Laplace Transforms for Integrals of Markov Processes
LLAPLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES
CLAUDIO ALBANESE AND STEPHAN LAWI
Abstract.
Laplace transforms for integrals of stochastic processes have been known in ana-lytically closed form for just a handful of Markov processes: namely, the Ornstein-Uhlenbeck,the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue oftheir analytical tractability, these processes are extensively used in modelling applications. Inthis paper, we construct broad extensions of these process classes. We show how the knownmodels fit into a classification scheme for diffusion processes for which Laplace transformsfor integrals of the diffusion processes and transitional probability densities can be evaluatedas integrals of hypergeometric functions against the spectral measure for certain self-adjointoperators. We also extend this scheme to a class of finite-state Markov processes related tohypergeometric polynomials in the discrete series of the Askey classification tree. Introduction
Let ( X t ) t ≥ be a time-homogenous, real-valued Markov process on the filtered probabilityspace (Ω , {F t } t ≥ , P) and consider the Laplace transform L T − t ( X t , ϑ ) defined as follows:(1.1) L T − t ( X t , ϑ ) = E P (cid:104) e − ϑ R Tt φ ( X s ) ds q ( X T ) F t (cid:105) where t ≤ T , ϑ ∈ C and φ, q : R → R two Borel functions. In this paper, we address thequestion of whether it is possible to compute the Laplace transform L T − t ( X t , ϑ ) in analyticallyclosed form. Our work builds upon several streams of research often motivated by applicationsto various fields of Physics and Finance, and unifies them to obtain a broad classification schemefor Laplace transforms expressible in analytic closed form.For q ≡
1, a class of examples for which analytic closed form solutions are available is repre-sented by the so called affine models which are characterized by a representation of the form(1.2) L T − t ( X t , ϑ ) = e m ( T − t,ϑ ) X t + n ( T − t,ϑ ) . The archetypical affine models are based on diffusion processes and are described by stochasticdifferential equations of the form(1.3) dX t = ( a − bX t ) dt + σX βt dW t where a, b, σ are constants and β = 0 or . The case β = 0 corresponds to the Gaussian Ornstein-Uhlenbeck process ( ? ) and the case β = corresponds to the Cox-Ingersoll-Ross (CIR) process( ? ). The case of the CIR process was generalized to bridges by Pitman and Yor in ( ? ). Ithas been shown in ( ? ), ( ? ) and in ( ? ) that any affine process which is a time-homogenous,nonnegative diffusion is necessarily of the CIR type. However, there are also affine processeswith jumps. General non-negative affine processes correspond to the so called conservative CBI-processes (continuous state branching processes with immigration) and have been well studied,among others, by Kawazu and Watanabe in ( ? ) and Filipovi´c in ( ? ).An extension of the affine class, known as the quadratic class, postulates the Laplace transformbeing of the form(1.4) L T − t ( X t , ϑ ) = e l ( T − t,ϑ ) X t + m ( T − t,ϑ ) X t + n ( T − t,ϑ ) . Date : November 4, 2018.Supported in part by the Natural Science and Engineering Council of Canada under grants RGPIN-171149. a r X i v : . [ m a t h . P R ] O c t CLAUDIO ALBANESE AND STEPHAN LAWI
The first examples of quadratic models appeared in the double square root model of Longstaffin ( ? ) and in the nonlinear equilibrium model by Beaglehole and Tenney in ( ? ). Rogers ( ? )also uses examples where the pricing kernel is a quadratic function of the Markov process. Mostrecently, Filipovi´c ( ? ) proved that if one represents the forward rate as a polynomial function ofthe diffusion process, the maximal consistent order of the polynomial is two. Consistency in thiscontext means that the interest rate model will produce forward rate curves belonging to theparameterized family. Finally, Leippold and Wu ( ? ) formulated a general asset and derivativepricing framework for the quadratic class.A separate class of models for which the Laplace transform can be expressed in analyticallyclosed form is represented by the exponential Brownian motion of equation(1.5) dX t = µX t dt + σX t dW t where µ, σ > ? ) who arrived toan expression involving a triple integral. An earlier related result for bond prices given in termsof an integral over modified Bessel functions was formulated by Dothan in ( ? ). As an alterna-tive, Geman and Yor ( ? ) derive a closed-form expression for the Laplace transform in termsof confluent hypergeometric functions (see Donati-Martin et al. ( ? ) and Yor ( ? ) for furtherreferences). For applications to finance, one needs to compute the inverse Laplace transform,for which numerical methods have been developed by Geman and Eydeland ( ? ), Fu et al. ( ? ),Craddock et al. ( ? ), Shaw ( ? ). Dufresne in ( ? ) and Linetsky in ( ? ) develop analytical methodsand alternative expansions.In this article, we obtain far reaching extensions of the representation formula for the Laplacetransform for the integrals of stochastic processes over geometric Brownian motions. The keyidea is to seek expansions of similar form as those in ( ? ) and ( ? ) but expressed in terms ofmore general hypergeometric functions, see ( ? ), as pioneered by Wong in ( ? ). Let us recall thatgeneral hypergeometric functions are denoted as follows:(1.6) p F q ( α , . . . , α p ; γ , . . . , γ q ; z )for p ≤ q + 1 , γ j ∈ C \ − Z + , and are represented by the following Taylor expansion around z = 0:(1.7) p F q ( α , . . . , α p ; γ , . . . , γ q ; z ) = ∞ (cid:88) n =0 ( α ) n . . . ( α p ) n ( γ ) n . . . ( γ q ) n z n n ! . The Kummer functions in the work by Geman and Yor ( ? ) are in the family F of the so calledconfluent hypergeometric functions. Gaussian hypergeometric functions are in the family F and admit the functions of type F as limits. Both Gaussian and confluent hypergeometricfunctions solve differential equations of the Fuchsian class, see ( ? ). Specifically(1.8) z (1 − z ) F (cid:48)(cid:48) ( α, β ; γ ; z ) + ( γ − (1 + α + β ) z ) F (cid:48) ( α, β ; γ ; z ) − αβ F ( α, β ; γ ; z ) = 0and(1.9) z F (cid:48)(cid:48) ( α ; γ ; z ) + ( γ − z ) F (cid:48) ( α ; γ ; z ) − α F ( α ; γ ; z ) = 0 . In general, higher order hypergeometric functions are not associated to a differential equation.However, in some particularly important cases, they provide solutions of finite difference equa-tions. The Askey classification scheme, see ( ? ) and ( ? ), gives a complete list of all orthogonalpolynomials solving either a differential or a finite difference equation and in addition satisfy arecurrence relation. All of these polynomials descend as particular or limiting cases from theso-called Racah polynomials, which are particular cases of the hypergeometric functions F .We first consider the case of diffusion processes and next the case of finite state Markov pro-cesses. In the diffusion case, we construct a classification scheme based on reduction to eigenvalueproblems admitting solutions within the class of Gaussian and confluent hypergeometric func-tions F and F . In the second case, the problem is more difficult for several reasons, as there isno discrete equivalent of a theory of Fuchs type equations and, in addition, the groups of confor-mal transformations and diffeomorphisms do not extend to lattices. What we do in the discrete APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 3 case is to take the moves from the Askey classification scheme for orthogonal polynomials andshow how to extend the previous spectral decomposition to the case of Meixner, dual Hahn andRacah polynomials, which are special cases of F , F and F hypergeometric functions.For a diffusion process, on a domain D x ⊂ R , of the form(1.10) dX t = µ ( X t ) dt + σ ( X t ) dW t , we define the transitional probability density p T − t ( x, y ) as the density of the Markov semigroupof the process X t :(1.11) E P (cid:2) f ( X T ) F t (cid:3) = (cid:90) D f ( y ) p T − t ( x, y ) dy. We are interested in building a classification scheme for the drift and volatility functions µ ( x ) , σ ( x )such that the calculation of both functions p and L can be reduced to computing an integralover hypergeometric functions. The transitional probability density and Laplace transform canbe computed in terms of the spectral resolution for the infinitesimal generator of the process X t (1.12) L = σ ( x ) ∂ ∂x + µ ( x ) ∂∂x and the Feynman-Kac operator(1.13) ˜ L = L − ϑφ ( x ) = σ ( x ) ∂ ∂x + µ ( x ) ∂∂x − ϑφ ( x ) . In fact, we have that(1.14) p T − t ( x, y ) = e ( T − t ) L ( x, y )and(1.15) L T − t ( x, ϑ ) = (cid:90) D q ( y ) e ( T − t ) ˜ L ( x, y ) dy. As we show in detail in Section 2, these operators are conjugated by a non-singular transformationto self-adjoint operators which admit a spectral resolution. The calculation of the transitionalprobability density in (1.14) and the Laplace transform in (1.15) is thus reduced to the resolutionof the differential eigenvalue problems(1.16) L f ( x ) = λ f ( x ) , ˜ L ¯ f ( x ) = ¯ λ ¯ f ( x ) . To properly define the classification scheme, we specify by what means the reduction can beaccomplished.
Definition 1.
