induced by ( λ, h ) does not explode. Refer topage 215 in Pinsky (1995). The martingale condition can be checked case-by-case, thus wedo not go further details. One of the main contributions of the present article is to investigate Assumption 5. Now, inthe set { ( λ, M λ ) ∈ C | λ ≤ λ } , we explore which tuples satisfy Assumption 5. For convenience,put U := { ( λ, M ( λ )) ∈ C | h M ( λ ) satiafies the usual conditions } . ere, h M ( λ ) is the function corresponding to the tuple ( λ, M λ ) . The notation U is inheritedfrom terminology “usual conditions”. Let L be the measure defined by the Radon-Nikodymderivative d L d Q (cid:12)(cid:12)(cid:12)(cid:12) F t = exp (cid:18) − Z t v ( X s ) ds − Z t v ( X s ) dW s (cid:19) , which is a martingale by Assumption 2. The L -dynamics of X t is dX t = ( b − vσ )( X t ) dt + σ ( X t ) dB t = k ( X t ) dt + σ ( X t ) dB t for a Brownian motion B t . Here, we used notation B t to be consistent with the notationused in Proposition 3.2.To investigate the set U , we need to employ the notion of boundary classification. Bothboundaries 0 and ∞ of ( X t ) t ≥ are inaccessible under Q if and only if those are inaccessibleunder L . It is because two measures Q and L are equivalent on each F T , T ≥ . FromAssumption 1, two boundaries 0 and ∞ are inaccessible under both measures Q and L . From now on, we discuss more detailed boundary classification under the measure L . Definition 3.3.
Let γ ( x ) = e − R xξ k ( s ) σ s ) ds ,Q ( x ) = 2 σ ( x ) γ ( x ) Z xξ γ ( s ) ds ,R ( x ) = γ ( x ) Z xξ σ ( s ) γ ( s ) ds . An endpoint is said to be inaccessible if R / ∈ L (0 , ξ ) . An inaccessible endpoint is said tobe ( entrance if Q ∈ L (0 , ξ ) , natural if Q / ∈ L (0 , ξ ) . The definitions of inaccessible, entrance and natural at the endpoint ∞ are defined in similarways. We now state main theorems of the paper, which describes the usual set U . The followingtheorem implies that the set U is a connected subset of R . Define λ := sup { λ | ( λ, M λ ) ∈ U } , then for all λ < λ , the tuple ( λ, M λ ) is in the usual set U . The endpoint ( λ, M λ ) may ormay not be in U . Theorem 3.5.
Assume δ < λ ≤ λ. Let g and h be the functions corresponding to tuple ( δ, M δ ) and ( λ, M λ ) , respectively. If h satisfies the usual conditions, then so does g. In otherwords, if ( λ, M λ ) is in U , then ( δ, M δ ) is also in U . Theorem 3.6.
Assume r ( · ) ≥ and r ( · ) is bounded on (0 , ξ ) . Then the set U is nonemptyif and only if is a natural boundary. In this case, for λ < r := inf x> r ( x ) , the tuple ( λ, M λ ) is in the set U . he above theorem states a sufficient and necessary condition for the existence of tupleswhich satisfies the usual conditions. Refer to Appendix A and B for proofs of Theorem 3.5and 3.6, respectively.For the remainder of this section, we find the usual set U when the short interest ratefunction r ( · ) is a constant r. By Theorem 3.6, U is nonempty if and only if 0 is a naturalboundary, thus we assume 0 is a natural boundary. For λ < r, the tuple ( λ, M λ ) is always in U . From equation (3.2), we know that λ ≥ r. The case of λ = r is relatively easy to find theset U . Since { ( λ, M λ ) | λ < r } ⊆ U ⊆ { ( λ, M λ ) | λ ≤ r } , The set U is determined by the solution corresponding to the tuple ( r, M r ) . Consider thesolution of the corresponding second-order differential equation12 σ ( x ) h ′′ ( x ) + k ( x ) h ′ ( x ) = 0 . By direct calculation, two linearly independent solutions are h ( x ) = 1 + c Z xξ e − R yξ k ( s ) σ s ) ds dy , h ( x ) = 1 . Clearly, h ( x ) = 1 does not satisfy the usual conditions. By considering the function h ( x ) , we have the following proposition. Recall that γ ( x ) := e − R xξ k ( s ) σ s ) ds . Propositioin 3.3.
Assume that λ = r and is a natural boundary. If R ∞ ξ γ ( x ) dx = ∞ and R ξ γ ( x ) dx < ∞ , then U = { ( λ, M λ ) | λ ≤ r } . Otherwise, U = { ( λ, M λ ) | λ < r } . Proof.