The problem of finding the transitional probability density and the Laplace trans-form for the process in (1.10) is said to be reducible to a spectral integral over hypergeometricfunctions if the two eigenvalue problems in (1.16) can be recast in the form of a differentialequation for hypergeometric functions such as either (1.9) or (1.8), by means of a combinationof the following three operations T i : (1) T Z : change of variable x (cid:55)→ z = Z ( x ) where Z ( x ) is a diffeomorphism Z : D x → D z such that (1.17) L x (cid:55)→ L z , ˜ L x (cid:55)→ ˜ L z , (2) T h : gauge transformation associated to a strictly positive function h such that (1.18) L (cid:55)→ h − L h , ˜ L (cid:55)→ h − ˜ L h, (3) T γ : left-multiplication by a strictly positive function γ such that (1.19) L (cid:55)→ γ L , ˜ L (cid:55)→ γ ˜ L . The third kind of transformations was first recognized in its generality by Natanzon in thearticle ( ? ) on integrable Schr¨odinger equations, see also Milson’s paper ( ? ). The followingtheorem gives a concise statement of our main classification result: CLAUDIO ALBANESE AND STEPHAN LAWI
Theorem 2 (First Classification Theorem) . The most general reducible diffusion process (up todiffeomorphism) according to Definition 1 can be constructed as follows: (1) four second order polynomials in x : A ( x ) , Q ( x, ϑ ) , R ( x ) , S ( x ) , such that A ( x ) belongsto the set { , x, x (1 − x ) , x + 1 } and R ( x ) ≥ ; (2) conditions for the stochastic process X t on the boundary of the domain D x , specifyingthe relative probability of reflection versus absorption upon hitting the boundary; (3) a solution of the following equation in D x for some ξ ∈ R : (1.20) A ( x ) R ( x ) h (cid:48)(cid:48) ( x ) + S ( x ) R ( x ) h ( x ) = ξh ( x ) The function h ( x ) is a linear combination of hypergeometric functions of the confluenttype F if A ( x ) ∈ { , x } and of the Gaussian type F if A ( x ) ∈ { x (1 − x ) , x + 1 } .The process associated to this choice is given by a generic solution to the following stochasticdifferential equation on the domain D x : (1.21) dX t = 2 h (cid:48) ( X t ) h ( X t ) A ( X t ) R ( X t ) dt + √ A ( X t ) (cid:112) R ( X t ) dW t , with the boundary conditions above. The Laplace transform is specified by (1.22) φ ( x ) = Q ( x, ϑ ) ϑR ( x ) . The proof of this theorem is in Section 2. These constructs are based on spectral analysistechniques for which we refer to the book by Reed and Simon ( ? ). We also make the spectralanalysis more explicit and list the expressions for the transitional probability density and Laplacetransforms in terms of the kernel of semigroups generated by integrable quantum Schr¨odingeroperators. In Section 3, we specialize further and re-discover the known cases of processes builtupon the geometric Brownian motion and on the Ornstein-Uhlenbeck and CIR processes, alongwith some interesting extensions.In Sections 4 and 5, we restrict the framework to the special case of hypergeometric polyno-mials. In the discrete case, studied in Section 5, the process X t takes on only a discrete set ofvalues, as opposed to following a diffusion process. In this class of models, we base our analysison the Askey-Wilson theory of orthogonal polynomials, see ( ? ) and ( ? ). We briefly review thebasic notions, following ( ? ). Definition 3. An orthogonal system of polynomials is given by a sequence of polynomials Q n ( x ) of order n for n ∈ N on the interval D ⊆ R which satisfies an orthogonality condition of theform (1.23) (cid:90) D Q n ( x ) Q m ( x ) ρ ( dx ) = d n δ nm , n, m ∈ N , where the d n are constants and ρ ( dx ) is a given measure. One distinguishes between continuouspolynomials whereby ρ ( dx ) is absolutely continuous with respect to the Lebesgue measure, i.e. (1.24) ρ ( dx ) = w ( x ) dx for some weight function w ( x ) , and discrete polynomials for which (1.25) ρ ( dx ) = N (cid:88) i =0 w ( x i ) δ ( x − i ) , N ∈ N . All orthogonal polynomials satisfy a three-term recurrence relation of the form(1.26) xQ n ( x ) = A n Q n +1 ( x ) − B n Q n ( x ) + C n Q n − ( x )where n ≥ A n > C n ≥ B n ∈ R . Together with the conditions Q − ( x ) = 0 and Q ( x ) = 1, all the Q n ( x ) can be determined based on this recurrence relation. The converse APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 5 is also true and is known as the Favard theorem, see ( ? ). Moreover, they satisfy the followingeigenvalue equation,(1.27) L Q n ( x ) = λ n Q n ( x ) , for L a second-order differential operator in the continuous case or a finite difference operatorin the discrete case.The reducibility condition in Definition 1 is mirrored by the following (inequivalent) one whichrefers to orthogonal polynomials in the continuous series as opposed to Gaussian hypergeometricfunctions: Definition 4.
The problem of finding the transitional probability density in (1.14) and theLaplace transform in (1.15) is said to be reducible to a spectral integral over orthogonal poly-nomials if the two eigenvalue problems in (1.16) have the same orthogonal polynomials as eigen-functions.
We first restrict the framework to continuous orthogonal polynomials that have as generatora second-order differential operator,(1.28) L = σ A ( x ) ∂ ∂x + ( a − bx ) ∂∂x , acting on the Hilbert space L ( D, ρ ), where A ( x ) ∈ { , x, x (1 − x ) } , a ∈ R and b, σ >
0. Thisclass consists of the Hermite, Laguerre and Jacobi polynomials up to diffeomorphism. Our mainresult concerning the latter continuous orthogonal polynomials can be stated as follows:
Theorem 5 (Second Classification Theorem) . The most general reducible diffusion process (upto diffeomorphism) in the sense of Definition 4, has infinitesimal generator L given by (1.28).Its transitional probability density can be expressed as (1.29) p T − t ( x, y ) = ∞ (cid:88) n =0 e λ n ( T − t ) d n Q n ( x ; a, b ) Q n ( y ; a, b ) w ( y ) . Furthermore, for some parameters ¯ a, ¯ b, C ∈ R , (1.30) φ ( x ) = C + (cid:0) ( a − bx ) − (¯ a − ¯ bx ) (cid:1) A (cid:48) ( x )2 ϑA ( x ) + (¯ a − ¯ bx ) − ( a − bx ) ϑσ A ( x ) and the Laplace transform is given by the following convergent series: (1.31) L T − t ( x, ϑ ) = exp (cid:18)(cid:90) x (¯ a − ¯ by ) − ( a − by ) σ A ( y ) dy (cid:19) ∞ (cid:88) n =0 e ¯ λ n ( T − t ) z n Q n ( x ; ¯ a, ¯ b ) . The coefficients z n are given by: (1.32) z n = 1¯ d n (cid:90) D x q ( x ) exp (cid:18) − (cid:90) x (¯ a − ¯ by ) − ( a − by ) σ A ( y ) dy (cid:19) Q n ( x ; ¯ a, ¯ b )¯ ρ ( dx ) . In Section 4 we present the proof of this alternative classification scheme based on orthogonalpolynomials of the continuous series. This discussion sets the premise for the extension of theresult to orthogonal polynomials in the discrete series. Discrete orthogonal polynomials arecharacterized by a finite difference generator on the Hilbert space l (Λ N , w ) with Λ N the set { , . . . , N } . Definition 6.
Let ∆ h and ∇ h + denote the difference operators defined as follows: (1.33) ∆ h y ( x ) = y ( x + h ) − y ( x ) + y ( x − h ) , ∇ h + y ( x ) = y ( x + h ) − y ( x ) . Using these operators, the finite difference generator L takes the form(1.34) L = − D ( x )∆ + (cid:0) D ( x ) − B ( x ) (cid:1) ∇ , CLAUDIO ALBANESE AND STEPHAN LAWI where B ( x ) and D ( x ) are rational functions of at most fourth order in the numerator and atmost second order in the denominator. Our main result in the discrete case can be stated asfollows: Theorem 7 (Third Classification Theorem) . The most general reducible discrete Markov processin the sense of Definition 4, has infinitesimal generator L given by (1.34). Its transitionalprobability density can be expressed as (1.35) p T − t ( x, y ) = N (cid:88) n =0 e λ n ( T − t ) d n Q n ( x ) Q n ( y ) w ( y ) .φ ( x ) must be of the form (1.36) φ ( x ) = B ( x ) + D ( x ) − ¯ B ( x ) − ¯ D ( x ) , where ¯ B ( x ) and ¯ D ( x ) are the same rational functions as B ( x ) and D ( x ) up to a multiplicativeconstant, but for different parameters, and satisfy the lattice condition (1.37) ¯ B ( x −
1) ¯ D ( x ) = B ( x − D ( x ) . The Laplace transform is given by the convergent series ( (cid:89) k =1 = 1 by convention): (1.38) L T − t ( x,
1) = x (cid:89) k =1 D ( k )¯ D ( k ) N (cid:88) n =0 e ¯ λ n ( T − t ) z n ¯ Q n ( x ) . The coefficients z n are as follows: (1.39) z n = 1¯ d n (cid:88) x ∈ Λ N x (cid:89) k =1 ¯ D ( k ) D ( k ) q ( x ) ¯ Q n ( x ) ¯ w ( x ) . In our notation, { ¯ Q n ( x ) } is the same set of orthogonal polynomials as { Q n ( x ) } expect for thevalue of the parameters. Section 5 gives a proof of the latter result, as well as explicit represen-tations for processes based on the Meixner, the dual Hahn and the Racah polynomials.This paper is organized as follows: Section 2 presents the proof of the first classificationscheme in the diffusion case and goes in more detail to provide a spectral representation formulafor transitional probability densities and the Laplace transform. Section 3 contains a discussionof the classical examples showing how the geometric Brownian motion, the Ornstein-Uhlenbeckand the CIR process fit in this classification scheme. Section 4 contains the proof of an alterna-tive classification scheme based on orthogonal polynomials of the continuous series, along withinteresting examples. This discussion sets the premise for Section 5 where we extend the resultto orthogonal polynomials in the discrete series. Finally, in Section 6 we discuss limiting rela-tion and establish connections between models corresponding to the discrete and the continuousseries, which is a useful result to construct numerical very stable discretization schemes. APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 7 Classification Theorem for Diffusion Processes
In this section, we prove our first classification result, Theorem 2, in the diffusion case. Westart by reviewing some background notions concerning Fuchsian differential equations and theso-called Bose invariants, and then proceed to the proof of the theorem.2.1.
Fuchsian Differential Equations.
Consider the second order partial differential equa-tions for the holomorphic function F ( z )(2.1) F (cid:48)(cid:48) ( z ) + p ( z ) F (cid:48) ( z ) + q ( z ) F ( z ) = 0for some holomorphic functions p ( z ) , q ( z ). Definition 8.
Let α ∈ C be an isolated singularity for the holomorphic function F ( z ) . Thesingularity in α is called regular if there is an exponent ρ ∈ C for which the function ( z − α ) − ρ F ( z ) admits a Laurent expansion with finitely many negative powers around z = α , i.e. (2.2) F ( z ) = ( z − α ) ρ (cid:88) n = − m ( z − α ) n for some m ∈ N . The point ∞ is a regular singularity of the function F ( z ) if z = 0 is a regularsingularity of the function F (cid:0) z (cid:1) . In ( ? ), Fuchs gives conditions on the coefficients p ( z ) and q ( z ) which ensure that solutionshave only regular singularities. Theorem 9 (Fuchs) . Let F ( z ) be a solution of equation (2.1) with singularities in the points α , . . . , α n and ∞ . Then these singularities are all regular if and only if the functions p ( z ) and q ( z ) have the form (2.3) p ( z ) = p ( z )( z − α ) . . . ( z − α n ) and (2.4) q ( z ) = q ( z )( z − α ) . . . ( z − α n ) where p ( z ) is a polynomial of order ( n − and q ( z ) is a polynomial of order n − . An alternative expression for the coefficient p ( z ) of an equation with only regular singularitiesis(2.5) p ( z ) = n (cid:88) i =1 δ i z − α i where the δ i , i = 1 , ..n , are constants. In particular, we have that(2.6) exp (cid:18) (cid:90) z p ( w ) dw (cid:19) = C n (cid:89) i =1 ( z − α i ) δi where C is a constant. The function(2.7) ¯ F ( y ) = n (cid:89) i =1 ( y − α i ) δi F ( y )solves the equation(2.8) ¯ F (cid:48)(cid:48) ( y ) + I ( y ) ¯ F ( y ) = 0where(2.9) I ( y ) = − p (cid:48) ( y ) + 14 p ( y ) + q ( y ) . CLAUDIO ALBANESE AND STEPHAN LAWI
Definition 10.
The function I ( y ) is called the Bose invariant of the equation (2.1). Notice that I ( y ) has the form (2.10) I ( y ) = I ( y )( y − α ) . . . ( y − α n ) where I ( y ) is a polynomial of order n − , without restrictions on the coefficients. Let us focus again on the case n = 2, assume that coefficients are real and that only real linear fractional transformations are allowed to move the regular singularities. In this situationwe have to distinguish between two different cases for Bose invariants: • Case I ( α = 0 , α = 1):(2.11) I ( y ) = s (1 − y ) + s y + s y (1 − y ) y (1 − y ) • Case II ( α = i, α = − i ):(2.12) I ( y ) = s + s y + s y ( y + 1) These cases reduce to the Gaussian hypergeometric equation (2.18) for the function F as isshown below. Furthermore, special cases for the Bose invariant occur in the limit when either α or α or both roots tend to ∞ , i.e. • Case III ( α = 0 , α = ∞ ):(2.13) I ( y ) = s + s y + s y y • Case IV ( α = ∞ , α = ∞ ):(2.14) I ( y ) = s + s y + s y Case III reduces to the confluent hypergeometric equation F and Case IV corresponds to thecase of triple confluence at infinity. Notice that the above four cases can all be captured by asingle expression as stated in the following: Remark 11.
The Bose invariants corresponding to the Gaussian hypergeometric function F and to its confluent limit can be reduced to the following normal form by means of a real valuedlinear fractional transformation: (2.15) I ( y ) = Q ( y ) A ( y ) where A ( y ) ∈ { y (1 − y ) , y + 1 , y, } and Q is a polynomial in y with deg Q ≤ . The first two cases correspond to three regular singularities at distinct points. In these cases,solutions can be expressed through Gaussian hypergeometric functions F . Fractional lineartransformations of the form(2.16) z (cid:55)→ az + bcz + d where a, b, c, d ∈ C and ad − bc (cid:54) = 0, are one-to-one maps of the extended complex line C ∪ ∞ into itself and map regular singularities into regular singularities. By applying a fractional lineartransformation, one can map the singularities α and α to 0 and 1, respectively. Furthermore,we have transformations of the form(2.17) F ( z ) (cid:55)→ ( z − α ) ρ ( z − α ) ρ F ( z ) . The combination of these two transformations allows one to reduce any Fuchsian differentialequation with three regular singular points to the form(2.18) z (1 − z ) F (cid:48)(cid:48) ( z ) + ( γ − (1 + α + β ) z ) F (cid:48) ( z ) − αβF ( z ) = 0 . The function F ( α, β ; γ ; z ) is an elementary solution of this equation along with F ( α, β ; 1 + α + β − γ ; 1 − z ). APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 9
Case III corresponds to the limit when a regular singularity merges with the regular singularityat ∞ while the other one stays at 0. This limit can be obtained starting from the equationcorresponding to two coinciding singularities at 0, i.e. α = α = 0:(2.19) F (cid:48)(cid:48) ( z ) + c + c zz F (cid:48) ( z ) + c + c z + c z z F ( z ) = 0 . By applying the coordinate transformation z (cid:55)→ z we find that(2.20) z F (cid:48)(cid:48) ( z ) + ((2 + c ) z + c z ) F (cid:48) ( z ) + ( c + c z + c z ) F ( z ) = 0By rescaling the independent variable z and rescaling the function so that F ( z ) (cid:55)→ e ρz f ( ωz ) thisequation reduces to the Kummer differential equation(2.21) zF (cid:48)(cid:48) ( z ) + ( γ − z ) F (cid:48) ( z ) − αF ( z ) = 0which admits F ( α ; γ ; z ) as a solution. In alternative, one can reduce equation (2.20) to theform(2.22) F (cid:48)(cid:48) ( z ) + (cid:18) −
14 + λz + − µ z (cid:19) F ( z ) = 0which is called the Whittaker differential equation . The case where all three singularities mergeat ∞ is also interesting and can be solved by rescaled confluent hypergeometric functions.2.2. Proof of the First Classification Theorem 2.
We start by presenting obvious factsabout the transformations T i in Definition 1. Remark 12.
The transformations T i are invertible with respective inverse: (2.23) T − Z = T X , T − h = T h , T − γ = T γ where X : D z → D x is the inverse of Z ( x ) . Remark 13.
The transformations T i commute with one another. The proof of the First Classification Theorem 2 for diffusion processes follows. The process X t has infinitesimal generator(2.24) L = A ( x ) R ( x ) ∂ ∂x + 2 h (cid:48) ( x ) h ( x ) A ( x ) R ( x ) ∂∂x . To solve the eigenvalue problem L f = λf , define the following left-multiplication T γ and gaugetransformation T h :(2.25) T γ L = A ( x ) R ( x ) L and T h L = 1 h L h, so that the eigenvalue equation then transforms into(2.26) T − γ T − h : L f = λf (cid:55)→ (cid:18) ∂ ∂x + S ( x ) A ( x ) (cid:19) f ( x ) = λ R ( x ) A ( x ) f ( x ) . As R ( x ) , S ( x ) are second order polynomials and A ( x ) ∈ { , x, x (1 − x ) , x } , the solution f ( x )can be expressed as a hypergeometric function with Bose invariant(2.27) I ( x ) = S ( x ) − λR ( x ) A ( x ) . The same operations can be applied to the eigenvalue problem for the Feynman-Kac operator,˜ L ¯ f = ¯ λ ¯ f with(2.28) ˜ L = L − ϑφ ( x ) , which leads to(2.29) T − γ T − h : ˜ L ¯ f = ¯ λ ¯ f (cid:55)→ (cid:18) ∂ ∂x + S ( x ) A ( x ) − ϑφ ( x ) (cid:19) ¯ f ( x ) = ¯ λ R ( x ) A ( x ) ¯ f ( x ) . For ¯ f ( x ) to be expressed as a hypergeometric function, we require(2.30) φ ( x ) = Q ( x, ϑ ) ϑA ( x ) for Q ( x, ϑ ) a second order polynomial in x . The Bose invariant for the equation is(2.31) ˜ I ( x ) = S ( x ) − Q ( x, ϑ ) − ¯ λR ( x ) A ( x ) . The converse follows from the facts that the transformations are invertible and the Bose invari-ants are both expressed in the most general form.
Remark 14.
The first transformation ( T Y : change of variable) has not been used in the proof. X t is therefore the most general reducible diffusion process only up to diffeomorphism. Spectral Resolutions.