Assume R ∞ ξ γ ( x ) dx = ∞ and R ξ γ ( x ) dx < ∞ . Then h ( x ) with c = 1 R ξ γ ( x ) dx is the function corresponding to ( r, M r ) . Clearly lim x → h ( x ) = 0 with this choice of c. Thus, this tuple is in U . The converse is trivial.It is noteworthy that the conditions R ∞ ξ γ ( x ) dx = ∞ and R ξ γ ( x ) dx < ∞ means that thediffusion process X t under L has the following property: L (cid:16) lim t →∞ X t = 0 (cid:17) = L (cid:16) sup ≤ t< ∞ X t < ∞ (cid:17) = 1 . Refer to page 345 in Karatzas and Shreve (2012).We now consider the case of λ > r.
Refer to Appendix C for proof of the followingtheorem.
Theorem 3.7.
Assume that λ > r and is a natural boundary. If R ∞ ξ γ ( x ) dx = ∞ , then U = { ( λ, M λ ) | λ ≤ λ } or { ( λ, M λ ) | λ < λ } . Moreover, if ∞ is a natural boundary, then U = { ( λ, M λ ) | λ ≤ λ } . If R ∞ ξ γ ( x ) dx < ∞ , then U = { ( λ, M λ ) | λ < r } . The authors conjecture that when R ∞ ξ γ ( x ) dx = ∞ , the right boundary ∞ is a naturalboundary if and only if U = { ( λ, M λ ) | λ ≤ λ } . .6 Admissible sets The purpose of this paper is to find the admissible set A under Assumption 1 - 5. Theadmissible set satisfies A = M ∩ U , thus A is a connected subset because M and U are connected subsets of { ( λ, M λ ) ∈ C | λ ≤ λ } . Define λ := inf { λ | ( λ, M λ ) ∈ M } , λ := sup { λ | ( λ, M λ ) ∈ U } . The endpoints λ and λ may or may not be in M and U , respectively. Assuming λ ≤ λ , weobtain that for λ between λ and λ , the tuple ( λ, M λ ) is an admissible tuple. In conclusion,one can recover the objective measures by the one-parameter family. We investigate how the previous results can be used for Ross recovery. In the continuous-timeconsumption-based CAPM, the state variable X t is the aggregate consumption (or income)process of the market. In a financial market, the aggregate income is equal to the aggregatedividend, thus we may assume that X t is the aggregate dividend. Let S t be a compositestock price index such as S&P 500 and assume that S t pays aggregate dividend which is afunction of S t , that is, X t = δ ( S t ) S t where δ ( S t ) is the dividend per one unit of the composite stock price index. The function δ ( s )is assumed to be known ex ante and is a nondecreasing function of s. Assume that the func-tion π ( s ) := δ ( s ) s is continuously twice differentiable with continuously twice differentiableinverse. By Appendix D, the transformed measure is invariant under the map π, thus wemay assume that the state variable is S t . Ross (2015) also used the dividend or the compositestock price index (S&P 500) in page 630-633 as the state variable.Let the numeraire G t be the wealth process induced the composite stock price process S t , that is, G t = e R t δ ( S u ) du S t . Assume that the state variable S t satisfies dS t = ( r ( S t ) − δ ( S t ) + σ ( S t )) S t dt + σ ( S t ) S t dW t . Then dG t G t = ( r ( S t ) + σ ( S t )) dt + σ ( S t ) dW t . The operator L corresponding to equation (3.1) is L h ( s ) = 12 σ ( s ) s h ′′ ( s ) + ( r ( s ) − δ ( s )) sh ′ ( s ) − r ( s ) h ( s ) . (4.1)Occasionally, Y t := log S t induces a simpler second-order equation. Let y = ln s anddefine κ ( y ) = r ( s ) − δ ( s ) , ν ( y ) = σ ( s ) and ρ ( y ) = r ( s ) . Then dY t = ( r ( S t ) − δ ( S t ) + 12 σ ( S t )) dt + σ ( S t ) dW t = ( κ ( Y t ) + 12 ν ( Y t )) dt + ν ( Y t ) dW t . The corresponding equation (4.1) becomes12 ν ( y ) g ′′ ( y ) + ( κ ( y ) − ν ( y )) g ′ ( y ) − ρ ( y ) g ( y ) = − λg ( y )where g ( y ) = h ( s ) . Examples
In this section, we explore examples of Ross recovery. Denote by S t the composite stock priceindex as discussed in Section 4. Occasionally, for convenience, we say S t is the stock pricewithout ambiguity. The classical Black-Scholes stock model and the exponential CIR stockmodel are discussed with constant short interest rate and constant dividend rate in Section5.1 and 5.2, respectively. The classical Black-Scholes stock model with log dividend rate isexplored in Section 5.3. The classical Black-Scholes stock model with constant short interest rate and constant divi-dend rate is discussed. The dividends of the stock are paid out continuously with rate δ dt.