Theorem 2 describes all processes with explicitly solvable transi-tional probability density and Laplace transform for the integral of the process. We wish now togive a closed form expression for both quantities. The following lemma allows one to determinethe nature of the spectrum of the operators L and ˜ L . The nature of the spectrum is indeed basedon the shape of the Schr¨odinger potential, coming out of the eigenvalue problem once reducedto a Schr¨odinger equation. Theorem 16 shows how this transformation operates on the kernelof the semigroup generated by the Schr¨odinger operators and gives a general spectral resolutionof the operators L and ˜ L . Lemma 15.
Let T = T g T Z T − h where the diffeomorphism Z : D x → D z is given by Z (cid:48) ( x ) = (cid:112) R ( x ) A ( x ) with inverse X and the gauge transformation T g by g ( z ) = (cid:18) A ( X ( z )) R ( X ( z )) (cid:19) / . Then theoperators L and ˜ L reduce to the following Schr¨odinger operators: T L = ∂ ∂z − U ( z ) ≡ − H T ˜ L = ∂ ∂z − U ( z ) ≡ − H (2.32) where the potentials are given by U ( z ) = (cid:18) g (cid:48) g (cid:19) − (cid:18) g (cid:48) g (cid:19) (cid:48) − S ( X ( z )) R ( X ( z ))˜ U ( z ) = (cid:18) g (cid:48) g (cid:19) − (cid:18) g (cid:48) g (cid:19) (cid:48) − S ( X ( z )) − Q ( X ( z ) , ϑ ) R ( X ( z ))(2.33) and (cid:48) denotes the derivative with respect to z .Proof. T g T Z T − h L = T g T Z (cid:18) A ( x ) R ( x ) ∂ ∂x + S ( x ) R ( x ) (cid:19) = T g (cid:18) ∂ ∂z + Z (cid:48)(cid:48) ( Z (cid:48) ) ∂∂z + S ( X ( z )) R ( X ( z )) (cid:19) = ∂ ∂z + g (cid:48)(cid:48) g − (cid:18) g (cid:48) g (cid:19) + S ( X ( z )) R ( X ( z ))and T g T Z T − h ˜ L = T g T Z T − h L − ϑφ ( X ( z )) . (cid:3) The two Schr¨odinger operators H and H defined in the previous lemma have a spectralresolution,(2.34) H i Φ ρ ( z ) = ρ Φ ρ ( z ) , APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 11 given by a complete set of normalized eigenfunctions Φ ρ ( z ) for ρ = − λ or ρ = − ¯ λ , i = 1 , σ pp ( H i ) and anabsolutely continuous spectrum σ ac ( H i ). The kernel of the semigroup generated by the respectiveSchr¨odinger operators has the general form(2.35) e − ( T − t ) H i ( z , z ) = (cid:88) ρ ∈ σ pp ( H i ) e − ( T − t ) ρ Φ ρ ( z )Φ ∗ ρ ( z ) + (cid:90) ρ ∈ σ ac ( H i ) e − ( T − t ) ρ Φ ρ ( z )Φ ∗ ρ ( z ) dk ( ρ )with dk ( ρ ) = dρ √ ρ − U − i and U − i the lowest limit of the potential U i ( z ) as z tends to the boundariesof the domain D z . Theorem 16.
The transitional probability density and the Laplace transform of any reducibleprocess described in Theorem 2 by the operator L and ˜ L is related to the kernels of the semigroupsgenerated by the respective Schr¨odinger operators as follows: e ( T − t ) L ( x, y ) = h ( y ) h ( x ) (cid:18) A ( x ) R ( x ) (cid:19) / (cid:18) R ( y ) A ( y ) (cid:19) / e − ( T − t ) H (cid:0) Z ( x ) , Z ( y ) (cid:1) ,e ( T − t ) ˜ L ( x, y ) = h ( y ) h ( x ) (cid:18) A ( x ) R ( x ) (cid:19) / (cid:18) R ( y ) A ( y ) (cid:19) / e − ( T − t ) H (cid:0) Z ( x ) , Z ( y ) (cid:1) . (2.36) Proof.
The transformation T defined in Lemma 15 extends to the kernels as follows: e ( T − t ) L ( x, y ) = e − ( T − t ) T h T − Z T − g H ( x, y )= h ( y ) h ( x ) dZdy g ( Z ( x )) g ( Z ( y )) e − ( T − t ) H (cid:0) Z ( x ) , Z ( y ) (cid:1) and similarly for ˜ L and H . Recall that g ( Z ( x )) = (cid:18) A ( x ) R ( x ) (cid:19) / and the Jacobian of T Z is givenby dZdy = (cid:112) R ( y ) A ( y ) , which concludes the proof. (cid:3) Examples of Solvable Diffusions
In this section, we show that the transitional probability density and the Laplace transformfor the integral of the geometric Brownian motion, the Ornstein-Uhlenbeck process and the CIRprocess arise as corollaries of the First Classification Theorem 2.3.1.
The geometric Brownian motion.Definition 17.
The geometric Brownian motion is defined by the solution of the followingstochastic differential equation: (3.1) dX t = µX t dt + σX t dW t with initial condition X t =0 = x . Corollary 18.
The transitional probability density for the geometric Brownian motion is givenby the following formula: (3.2) p T − t ( x, y ) = 1 y (cid:112) πσ ( T − t ) exp − (cid:16) ln( yx ) − ( µ − σ )( T − t ) (cid:17) σ ( T − t ) . The Laplace transform is explicitly solvable if and only if (3.3) φ ( x ) = σ ϑ (cid:16) µσ (cid:16) − µσ (cid:17) − t − t x + t x (cid:17) , where t , t ∈ R and t > could depend on ϑ . It is then given in terms of the Laguerrepolynomials L ( δ ) n and the Whittaker function M λ,µ (by convention, N (cid:88) n =0 = 0 if N < ): L T − t ( x, ϑ ) = x − µσ + e −√ t x N (cid:88) n =0 z n e ( T − t ) λ n x δn L ( δ n ) n (2 √ t x )+ x − µσ (cid:90) ∞ z k e − ( T − t )( k + U − ) M t √ t , δk (2 √ t x ) dk (3.4) where N ≡ (cid:100) t √ t − (cid:101) ( (cid:100) t (cid:101) denotes the integer part of t ), δ n ≡ − n − t √ t , U − ≡ σ ( − t ) and δ k ≡ i (cid:113) σ k . For n = 0 , , . . . , N , the discrete eigenvalues are given by (3.5) λ n = σ (cid:32)(cid:18) n + 12 − t √ t (cid:19) + t − (cid:33) . The coefficients z n and z k are respectively (3.6) z n = (2 √ t ) δ n Γ( δ n ) (cid:90) ∞ q ( x ) x µσ + δn − e −√ t x L ( δ n ) n (2 √ t x ) dx and (3.7) z k = 12 π (cid:114) σ t (cid:90) ∞ q ( x ) x µσ − M t √ t , − δk (2 √ t x ) dx. Proof.
The infinitesimal generator is(3.8) L = σ x ∂ ∂x + µx ∂∂x whereas the Feynman-Kac operator has the form(3.9) ˜ L = σ x ∂ ∂x + µx ∂∂x + ϑφ ( x ) . APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 13
This case fits the classification scheme in Theorem 2 if one selects(3.10) A ( x ) = x , R ( x ) = 2 σ , h (cid:48) h = µσ x . This choice sets the shape of the polynomial S ( x ), from (1.20) in Theorem 2, to be(3.11) S ( x ) = 1 σ (cid:16) ξ + µ (cid:0) − µσ (cid:1)(cid:17) , which in turn defines the Bose invariant as(3.12) I ( x ) = 2( ξ − λ ) + µ (cid:0) − µσ (cid:1) σ x . The Schr¨odinger potential, given by(3.13) U ( z ) = σ − ξ − µ (cid:16) − µσ (cid:17) , is constant for all z . Hence the spectrum is absolutely continuous. The normalized eigenfunctionsfor the Schr¨odinger equations (2.34) are(3.14) Φ ρ ( z ) = 1 √ π e ± ik ( ρ ) z where k = ρ − σ + µ (cid:0) − µσ (cid:1) . Theorem 16 yields the kernel for the semigroup generated bythe operator L :(3.15) e ( T − t ) L ( x, y ) = 1 y (cid:112) πσ ( T − t ) exp − (cid:16) ln( yx ) − ( µ − σ )( T − t ) (cid:17) σ ( T − t ) which is p T − t ( x, y ) and in which one recognizes the transitional probability density of the geo-metric Brownian motion.The kernel of the semigroup generated by ˜ L is more general, since there is no restrictions onthe polynomial Q ( x, ϑ ), which, for t , t , t ∈ R , can be written in the form(3.16) Q ( x, ϑ ) = S ( x ) − t − t x + t x . The Bose invariant in this case is(3.17) ˜ I ( x ) = − t + t x + t − λσ x which gives rise to two independent solutions to the eigenvalue problem u (cid:48)(cid:48) + Iu = 0: u ± ( x ) = M t √ t , ± δ (cid:0) √ t x (cid:1) = (2 √ t x ) ± δ e −√ t x F (cid:32) ± δ +12 − t √ t ± δ + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ t x (cid:33) (3.18)where δ = − t + λσ . The Schr¨odinger potential, given by(3.19) ˜ U ( z ) = σ (cid:18) − t − t e q σ z + t e q σ z (cid:19) , has no singularities and is bounded from below if t >
0. It indicates that the spectrum is notstrictly discrete, since ˜ U ( z ) → ∞ as z → ∞ but ˜ U ( z ) → U − ≡ σ ( − t ) as z → −∞ . Hence,the description of the spectrum separates in two cases: • If t ≤
0, then ˜ U ( z ) is monotonously increasing on D z = R and the spectrum is contin-uous: ρ = − λ ≥ U − . • If t >
0, then ˜ U ( z ) has a minimum, U ≡ U − − σ t t , at z = (cid:113) σ ln t t and thespectrum is discrete for U < ρ < U − and continuous for ρ ≥ U − . Both cases can however be solved simultaneously. The solution to the Schr¨odinger equation,Φ ρ ( z ), is related to u ± ( x ), via the diffeomorphism x = X ( z ) = e q σ z and the gauge transfor-mation g ( z ) = ( σ ) e q σ z , as follows:(3.20) Φ ρ ( z ) = g − ( z ) u ± ( X ( z )) . The asymptotic behavior as z → −∞ , Φ ρ ( z ) ≈ √ π e ± ikz with k = (cid:112) ρ − U − ≥
0, enforces thenormalization condition for Φ ρ ( z ) to the following:(3.21) Φ ρ ( z ) = 1 √ π (2 √ t ) − ± δ e − q σ z u ± ( X ( z )) . The continuous spectrum appears for ρ greater than the lowest limit U − , implying that δ k ≡ δ ( k ) = i (cid:113) σ k is imaginary. Whether or not the discrete part of the spectrum has an infinitenumber of discrete levels depends on the asymptotic behavior of the potential. As z → −∞ , x → U ( Z ( x )) develops up to second order:(3.22) ˜ U ( Z ( x )) = σ (cid:18) − t − t x + t x (cid:19) . Since t > U − as x → x , which implies that there is only a finite number of bound states.The hypergeometric functions in the solutions u ± ( x ) reduce to polynomials for x ∈ [0 , ∞ ) ifrespectively(3.23) ± δ ( n ) + 12 − t √ t = − n ∈ − N . We set δ n ≡ δ ( n ) to emphasize the dependance on n . The solution u − ( x ) is not L -normalizable,whereas u + ( x ) is given in terms of the Laguerre polynomials(3.24) u n ( x ) = C (2 √ t x ) δn +12 e −√ t x L ( δ n ) n (2 √ t x )with normalization constant C . The normalization condition (cid:90) ∞−∞ | Φ n ( z ) | dz = 1 fixes theconstant C = (cid:115) σ √ t Γ( δ n ) (cf. ( ? ), p. 462). The discrete eigenvalues ρ = − λ n are given by(3.5), where n is restricted to { , , . . . , N } by the condition − λ n > U , and N ≡ (cid:100) t √ t − (cid:101) .Hence, from Theorem 16, the kernel of the semigroup generated by ˜ L is given by the followingspectral resolution: e ( T − t ) ˜ L ( x, y ) = x − µσ + y µσ − e −√ t ( x + y ) (3.25) · N (cid:88) n =0 e ( T − t ) λ n (2 √ t ) δ n Γ( δ n ) ( xy ) δn L ( δ n ) n (2 √ t x ) L ( δ n ) n (2 √ t y )+ 12 π (cid:114) σ t x − µσ y µσ − · (cid:90) ∞ e − ( T − t )( k + U − ) M t √ t , δk (2 √ t x ) M t √ t , − δk (2 √ t y ) dk. Finally the Laplace transform of the latter kernel is given by integration over D x = [0 , ∞ ) andyields (3.4). (cid:3) APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 15
The Ornstein-Uhlenbeck process.
In this subsection, we restrict the framework to theaffine models, i.e. we set(3.26) q ( x ) = exp (cid:0) ωφ ( x ) (cid:1) for some ω ∈ R . We show that in the special case of the Ornstein-Uhlenbeck process, both thetransitional probability density and the Laplace transform can be expressed as summations overHermite polynomials. Definition 19.
The Ornstein-Uhlenbeck process is defined by the solution of the following sto-chastic differential equation: (3.27) dX t = ( a − bX t ) dt + σdW t with b > and initial condition X t =0 = x . Corollary 20.
The transitional probability density is given by the following formula: (3.28) p T − t ( x, y ) = (cid:114) bσ π (cid:16) − e − b ( T − t ) (cid:17) − / exp (cid:34) − (cid:0) z ( y ) − z ( x ) e − b ( T − t ) (cid:1) − e − b ( T − t ) (cid:35) where z ( x ) = (cid:113) bσ (cid:0) x − ab (cid:1) . The Laplace transform is explicitly solvable if and only if (3.29) φ ( x ) = σ ϑ (cid:18) bσ − a σ − t + 2 ab − ¯ a ¯ bσ x + ¯ b − b σ x (cid:19) , with t , ¯ a, ¯ b ∈ R and could depend on ϑ . It is of the quadratic form (3.30) L T − t ( x, ϑ, ω ) = e m ( T − t ) − n ( T − t ) x − l ( T − t ) x where the functions m ( τ ) , n ( τ ) and l ( τ ) are as follows m ( τ ) = 12 ln (cid:18) ¯ bσ (cid:19) −
12 ln (cid:18) p − ( p − ¯ bσ ) e − bτ (cid:19) − ¯ a σ ¯ b − τ (cid:18) ¯ b − ¯ a σ − σ t (cid:19) + ω ϑ (cid:18) b − a σ − σ t (cid:19) + p (cid:16) q + ¯ a ¯ b (cid:17) − pq ¯ aσ e − ¯ bτ + (cid:16) ( p − ¯ bσ ) ¯ a ¯ bσ + pq bσ (cid:17) e − bτ p − ( p − ¯ bσ ) e − bτ n ( τ ) = − ¯ a − aσ − pq ¯ bσ e − ¯ bτ + ( p − ¯ bσ )2 ¯ aσ e − bτ p − ( p − ¯ bσ ) e − bτ l ( τ ) = ¯ b − b σ + ( p − ¯ bσ ) ¯ bσ e − bτ p − ( p − ¯ bσ ) e − bτ with p = (¯ b + b )( ω ( b − ¯ b ) + ϑ )2 σ ϑ , q = ϑ ( a + ¯ a ) + ω ( ab − ¯ a ¯ b )(¯ b + b )( ϑ − ω (¯ b − b )) − ¯ a ¯ b . Proof.
The infinitesimal generator of the Ornstein-Uhlenbeck process is(3.31) L = σ ∂ ∂x + ( a − bx ) ∂∂x and the Feynman-Kac operator has the form(3.32) ˜ L = σ ∂ ∂x + ( a − bx ) ∂∂x − ϑφ ( x ) . This case fits the classification scheme in Theorem 2 if one selects(3.33) A ( x ) = 1 , R ( x ) = 2 σ , h ( x ) = e aσ x − b σ x . This choice implies that the shape of the polynomial S ( x ), from (1.20) in Theorem 2, must be(3.34) S ( x ) = 1 σ (2 ξ + b − a σ ) + 2 abσ x − b σ x , which in turn defines the Bose invariant as(3.35) I ( x ) = 1 σ (2 ξ − λ + b − a σ ) + 2 abσ x − b σ x . The Schr¨odinger potential, given by(3.36) U ( z ) = − ξ − b a σ − ab √ σ z + b z with z = (cid:113) σ x , goes to ∞ as z → ±∞ . Hence, the spectrum displays discrete eigenvalues ofthe form λ n = − bn for n ∈ N . The corresponding normalized eigenfunctions that satisfy theSchr¨odinger equation (2.34) are(3.37) Φ n ( Z ( x )) = 1 √ n !2 n (cid:18) b π (cid:19) / e − b σ ( x − ab ) H n (cid:32)(cid:114) bσ (cid:0) x − ab (cid:1)(cid:33) . Theorem 16 yields the kernel for the semigroup generated by L as a summation over Hermitepolynomials. Using Mehler’s formula (cf. ( ? ), p. 710) and the notation z ( x ) = (cid:113) bσ (cid:0) x − ab (cid:1) ,the kernel re-sums into(3.38) e ( T − t ) L ( x, y ) = (cid:114) bσ π (cid:16) − e − b ( T − t ) (cid:17) − / exp (cid:34) − (cid:0) z ( y ) − z ( x ) e − b ( T − t ) (cid:1) − e − b ( T − t ) (cid:35) , which is p T − t ( x, y ) and in which one recognizes the probability density of the Ornstein-Uhlenbeckprocess.For convenience and without loss of generality, we set the form of the polynomial Q ( x, ϑ ) to(3.39) Q ( x, ϑ ) = S ( x ) − t − a ¯ bσ x + ¯ b σ x . where t , ¯ a, ¯ b could depend on ϑ . The Bose invariant in this case is(3.40) ˜ I ( x ) = t − λσ + 2¯ a ¯ bσ x − ¯ b σ x . The Schr¨odinger potential, given by(3.41) ˜ U ( z ) = − t − ¯ a ¯ bσ (cid:114) σ z + ¯ b z is very similar to U ( z ) and indicates that the spectrum is again discrete since ˜ U ( z ) → ∞ as z → ±∞ . The solution Φ n ( y ) is given in terms of the Hermite polynomials(3.42) Φ n ( Z ( x )) = 1 √ n !2 n (cid:18) ¯ b π (cid:19) / e − ¯ b σ ( x − ¯ a ¯ b ) H n (cid:32)(cid:114) ¯ bσ (cid:0) x − ¯ a ¯ b (cid:1)(cid:33) . for the eigenvalues λ n = − ¯ bn − ¯ b + ¯ a σ + σ t . From Theorem 16, the kernel of the semigroupgenerated by ˜ L is given by the following spectral resolution: e ( T − t ) ˜ L ( x, y ) = (cid:114) ¯ bσ π exp (cid:18) − ¯ a σ ¯ b + yσ (¯ a + a ) − y σ (¯ b + b ) + xσ (¯ a − a ) − x σ (¯ b − b ) (cid:19) · ∞ (cid:88) n =0 e λ n ( T − t ) n n ! H n (cid:32)(cid:114) ¯ bσ (cid:0) x − ¯ a ¯ b (cid:1)(cid:33) H n (cid:32)(cid:114) ¯ bσ (cid:0) y − ¯ a ¯ b (cid:1)(cid:33) . (3.43)The integration of the latter yields the Laplace transform for q ( x ) = exp( ωx ) as a convergentseries in terms of the Hermite polynomials which re-sums to the formula (3.30) (cf. ( ? ), p.488(16) and p. 710(1)). (cid:3) APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 17
Remark 21.