Suppose S t follows a geometric Brownian motion dS t = ( r − δ + σ ) S t dt + σS t dW t , S = 1and the numeraire is G t = S t e δt . The corresponding second-order equation is L h ( s ) = 12 σ s h ′′ ( s ) + ( r − δ ) sh ′ ( s ) − rh ( s ) = − λh ( s ) . By direct calculation, we have λ = ( σ − r − δσ ) + r. For λ ≤ λ, it can be easily shown thatthe value M λ is M λ = 12 − r − δσ + s(cid:18) − r − δσ (cid:19) + 2( r − λ ) σ and the function corresponding to the tuple ( λ, M λ ) is h λ ( s ) := s − r − δσ + r(cid:16) − r − δσ (cid:17) + r − λ ) σ . The function γ ( s ) is s − r − δ ) σ . We find the admissible set A . It can be easily checked that every candidate pair is ad-missible by using the method in Section 3.4, thus A = U . As is well-known, both endpoints0 and ∞ of the geometric Brownian motion are natural boundaries. Applying Theorem 3.6,Proposition 3.3 and Theorem 3.7, we obtain A = { ( λ, M λ ) | λ ≤ λ } if 2( r − δ ) < σ , A = { ( λ, M λ ) | λ < r } if 2( r − δ ) ≥ σ . We explore an example of stock model with inaccessible entrance 0 boundary. By Theorem3.6, the usual set is empty, that is, U = ∅ . Even though U is empty, it would be interesting to find the set M . Assume that the shortinterest rate r and the dividend rate δ are constants. Put θ := r − δ. Let Y t be an extendedCIR process given by dY t = ( θ + 12 σ Y t ) dt + σ p Y t dW t ith 2 θ ≥ σ . It is well known that the left boundary 0 and the right boundary ∞ areentrance and natural, respectively. Assume the stock price follows S t = e Y t so that dS t = ( θ + σ ln S t ) S t dt + σ p ln S t S t dW t . The corresponding second-order differential equation is12 σ ln( s ) s h ′′ ( s ) + θsh ′ ( s ) − rh ( s ) = − λh ( s ) . However, the process Y t := log S t induces a simpler second-order equation12 σ yg ′′ ( y ) + ( θ − σ y ) g ′ ( y ) − rg ( y ) = − λg ( y ) . It can be easily checked that g λ ( y ) := M (cid:18) r − λ ) σ , θσ , y (cid:19) is a solution corresponding to ( λ, M λ ) for λ ≤ r = λ. It is known that the confluent hyperge-ometric function M ( α, β, y ) is positive if and only if α ≤ . Refer to Qin and Linetsky (2014)for more details. We obtain that dY t = (cid:18) θ + 12 σ Y t + σ Y t g ′ λ ( Y t ) g λ ( Y t ) (cid:19) dt + σ p Y t dB t where B t is a Brownian motion under the corresponding transformed measure.It can be easily checked that e λt g λ ( Y t ) G − t is a martingale. By considering the asymptoticbehavior M ( α, β, y ) ∼ e y y α − β / Γ( α ) as y → ∞ and the fact that M ′ ( α, β, y ) = ( α/β ) M ( α +1 , β + 1 , y ) , we obtain as y → ∞ , g ′ λ ( y ) g λ ( y ) ∼ σ θ . The drift of equation 5.2 has linear growth rate, as the CIR model, Y t does not explodeby the criteria in equation (3.4). In conclusion, the function corresponding to ( λ, M λ ) is h λ ( s ) := g λ (ln s ) and M = (cid:26) (cid:18) λ , h ′ λ ( S ) h λ ( S ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) λ ≤ r (cid:27) = λ , r − λθS · M (cid:16) r − λ ) σ + 1 , θσ + 1 , ln S (cid:17) M (cid:16) r − λ ) σ , θσ , ln S (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ≤ r . In this section, we explore the possibility of recovering when the dividends of the stock arepaid out continuously with rate b log S t dt. Suppose S t has a constant volatility σ and theshort interest rate is a constant r.dS t = ( r + σ − b log S t ) S t dt + σS t dW t . It can be easily shown that S t = e Y t where dY t = ( r + 12 σ − bY t ) dt + σ dW t . he corresponding second-order differential equation is12 σ s h ′′ ( s ) + ( r − b log s ) sh ′ ( s ) − rh ( s ) = − λh ( s ) . Substituting s = e y and h ( s ) = g ( y ) , it follows that12 σ g ′′ ( y ) + ( r − σ − by ) g ′ ( y ) − rg ( y ) = − λg ( y ) . One can check that for some normalizing constant c,g λ ( y ) = c M ( r − λ b , , bσ ( y − κ ) )Γ( + r − λ b ) + 2( y − κ ) r bσ M ( r − λ b + , , bσ ( y − κ ) )Γ( r − λ b ) ! is an admissible function (i.e., positive increasing solution) with g λ ( −∞ ) = 0 , g λ ( ∞ ) = ∞ for λ ≤ r = λ where κ = rb − σ b