Computing the Laplace transform L in the case where φ ( x ) is affine, i.e. φ ( x ) = x ,is a direct consequence of the previous corollary. Setting the parameters to (3.44) t = bσ − (cid:16) aσ (cid:17) , ¯ a = a − σ ϑb , ¯ b = b proves the following proposition: Remark 22.
The Laplace transform for the affine Ornstein-Uhlenbeck process is as follows: (3.45) L T − t ( x, ϑ, ω ) = e m ( T − t ) − n ( T − t ) x where n ( τ ) = ϑ − ( ϑ + ωb ) e − bτ bm ( τ ) = (cid:0) n ( τ ) + ω − ϑτ (cid:1) ( ab − ϑ σ ) b − σ b (cid:0) n ( τ ) − ω (cid:1) . (3.46)3.3. The CIR process.
In this subsection, we focuss again on the affine models, i.e. we set(3.47) q ( x ) = exp( ωx )for some ω ∈ R . We show how to derive from the First Classification Theorem 2, the transitionalprobability density for the CIR process and the Laplace transform in the affine case. The maintool is the use of the Laguerre polynomials as eigenfunctions for both the infinitesimal generatorand the Feynman-Kac operator. Definition 23.
The CIR process is defined as the solution of the following stochastic differentialequation on D x = R + : (3.48) dX t = ( a − bX t ) dt + σ (cid:112) X t dW t with a, b > and initial condition X t =0 = x . Corollary 24.
The transitional probability density of the CIR process is given in terms of themodified Bessel function I α as follows: p T − t ( x, y ) = c (cid:18) ye b ( T − t ) x (cid:19) ( aσ − ) exp (cid:104) − c (cid:16) y + xe − b ( T − t ) (cid:17)(cid:105) I aσ − (cid:16) c (cid:112) xye − b ( T − t ) (cid:17) (3.49) with c ≡ c ( T − t ) = bσ (1 − e − b ( T − t ) ) − . The Laplace transform is computable in closed form ifand only if (3.50) φ ( x ) = (cid:18) a (cid:0) − aσ (cid:1) − σ t (cid:19) x + (cid:18) abσ − σ t (cid:19) + (cid:18) σ t − b σ (cid:19) x, where t , t , t ∈ R could depend on ϑ .In the affine case, where φ ( x ) = x , the Laplace transform is of the closed form (3.51) L T − t ( x, ϑ, ω ) = e m ( T − t ) − n ( T − t ) x where m ( τ ) = 2 aσ ln (cid:34) ¯ be bτ/ ¯ b cosh( ¯ bτ ) + ( b − ωσ ) sinh( ¯ bτ ) (cid:35) n ( τ ) = − ω + ¯ b − ( b − ωσ ) σ sinh( ¯ bτ )¯ b cosh( ¯ bτ ) + ( b − ωσ ) sinh( ¯ bτ )(3.52) and ¯ b = √ ϑσ + b . Proof.
The infinitesimal generator of the CIR process is(3.53) L = σ x ∂ ∂x + ( a − bx ) ∂∂x and the Feynman-Kac operator has the form(3.54) ˜ L = σ x ∂ ∂x + ( a − bx ) ∂∂x − ϑφ ( x ) . This case fits the classification scheme in Theorem 2 if one selects(3.55) A ( x ) = x , R ( x ) = 2 xσ , h ( x ) = x a/σ e − bσ x . This choice sets the form of the polynomial S ( x ), from (1.20) in Theorem 2, to be(3.56) S ( x ) = aσ (cid:16) − aσ (cid:17) + 2 σ (cid:18) ξ + abσ (cid:19) x − b σ x , which in turn defines the Bose invariant as(3.57) I ( x ) = − b σ + 2 σ (cid:18) ξ − λ + abσ (cid:19) x + aσ (cid:16) − aσ (cid:17) x . The Schr¨odinger potential, given for z = (cid:113) xσ by(3.58) U ( z ) = (cid:18) − aσ (cid:16) − aσ (cid:17)(cid:19) z − ξ − abσ + b z goes to ∞ as z → ∞ and has a singularity at z = 0. The asymptotic behavior as z → aσ (cid:0) − aσ (cid:1) ≤ λ n = − bn for n ∈ N . The corresponding normalized eigenfunctions thatsatisfy the Schr¨odinger equation (2.34) are given in terms of the Laguerre polynomials(3.59) Φ n ( Z ( x )) = (cid:115) n !Γ( n + aσ ) (cid:18) σ x (cid:19) / (cid:18) bxσ (cid:19) a/σ e − bσ x L ( aσ − n (cid:18) bσ x (cid:19) . Theorem 16 yields the kernel for the semigroup generated by the operator L as a summationover Laguerre polynomials, which can be re-summed into (cf. ( ? ), p. 705(7)): e ( T − t ) L ( x, y ) = c (cid:18) ye b ( T − t ) x (cid:19) ( aσ − ) exp (cid:104) − c (cid:16) y + xe − b ( T − t ) (cid:17)(cid:105) I aσ − (cid:16) c (cid:112) xye − b ( T − t ) (cid:17) (3.60)with c ≡ c ( T − t ) = bσ (1 − e − b ( T − t ) ) − and in which one recognizes the transitional probabilitydensity of the CIR process.For the Laplace transform L , we specialize to the case where φ ( x ) is affine, i.e. φ ( x ) = x .Therefore, (1.22) sets the shape of the polynomial Q ( x, ϑ ) to(3.61) Q ( x, ϑ ) = S ( x ) + 2 ϑσ x . The Bose invariant in this case is(3.62) ˜ I ( x ) = − b σ − ϑσ + 2 σ (cid:18) ξ − λ + abσ (cid:19) x + aσ (cid:16) − aσ (cid:17) x . whereas the Schr¨odinger potential, given by(3.63) ˜ U ( z ) = (cid:18) − aσ (cid:16) − aσ (cid:17)(cid:19) z − ξ − abσ + b + 2 ϑσ z corresponds to a discrete spectrum, following the same reasoning as for U ( z ). The solution Φ n ( z )is given in terms of the Laguerre polynomials(3.64) Φ n ( Z ( x )) = (cid:115) n !Γ( n + aσ ) (cid:18) σ x (cid:19) / (cid:18) bxσ (cid:19) a/σ e − ¯ bσ x L ( aσ − n (cid:18) bσ x (cid:19) with ¯ b = √ ϑσ + b and for the corresponding eigenvalues λ n = − ¯ bn − aσ (¯ b − b ). The Laplacetransform L in (1.15) is easily integrated and yields a convergent series in terms of the Laguerrepolynomials which re-sums to the famous formula (3.51) (cf. ( ? ), p. 462(3) and p. 705(7)). (cid:3) Processes related to continuous orthogonal polynomials
Proof of the Second Classification Theorem.
We give a constructive proof of Theorem5, independent of Theorem 2. In the following remark, we show that Theorem 5 can actually beregarded as a corollary of Theorem 2.The reducibility condition implies that the infinitesimal generator L must be of the form(4.1) L = σ A ( x ) ∂ ∂x + ( a − bx ) ∂∂x for coefficients a ∈ R and b >
0. The transitional probability density of the process, p T − t ( x, y ),satisfies the backward Kolmogorov equation:(4.2) ∂p∂t + ( a − bx ) ∂p∂x + σ A ( x ) ∂ p∂x = 0with final time condition(4.3) lim t → T p T − t ( x, y ) = δ ( x − y ) . By the reducibility assumption, a general solution to this equation is given by the followingeigenfunction expansion in terms of the orthogonal polynomials Q n ( x ; a, b )(4.4) p = ∞ (cid:88) n =0 h n ( t ) Q n ( x ; a, b ) . According to (1.27), the functions of time h n ( t ) satisfy the ordinary differential equations(4.5) ˙ h n + λ n h n = 0which admits the general solution, for t ≤ T ,(4.6) h n ( t ) = z n e λ n ( T − t ) . The coefficients z n are given by the final time condition (4.3):(4.7) ∞ (cid:88) n =0 z n Q n ( x ; a, b ) = δ ( x − y ) . Hence, multiplying on both sides by Q m ( x ; a, b ) and the invariant measure ρ ( dx ), before inte-grating over the domain of x , leads to the result (1.29) by the orthogonality property (1.23).The Laplace transform L T − t ( x, ϑ ) satisfies the backward Kolmogorov equation with differen-tial operator given by (1.13):(4.8) ∂L∂t + ( a − bx ) ∂L∂x + σ A ( x ) ∂ L∂x = ϑφ ( x ) L with final time condition(4.9) lim t → T L T − t ( x, ϑ ) = q ( x ) . For the sake of having a clearer constructive proof, consider the following ansatz for the Laplacetransform(4.10) L = V ( x ) ¯ L. The previous equation then reads(4.11)
V ∂ ¯ L∂t + ( a − bx )( V (cid:48) ¯ L + V ¯ L (cid:48) ) + σ | A ( x ) | ( V (cid:48)(cid:48) ¯ L + 2 V (cid:48) ¯ L (cid:48) + V ¯ L (cid:48)(cid:48) ) = ϑφV ¯ L, where the symbol (cid:48) denotes differentiation in the x -variable. The function V is chosen to satisfy(4.12) ( a − bx ) V + 2 σ A ( x ) V (cid:48) = (¯ a − ¯ bx ) V APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 21 for some parameters ¯ a ∈ R and ¯ b >
0, which implies(4.13) V ( x ) = exp (cid:18)(cid:90) x (¯ a − ¯ by ) − ( a − by ) σ | A ( y ) | dy (cid:19) . The function φ ( x ) is specified as follows:(4.14) ϑφ ( x ) = ( a − bx ) V (cid:48) V + σ A ( x ) V (cid:48)(cid:48) V , which is equivalent to (1.30) with the assumption that C regroups all the constant terms. Thischoice yields the following partial differential equation for the function ¯ L :(4.15) ∂ ¯ L∂t + (¯ a − ¯ bx ) ∂ ¯ L∂x + σ A ( x ) ∂ ¯ L∂x = 0 . Following the same reasoning as for the transitional probability denstiy, a general solution tothis equation is given by the following eigenfunction expansion in terms of the same orthogonalpolynomials Q n ( x ; ¯ a, ¯ b ), but with different coefficients,(4.16) ¯ L = ∞ (cid:88) n =0 h n ( t ) Q n ( x ; ¯ a, ¯ b ) . According to (1.27), the functions of time h n ( t ) satisfy the ordinary differential equations(4.17) ˙ h n + ¯ λ n h n = 0where ¯ λ n are the eigenvalues corresponding to Q n ( x ; ¯ a, ¯ b ). The general solution, for t ≤ T , is(4.18) h n ( t ) = z n e ¯ λ n ( T − t ) where the z n are constants. The latter equation for h n ( t ) with the explicit form of V ( x ) in (4.13)and the expression of ¯ L in (4.16) gives the expected result (1.31) for the Laplace transform.The coefficients z n are given by the final time condition (4.9):(4.19) ∞ (cid:88) n =0 z n Q n ( x ; ¯ a, ¯ b ) = q ( x ) exp (cid:18) − (cid:90) x (¯ a − ¯ by ) − ( a − by ) σ | A ( y ) | dy (cid:19) . Hence, multiplying on both sides by Q m ( x ) and the invariant measure ρ ( dx ), before integratingover the domain of x , leads to the final result (1.32) by the orthogonality property (1.23) andconcludes the proof of Theorem 5. Remark 25.
The reducibility condition implies that the operators L and ˜ L have the form L = σ A ( x ) ∂ ∂x + ( a − bx ) ∂∂x ˜ L = σ A ( x ) ∂ ∂x + (¯ a − ¯ bx ) ∂∂x (4.20) for possibly different coefficients a, ¯ a ∈ R and b, ¯ b > . Hence, setting (4.21) R ( x ) = 2 A ( x ) σ and h ( x ) = exp (cid:18)(cid:90) x a − byσ A ( y ) dy (cid:19) in the First Classification Theorem 2 yields the Second Classification Theorem 5. Moreover,consider the gauge transformation T ¯ h defined by (4.22) ¯ h ( x ) = exp (cid:18)(cid:90) x ¯ a − ¯ byσ A ( y ) dy (cid:19) . Then we have the following relation: (4.23) T h T − h ˜ L = σ A ( x ) ∂ ∂x + ( a − bx ) ∂∂x + σ Q ( x, ϑ ) ϑA ( x ) . Notice further that the function V ( x ) of the previous proof is related to the latter two gaugetransformations by V ( x ) = ¯ h ( x ) h ( x ) . The Ornstein-Uhlenbeck process.Corollary 26.
Assume that L and ˜ L are reducible in the sense of Definition 4 to Hermitepolynomials. Assume also that the function φ ( x ) = x . Then the transitional probability densityof the Ornstein-Uhlenbeck process is given by: (4.24) p T − t ( x, y ) = (cid:114) bσ π e − bσ ( y − ab ) ∞ (cid:88) n =0 e − bn ( T − t ) n !2 n H n (cid:0) z ( x ) (cid:1) H n (cid:0) z ( y ) (cid:1) and the Laplace transform is given by the following convergent series: (4.25) L T − t ( x, ϑ ) = e ϑb x ∞ (cid:88) n =0 e − bn ( T − t ) z n H n (cid:0) ¯ z ( x ) (cid:1) , where z ( x ) = (cid:113) bσ (cid:0) x − ab (cid:1) and ¯ z ( x ) = (cid:113) bσ (cid:0) x − ¯ ab (cid:1) . The coefficients z n are given by: (4.26) z n = 1 n !2 n (cid:114) bσ π (cid:90) ∞−∞ q ( x ) e − ϑb x H n (cid:0) ¯ z ( x ) (cid:1) e − bσ ( x − ¯ ab ) dx. Proof.
This case fits the classification scheme in Theorem 5 if one selects A ( x ) = 1. This choiceimplies(4.27) φ ( x ) = C + 12 ϑσ (cid:0) (¯ a − ¯ bx ) − ( a − bx ) (cid:1) . In order to reduce φ ( x ) to an affine function, we choose ¯ b = b , ¯ a = a − ϑσ b and C = a − ¯ a ϑσ . Theoperator L , which has the form(4.28) L = σ ∂ ∂x + ( a − bx ) ∂∂x , has the Hermite polynomials H n (cid:0) z ( x ) (cid:1) as eigenfunctions with eigenvalues λ n = − bn . Theinvariant measure density and the normalization factor are respectively(4.29) w ( x ) = (cid:114) bσ π e − bσ ( x − ab ) and d n = n !2 n which leads, by Theorem 5, to the formulation of the kernel of the semigroup generated by L asthe convergent series in (4.24). The latter series re-sums using Mehler’s formula to give (3.28).By Theorem 5, the Laplace transform is given by (4.25). The coefficients z n are given by (4.26)and lead to the result (3.45), once integrated and re-summed. (cid:3) The CIR process.Corollary 27.
Assume that L and ˜ L are reducible in the sense of Definition 4 to Laguerrepolynomials. Assume also that the function φ ( x ) = x . Then the transitional probability densityof the CIR process is given by: (4.30) p T − t ( x, y ) = (cid:18) bσ (cid:19) aσ y aσ − e − bσ y ∞ (cid:88) n =0 n ! e − bn ( T − t ) Γ( n + aσ ) L ( aσ − n (cid:0) bσ x (cid:1) L ( aσ − n (cid:0) bσ y (cid:1) and the Laplace transform is given by the following convergent series: (4.31) L T − t ( x, ϑ ) = e b − ¯ bσ x ∞ (cid:88) n =0 e − ¯ bn ( T − t ) z n L ( aσ − n (cid:0) bσ x (cid:1) . The coefficients z n are given by: (4.32) z n = n !Γ( n + aσ ) (cid:18) bσ (cid:19) aσ (cid:90) ∞ q ( x ) x aσ − e − b +¯ bσ x L ( aσ − n (cid:0) bσ x (cid:1) dx. APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 23
Proof.
This case also fits the classification scheme in Theorem 5 if one selects A ( x ) = x . Thischoice implies(4.33) φ ( x ) = C + 12 ϑx (cid:0) ( a − bx ) − (¯ a − ¯ bx ) (cid:1) + 12 ϑσ x (cid:0) (¯ a − ¯ bx ) − ( a − bx ) (cid:1) . In order to reduce φ ( x ) to an affine function, we choose ¯ a = a , ¯ b = √ ϑσ + b and C = ¯ b − bϑ (cid:0) aσ − (cid:1) . The operator L , which has the form(4.34) L = σ x ∂ ∂x + ( a − bx ) ∂∂x , has the Laguerre polynomials L ( aσ − n (cid:0) bσ x (cid:1) as eigenfunctions with eigenvalues λ n = − bn if a >
0. The invariant measure density and the normalization factor are respectively(4.35) w ( x ) = (cid:18) bσ (cid:19) aσ x aσ − e − bσ x and d n = Γ( n + aσ ) n !which leads, by Theorem 5, to the formulation of the kernel of the semigroup generated by L as the convergent series in (4.30) which re-sums to give (3.49). The Laplace transform is givenby (4.31) and the coefficients z n are given by (4.32). The latter results lead to (3.51), onceintegrated and re-summed. (cid:3) The Jacobi process.Definition 28.
The Jacobi polynomials P ( α,β ) n ( x ) are defined by the following Gaussian hyper-geometric function for x ∈ [ − , : (4.36) P ( α,β ) n ( x ) = ( α + 1) n n ! F (cid:32) − n, n + α + β + 1 α + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − x (cid:33) , n = 0 , , , . . . Definition 29.
The Jacobi process is solution to the following equation: (4.37) dX t = ( a − bX t ) dt + σ (cid:112) X t (1 − X t ) dW t with initial condition X t =0 = x ∈ (0 , . Corollary 30.