and M ( · , · , · ) is the confluent hypergeometric function.The corresponding transformed measure P is d P d Q (cid:12)(cid:12)(cid:12)(cid:12) F t = e λt g λ ( Y t ) G − t = e λt g λ (log S t ) G − t under which the dynamics of Y t is dY t = (cid:18) r + 12 σ − bY t + σ g ′ λ ( Y t ) g λ ( Y t ) (cid:19) dt + σ dB t . (5.1)Thus, we obtain the P -dynamics of S t = e Y t . It can be easily checked that e λt g λ ( Y t ) G − t is a martingale. By considering the asymptoticbehaviors of M ( · , · , · ) and M ′ ( · , · , · ) as in Section 5.3, we obtain that as | y | → ∞ ,g ′ λ ( y ) g λ ( y ) ∼ b σ | y | . Because the drift of equation (5.1) has linear growth rate, by the criteria in equation (3.4),we know Y t does not explode with the dynamics of Y t in equation (5.1). In conclusion, weget A = (cid:26) (cid:18) λ , h ′ λ ( S ) h λ ( S ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) λ ≤ r (cid:27) where h λ ( s ) = g λ (ln s ) . This paper determines a representative agent model from a risk-neutral measure in a continuous-time setting. One of the key ideas of the argument is that the reciprocal of the pricing kernelis expressed by the transition independent form e βt φ ( X t )for a constant β and a positive function φ. This form is originated from the continuous-timeconsumption-based asset pricing model, which is a well-known asset pricing theory. Basedon the theory, several conditions such as the martingale condition, divergence to infinity and he usual conditions are assumed on the function φ and the underlying process X t . The pair( β, φ ) satisfying these conditions was called an admissible pair.The main purpose of this paper is to investigate the admissible pairs. A necessary andsufficient condition for the existence of admissible pairs was explored. Moreover, we showedthat, if it exists, the set of admissible pairs is expressed by a one-parameter family. Theadmissible set is determined by the lower bound of the martingale set M and the upperbound of the usual set U . As a special case, when the short interest rate is a constant, theset U was presented.The following extensions for future research are suggested. First, it would be interestingto extend the recovery to multi-dimensional state variables. In this case, the correspondingSturm-Liouville equation is a second-order partial differential equation. Second, it would bevaluable to find economically meaningful methods to determine β. We could not provide suchmethods in this article. Finally, much work remains to be conducted on the implementationand empirical testing of recovery theory in future research.
A Proof of Theorem 3.5
Lemma A.1.
Assume δ < λ ≤ λ. Let g and h be the functions corresponding to tuple ( δ, M δ ) and ( λ, M λ ) , respectively. Then we have g ′ g − > h ′ h − . For proof, see Lemma E.2 in Park (2016).Now we prove Theorem 3.5.
Proof.
Assume that h satisfies the usual conditions. From Lemma A.1, it is straightforwardthat g ′ > . We now show that lim x →∞ g ( x ) = ∞ . It is enough to prove that g ( x ) > h ( x )for x > ξ. Integrating by R xξ to the inequality g ′ g − > h ′ h − in the Lemma A.1, we haveln g ( x ) − ln g ( ξ ) > ln h ( x ) − ln h ( ξ ) . Since g ( ξ ) = h ( ξ ) = 1 , it follows that g ( x ) > h ( x ) for x > ξ. In a similar way, one can prove that lim x → g ( x ) = 0 by showing g ( x ) < h ( x ) for0 < x < ξ. B Proof of Theorem 3.6
The following lemmas will be used to prove Theorem 3.6 when r ( x ) is bounded near 0 . Recallthe definition r := inf x> r ( x ) . Lemma B.1.
Assume that r ( · ) ≥ and λ < r. Let h be a solution of L h = − λh. Then h can attain nether a positive local maximum nor a negative local minimum.Proof. Suppose that h has a positive local maximum at x . Then h ′ ( x ) = 0 and h ′′ ( x ) ≤ . From L h = − λh, it follows that12 σ ( x ) h ′′ ( x ) = 12 σ ( x ) h ′′ ( x ) + k ( x ) h ′ ( x ) = ( r ( x ) − λ ) h ( x ) > , which is a contradiction. In a similar way, one can show that h cannot attain a negative localminimum. Lemma B.2.
Assume r ( · ) ≥ and λ < r. Let b and c be two numbers with c < b, notnecessarily to be between m λ and M λ , and let h b and h c be the functions corresponding totuples ( λ, b ) and ( λ, c ) , respectively. Then h b ( x ) > h c ( x ) for x > ξ and h b ( x ) < h c ( x ) for < x < ξ. roof. Since h ′ c ( ξ ) = c < b = h ′ b ( ξ ) and h b ( ξ ) = h c ( ξ ) = 1 , there exists an interval ( ξ, x ) inwhich h c < h b . Suppose x < ∞ . Then h b ( x ) = h c ( x ) . This means that g := h b − b c is asolution of L g = − λg and g has two zeros at ξ and x . By Lemma B.1, since g can attainnether a positive local maximum nor a negative local minimum, g should be identically zero.This leads us a contradiction. In a similar way, it can be shown that h b ( x ) < h c ( x ) for0 < x < ξ. Lemma B.3.
Assume that r ( · ) ≥ and λ < r. Let h M ( λ ) and h m ( λ ) be the functionscorresponding to ( λ, M λ ) and ( λ, m λ ) , respectively. Then h ′ M ( λ ) > and h ′ m ( λ ) < . Proof.
We show that h ′ M ( λ ) > . Let b be a real number with b > M λ and denote by h b thefunction corresponding to ( λ, b ) . We prove that h b is a monotone increasing function. Bydefinition of M λ , h b has a zero at a point x > . It follows that x < ξ because h M ( λ ) ispositive and h b ( x ) > h M ( λ ) ( x ) for x > ξ by Lemma B.2. We have that h b ( x ) is monotoneincreasing on x > x since h b ( x ) = 0 and h b ( ξ ) = 1 , otherwise h b has a positive localmaximum, which is a contradiction to Lemma B.1. Clearly, h ′ b ( x ) > h ′ b ( x ) = 0 ,h b is identically zero. h b is strictly increasing near x , and thus h b is monotone increasing on x < x since h b has no negative local minimum. In conclusion, h b is monotone increasing on(0 , ∞ ) . We now show that h ′ M ( λ ) ( x ) > x > . It can be easily shown that h M ( λ ) ( x ) = lim b → M ( λ )+ h b ( x )because h b can be expressed by the linear combination of h M ( λ ) and h M ( λ ) : h b = b − mM − m h M + M − bM − m h m . Here, for a moment, we used M and m instead of M ( λ ) and m ( λ ) , respectively, to avoid theheavy notions. Since h M ( λ ) is the limit of monotone increasing functions, h M ( λ ) is a monotoneincreasing function, that is, h ′ M ( λ ) ( x ) ≥ x > . Suppose that h ′ M ( λ ) ( x ) = 0 at somepoint x > . Then12 σ ( x ) h ′′ ( x ) = 12 σ ( x ) h ′′ ( x ) + k ( x ) h ′ ( x ) = ( r ( x ) − λ ) h ( x ) > . Here, for a moment, we used h instead of h M ( λ ) to avoid the heavy notions. Thus, h ′′ M ( λ ) ( x ) > , which contradicts to the fact that h M ( λ ) is monotone increasing. In conclusion, it isobtained that h ′ M ( λ ) ( x ) > x > . It can be shown that h ′ m ( λ ) ( x ) < x > Proof. ( ⇐ ) Suppose that 0 is a natural boundary. Note that by Assumption 1, ∞ is a naturalor entrance boundary. Fix λ < r and let h be the function corresponding to the tuple ( λ, M λ ) . We show that h satisfies the usual conditions. By Lemma B.3, it is obtained that h ′ > , which is one of the usual conditions. Now we show that lim x →∞ h ( x ) = ∞ . Suppose thatlim x →∞ h ( x ) is finite. Recall the function of γ from Definition 3.3. By direct calculation, (cid:18) h ′ γ (cid:19) ′ = 2( r − λ ) hσ γ . (B.1) t follows that h ′ ( x ) = γ ( x ) (cid:18) h ′ ( ξ ) + Z xξ r ( s ) − λ ) h ( s ) σ ( s ) γ ( s ) ds (cid:19) . (B.2)The two terms in the right-hand side are in L ( ξ, ∞ ) because they are positive and the integra-tion of the left hand side R ∞ ξ h ′ ( y ) dy = lim x →∞ h ( x ) − h ( ξ ) is finite. Thus, γ ( x ) R xξ h ( s ) σ ( s ) γ ( s ) ds is in L ( ξ, ∞ ) since r ( x ) − λ ≥ r − λ > . On the other hand, since h is increasing and h ( ξ ) = 1 , we know R ( x ) = γ ( x ) Z xξ σ ( s ) γ ( s ) ds ≤ γ ( x ) Z xξ h ( s ) σ ( s ) γ ( s ) ds . Therefore, R ∈ L ( ξ, ∞ ) , which contracts the assumption that ∞ is inaccessible.Now we prove that lim x → h ( x ) = 0 . Suppose lim x → h ( x ) > . Since h ′ /γ is increasingby equation (B.1) and h ′ is positive, the limit existslim x → h ′ ( x ) γ ( x ) = C ≥ . (B.3)Integrating the equation (B.1) by R x , we have h ′ ( x ) = γ ( x ) (cid:18) C + Z x r ( s ) − λ ) h ( s ) σ ( s ) γ ( s ) ds (cid:19) . (B.4)The two terms in the right-hand side are in L (0 , ξ ) because they are nonnegative and R ξ h ′ ( x ) dx = h ( ξ ) − lim x → h ( x ) is finite. It follows that γ ( x ) R x h ( s ) σ ( s ) γ ( s ) ds is in L (0 , ξ )since r ( x ) − λ ≥ r − λ > . Because h is an increasing function and lim x → h ( x ) > , wehave that γ ( x ) R x σ ( s ) γ ( s ) ds is in L (0 , ξ ) . By the Fubini theorem, Z ξ γ ( x ) Z x σ ( s ) γ ( s ) ds dx = Z ξ σ ( s ) γ ( s ) Z ξs γ ( x ) dx ds is finite. Thus, Q is in L (0 , ξ ) , which means that 0 is not a natural boundary. This is acontradiction.( ⇒ ) Assume that the set U is nonempty. Let ( λ, M λ ) be an element in the set and denoteby h the corresponding function. By Thorem 3.5, we may assume λ < r. We now prove that0 is a natural boundary. Suppose that 0 is an entrance boundary, that is, Q ∈ L (0 , ξ ) . Bythe Fubini theorem, we know that γ ( x ) Z x σ ( s ) γ ( s ) ds ∈ L (0 , ξ ) . (B.5)Since h is bounded on (0 , ξ ) and r is assumed to be bounded near 0 , it is obtained that γ ( x ) Z x r ( s ) − λ ) h ( s ) σ ( s ) γ ( s ) ds ∈ L (0 , ξ ) . (B.6)On the other hand, from equation (B.3), it follows that lim x → h ′ ( x ) γ ( x ) = C ≥ . Weshow that C = 0 . Suppose C = 0 . Then, from equation (B.4) and (B.6), γ is in L (0 , ξ ) . For x < ξ/ , we know0 ≤ σ ( x ) γ ( x ) = 2 σ ( x ) γ ( x ) R ξx γ ( s ) ds R ξx γ ( s ) ds ≤ σ ( x ) γ ( x ) R ξx γ ( s ) ds R ξξ/ γ ( s ) ds = − Q ( x ) R ξξ/ γ ( s ) ds , o we have that c := R ξ σ ( x ) γ ( x ) dx is finite since Q ∈ L (0 , ξ ) . It follows that R ∈ L (0 , ξ )because for x < ξ, ≤ − R ( x ) = γ ( x ) Z ξx σ ( s ) γ ( s ) ds ≤ γ ( x ) Z ξ σ ( s ) γ ( s ) ds = c γ ( x ) ∈ L (0 , ξ ) . This is a contradiction because ∞ is an inaccessible boundary. Thus C = 0 . Now we prove that the hypothesis Q ∈ L (0 , ξ ) induce a contradiction. Since C = 0 and h is increasing, from equation (B.4) and (B.5), we know h ′ ( x ) h ( x ) = γ ( x ) h ( x ) Z x r ( s ) − λ ) h ( s ) σ ( s ) γ ( s ) ds ≤ γ ( x ) Z x r ( s ) − λ ) σ ( s ) γ ( s ) ds ≤ c γ ( x ) Z x σ ( s ) γ ( s ) ds ∈ L (0 , ξ )where c := | λ | + sup ≤ x ≤ ξ r ( x ) . This implies that lim x → h ( x ) > − lim x → ln h (0) = ln h ( ξ ) − lim x → ln h (0) = Z ξ h ′ ( x ) h ( x ) dx is finite. This a contradiction to the assumption that h satisfies the usual condition. C Proof of Theorem 3.7
Theorem 3.7 will be shown. For this purpose, we first prove Theorem C.1, Lemma C.1 andC.2 stated below. In this appendix, assume r < λ.
Theorem C.1.
Let r < λ ≤ λ. Denote by h the function corresponding to ( λ, M λ ) . Then R ∞ ξ γ ( x ) dx = ∞ if and only if h ′ > . Proof.
Suppose that R ∞ ξ γ ( x ) dx = ∞ . Recall from equation (B.2) that h ′ ( x ) = γ ( x ) (cid:18) h ′ ( ξ ) − λ − r ) Z xξ h ( s ) σ ( s ) γ ( s ) ds (cid:19) . (C.1)If h ′ ( x ) < x , then for x > x ,h ′ ( x ) = γ ( x ) (cid:18) h ′ ( ξ ) − λ − r ) Z xξ h ( s ) σ ( s ) γ ( s ) ds (cid:19) ≤ γ ( x ) (cid:18) h ′ ( ξ ) − λ − r ) Z x ξ h ( s ) σ ( s ) γ ( s ) ds (cid:19) = γ ( x ) γ ( x ) h ′ ( x ) . Since R ∞ ξ γ ( x ) dx = ∞ and h ′ ( x ) < , it follows that h ( x ) → −∞ as x → ∞ , which is acontradiction to the fact that h is positive. Therefore, we obtain that h ′ ≥ . We now showthat h ′ > . Suppose that h ′ ( x ) = 0 at some point. Then12 σ ( x ) h ′′ ( x ) = 12 σ ( x ) h ′′ ( x ) + k ( x ) h ′ ( x ) = − ( λ − r ) h ( x ) < , thus h has local maximum at x . This is a contradiction to the fact that h ′ ≥ . e now prove the converse. Assume that h ′ > . From equation (C.1), we know h ′ ( ξ ) Z xξ γ ( y ) dy = h ( x ) − h ( ξ ) + 2( λ − r ) Z xξ γ ( y ) Z yξ h ( s ) σ ( s ) γ ( s ) ds dy . To show that R ∞ ξ γ ( x ) dx = ∞ , since the first term h ( x ) of the right-hand side is positive, itis enough to show that Z ∞ ξ γ ( x ) Z xξ h ( s ) σ ( s ) γ ( s ) ds dx = ∞ . On the other hand, because ∞ is an inaccessible boundary, we know Z ∞ ξ γ ( x ) Z xξ σ ( s ) γ ( s ) ds dx = ∞ . Since h is positive and increasing, we obtain the desired result. Lemma C.1.
Let r < λ ≤ λ. If h is a positive and increasing solution of L h = − λh, then h (0) := lim x → h ( x ) = 0 . Proof.
It can be easily shown that e ( λ − r ) t h ( X t ) is a local martingale. Since a positive localmartingale is a supermartingale, it follows that E [ e ( λ − r ) t h ( X t )] ≤ h ( X ) = h ( ξ ) . Since h is increasing, we have e ( λ − r ) t h (0) ≤ E [ e ( λ − r ) t h ( X t )] ≤ h ( ξ ) . Thus, h (0) ≤ e − ( λ − r ) t h ( ξ ) . Letting t → ∞ , we obtain the desired result. Lemma C.2.
Let λ < λ.
Let h be the function corresponding to ( λ, M λ ) , respectively. If ( λ, h ) satisfies the martingale condition and R ∞ ξ γ ( x ) dx = ∞ , then h is unbounded.Proof. Let h and g be the functions corresponding to ( λ, M λ ) and ( λ, M λ ) . Since ( λ, h )satisfies the martingale condition, so does ( λ, M λ ) by Theorem 3.4. Let P be the transformedmeasure with respect to ( λ, M λ ) . Then h ( ξ ) = E Q [ e ( λ − r ) t h ( X t )] = E P [( g − h )( X t )] e − ( λ − λ ) t g ( ξ ) . (C.2)The first equality holds because ( λ, h ) satisfies the martingale condition. Suppose that h isbounded, that is, h < c for some constant c. By Lemma A.1, as in the proof of Theorem 3.5,we know that h ( x ) < g ( x ) for 0 < x < ξ and h ( x ) > g ( x ) for x > ξ, that is, ( g − h )( x ) < < x < ξ and ( g − h )( x ) < cg − ( x ) for x > ξ. Since R ∞ ξ γ ( x ) dx = ∞ , g is increasingby Theorem C.1, so we know that ( g − h )( x ) < cg − ( x ) < cg − ( ξ ) = c for x > ξ. Thus, it isobtained that ( g − h )( x ) < c + 1 for all x > . From equation (C.2), we have h ( ξ ) = E P [( g − h )( X t )] e − ( λ − λ ) t g ( ξ ) ≤ ( c + 1) e − ( λ − λ ) t g ( ξ ) . Letting t → ∞ , it follows that h ( ξ ) ≤ . This leads us a contradiction. roof. We now prove Theorem 3.7. Suppose that R ∞ ξ γ ( x ) dx = ∞ . Fix any λ with r < λ < λ and let h be the function corresponding to ( λ, M λ ) . By Theorem C.1, it follows that h ′ > . By Lemma C.1, since h is positive and increasing, lim x → h ( x ) = 0 . By Lemma C.2, wehave lim x →∞ h ( x ) = ∞ . Thus, h satisfies the usual condition, that is, ( λ, M λ ) is in U . ByTheorem 3.5, it is obtained that { ( λ, M λ ) | λ < λ } ⊆ U , which is the desired result.We now prove that if ∞ is a natural boundary, then U = { ( λ, M λ ) | λ ≤ λ } . Let h be the function corresponding to ( λ, M λ ) . By the same argument, we have h ′ > x → h ( x ) = 0 . It suffices to show that lim x →∞ h ( x ) = ∞ . From equation (B.2), we have h ′ ( x ) = γ ( x ) (cid:18) h ′ ( ξ ) − λ − r ) Z xξ h ( s ) σ ( s ) γ ( s ) ds (cid:19) . Because the term inside the parenthesis is a decreasing function of x and the left-hand side h ′ ( x ) is positive for all x, by letting x → ∞ , it follows that C := h ′ ( ξ ) − λ − r ) R ∞ ξ h ( s ) σ ( s ) γ ( s ) ds is nonnegative. The above equation can be written by h ′ ( x ) = γ ( x ) (cid:18) C + 2( λ − r ) Z ∞ x h ( s ) σ ( s ) γ ( s ) ds (cid:19) ≥ λ − r ) γ ( x ) Z ∞ x h ( s ) σ ( s ) γ ( s ) ds . (C.3)On the other hand, applying the Fubini theorem, we know that Z ∞ ξ γ ( x ) Z ∞ x σ ( s ) γ ( s ) ds dx = Z ∞ ξ σ ( s ) γ ( s ) Z sξ γ ( x ) dx ds = ∞ because ∞ is a natural boundary. Thus, the integration of the right hand side of equation(C.3) becomes R ∞ ξ γ ( x ) R ∞ x h ( s ) σ ( s ) γ ( s ) ds dx = ∞ since h is an increasing function. This impliesthat lim x →∞ h ( x ) = ∞ . Now suppose that R ∞ ξ γ ( x ) dx < ∞ . By Theorem C.1, for any λ with r < λ ≤ λ, thecorresponding function h does not satisfy the condition h ′ > . Thus,
U ⊆ { ( λ, M λ ) | λ ≤ r } . When λ = r, by the same argument in the proof of Proposition 3.3, we know that ( r, M r )is not in U , so U ⊆ { ( λ, M λ ) | λ < r } . On the other hand, { ( λ, M λ ) | λ < r } ⊆ U is clear byTheorem 3.6. Therefore, U = { ( λ, M λ ) | λ < r } . D An invariant property
Let X t be a diffusion process satisfying dX t = k ( X t ) dt + σ ( X t ) dW t with the killing rate r ( x ) . Then the corresponding generator is L h ( x ) = 12 σ ( x ) h ′′ ( x ) + k ( x ) h ′ ( x ) − r ( x ) h ( x ) . Fix an admissible pair ( λ, h ) of the generator L . Let ( a, b ) an open interval in R . Suppose that π : (0 , ∞ ) ( a, b ) is a continuouslytwice differentiable bijective map with continuously twice differentiable inverse. Then π isincreasing or decreasing, so we may assume that π is increasing. Define Y t = π ( X t ) and H ( y ) = h ( π − ( y )) . Then Y t satisfies dY t = ( kπ ′ + 12 σ π ′′ )( X t ) dt + ( σπ ′ )( X t ) dW t = ( kπ ′ + 12 σ π ′′ ) ◦ ( π − ( Y t )) dt + ( σπ ′ ) ◦ ( π − ( Y t )) dW t ith the killing rate r ( π − ( y )) . The corresponding generator is L π H ( y ) = 12 ( σπ ′ ) ◦ ( π − ( y )) H ′′ ( y ) + ( kπ ′ + 12 σ π ′′ ) ◦ ( π − ( y )) H ′ ( y ) − r ( π − ( y )) H ( y ) . The pair ( λ, H ) is a solution pair of L π H = − λH. In addition, we have d P h d Q (cid:12)(cid:12)(cid:12)(cid:12) F t = e λt h ( X t ) G − t = e λt H ( Y t ) G − t = d P H d Q (cid:12)(cid:12)(cid:12)(cid:12) F t , thus ( λ, h ) and ( λ, H ) induce the same transformed measures. Since X t approaches to in-finity as t → ∞ under P h (by the definition of an admissible pair), the process Y t = π ( X t )approaches to the right boundary b as t → ∞ under P H . It is clear that the function H satisfies the usual condition (i),(ii) and (iii), (iv) replaced the limitslim x → h ( x ) = 0 , lim x →∞ h ( x ) = ∞ by lim y → a + H ( y ) = 0 , lim y → b − H ( y ) = ∞ . References
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