Assume that L and ˜ L are reducible in the sense of Definition 4 to Jacobi poly-nomials. Then: (4.38) φ ( x ) = σ ϑ (cid:18) ¯ α − α x + ¯ β − β − x (cid:19) , for α = aσ − > − , β = σ ( b − a ) − > − and ¯ α = aσ − > − , ¯ β = σ (¯ b − ¯ a ) − > − .The transitional probability density of the Jacobi process is given by: (4.39) p T − t ( x, y ) = y α (1 − y ) β ∞ (cid:88) n =0 e − σ n ( n + α + β +1)( T − t ) d n P ( α,β ) n (1 − x ) P ( α,β ) n (1 − y ) with normalization constant (4.40) d n = Γ( n + α + 1)Γ( n + β + 1)(2 n + α + β + 1)Γ( n + α + β + 1) n ! . The Laplace transform is given by the following convergent series: (4.41) L T − t ( x, ϑ ) = x ¯ α − α (1 − x ) ¯ β − β ∞ (cid:88) n =0 e − σ n ( n +¯ α + ¯ β +1)( T − t ) z n P (¯ α, ¯ β ) n (1 − x ) . The coefficients z n are given by: (4.42) z n = 1¯ d n (cid:90) q ( x ) P (¯ α, ¯ β ) n (1 − x ) x α +¯ α (1 − x ) β +¯ β dx, with normalization constant (4.43) ¯ d n = Γ( n + ¯ α + 1)Γ( n + ¯ β + 1)(2 n + ¯ α + ¯ β + 1)Γ( n + ¯ α + ¯ β + 1) n ! . Proof.
This case also fits the classification scheme in Theorem 5 if one selects A ( x ) = x (1 − x ).This choice implies(4.44) φ ( x ) = C + σ ϑ (cid:18) ¯ α − α x + ¯ β − β − x + ( α + β ) − (¯ α + ¯ β ) (cid:19) . Since C is an arbitrary constant, we set it to C = (¯ α + ¯ β ) − ( α + β ) . The infinitesimal generator L , which has the form(4.45) L = σ x (1 − x ) ∂ ∂x + ( a − bx ) ∂∂x , has the Jacobi polynomials P ( α,β ) n (1 − x ) as eigenfunctions with eigenvalues λ n = − σ n ( n + α + β + 1) if α > − β > −
1. The invariant measure density and the normalization factorare respectively(4.46) w ( x ) = x α (1 − x ) β and d n = Γ( n + α + 1)Γ( n + β + 1)(2 n + α + β + 1)Γ( n + α + β + 1) n ! , which concludes the proof by Theorem 5. (cid:3) The Dual Jacobi process.
We introduce the dual Jacobi polynomials by applying thetransformation x (cid:55)→ Z ( x ) = x (2 − x ) to the Jacobi polynomials defined in the previous subsection. Definition 31.
The dual Jacobi polynomials D ( α,β ) n ( x ) ≡ P ( α,β ) n (1 − Z ( x )) are defined asfollows: (4.47) D ( α,β ) n ( x ) = ( α + 1) n n ! F (cid:32) − n, n + α + β + 1 α + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (2 − x ) (cid:33) , n = 0 , , , . . . They also satisfy an orthogonality relation (1.23) on (0 ,
1) with normalization constants andcontinuous measure density:(4.48) w ( x ) = 2 (cid:0) x (2 − x ) (cid:1) α (1 − x ) β +1 , d n = Γ( n + α + 1)Γ( n + β + 1)(2 n + α + β + 1)Γ( n + α + β + 1) n ! . They are solutions to the eigenvalue problem (1.27) with generator:(4.49) L = σ x (2 − x ) ∂ ∂x + 2 a − (2 b − σ ) x (2 − x )4(1 − x ) ∂∂x and eigenvalues λ n = − σ n ( n + α + β + 1), still conditioned to α = aσ − > − β = σ ( b − a ) − > −
1. Hence, we have the definition of the dual Jacobi process as follows:
Definition 32.
The dual Jacobi process is solution to the following equation: (4.50) dX t = 2 a − (2 b − σ ) X t (2 − X t )4(1 − X t ) dt + σ (cid:112) X t (2 − X t ) dW t with initial condition X t =0 = x ∈ (0 , . We obtain yet another corollary to Theorem 5:
Corollary 33.
Assume that L and ˜ L are reducible in the sense of Definition 4 to the dual Jacobipolynomials. Then: (4.51) φ ( x ) = σ ϑ (cid:18) ¯ α − α x (2 − x ) + ¯ β − β (1 − x ) (cid:19) , APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 25 for ¯ α = aσ − > − , ¯ β = σ (¯ b − ¯ a ) − > − . The transitional probability density of the dualJacobi process is given by: (4.52) p T − t ( x, y ) = 2 (cid:0) y (2 − y ) (cid:1) α (1 − y ) β +1 ∞ (cid:88) n =0 e λ n ( T − t ) d n D ( α,β ) n ( x ) D ( α,β ) n ( y ) . The Laplace transform can be expressed by the following convergent series: (4.53) L T − t ( x, ϑ ) = 2 (cid:0) x (2 − x ) (cid:1) ¯ α − α (1 − x ) ¯ β − β ∞ (cid:88) n =0 e λ n ( T − t ) z n D (¯ α, ¯ β ) n ( x ) where the coefficients z n are given by: (4.54) z n = 1¯ d n (cid:90) q ( x ) D (¯ α, ¯ β ) n ( x ) (cid:0) x (2 − x ) (cid:1) α +¯ α (1 − x ) β + ¯ β +1 dx. with normalization constant (4.55) ¯ d n = Γ( n + ¯ α + 1)Γ( n + ¯ β + 1)(2 n + ¯ α + ¯ β + 1)Γ( n + ¯ α + ¯ β + 1) n ! . Proof.
The proof follows from Corollary 30 and the transformation x (cid:55)→ Z ( x ) = x (2 − x ). (cid:3) Processes related to discrete orthogonal polynomials
The proof of Theorem 7 follows a very similar reasoning as the proof of Theorem 5, itscontinuous version.5.1.
Proof of the Third Classification Theorem.
The reducibility condition implies thatthe infinitesimal generator L must be of the form(5.1) L = − D ( x )∆ + (cid:0) D ( x ) − B ( x ) (cid:1) ∇ . The transitional probability density p T − t ( x, y ) satisfies the backward Kolmogorov equation withgenerator given by (1.12):(5.2) ∂p∂t − D ( x )∆ p + (cid:0) D ( x ) − B ( x ) (cid:1) ∇ p = 0with final time condition(5.3) lim t → T p T − t ( x, y ) = δ ( x − y ) . By the reducibility assumption, a general solution to this equation is given by the followingeigenfunction expansion in terms of the discrete orthogonal polynomials Q n ( x )(5.4) p = ∞ (cid:88) n =0 h n ( t ) Q n ( x ) . According to (1.27), the functions of time h n ( t ) satisfy the ordinary differential equations(5.5) ˙ h n + λ n h n = 0which admits the general solution, for t ≤ T ,(5.6) h n ( t ) = z n e λ n ( T − t ) . The coefficients z n are given by the final time condition (5.3):(5.7) ∞ (cid:88) n =0 z n Q n ( x ) = δ ( x − y ) . Hence, multiplying on both sides by Q m ( x ) and the weight w ( x ), before summing over the latticeΛ N , leads to the result (1.35) by the orthogonality property (1.23).The Laplace transform L T − t ( x,
1) satisfies this time a finite difference version of the BackwardKolmogorov equation(5.8) ∂L∂t − D ( x )∆ L + (cid:0) D ( x ) − B ( x ) (cid:1) ∇ L = ϑφL with the same final time condition(5.9) lim t → T L T − t ( x, ϑ ) = q ( x ) . Consider the ansatz for the Laplace transform(5.10) L = V ( x ) ¯ L. The latter finite difference equation reads(5.11) ∂ ¯ L∂t − B ( x ) V ( x + 1) V ( x ) (cid:124) (cid:123)(cid:122) (cid:125) ¯ B ( x ) ¯ L ( x + 1) + (cid:0) B ( x ) + D ( x ) − ϑφ (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) ¯ B ( x )+ ¯ D ( x ) ¯ L ( x ) − D ( x ) V ( x − V ( x ) (cid:124) (cid:123)(cid:122) (cid:125) ¯ D ( x ) ¯ L ( x −
1) = 0 . ¯ B ( x ) and ¯ D ( x ) are defined such that they satisfy the relations(5.12) V ( x ) V ( x −
1) = ¯ B ( x − B ( x −
1) = D ( x )¯ D ( x ) APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 27 which implies condition (1.37) and solves iteratively to give(5.13) V ( x ) = x (cid:89) k =1 D ( k )¯ D ( k ) . The function φ ( x ) is specified as follows:(5.14) φ ( x ) = 1 ϑ (cid:0) B ( x ) + D ( x ) − ¯ B ( x ) − ¯ D ( x ) (cid:1) . But φ ( x ) , B ( x ) , D ( x ) are all by definition independent of the parameter ϑ , so we are bound toset ϑ = 1. This choice yields the following finite difference equation for the function ¯ L :(5.15) ∂ ¯ L∂t − ¯ D ( x )∆ ¯ L + (cid:0) ¯ D ( x ) − ¯ B ( x ) (cid:1) ∇ ¯ L = 0 . A general solution to this equation is given by the following eigenfunction expansion in terms ofdiscrete orthogonal polynomials:(5.16) ¯ L = N (cid:88) n =0 h n ( t ) ¯ Q n ( x ) . The functions of time h n ( t ) satisfy the ordinary differential equations(5.17) ˙ h n + ¯ λ n h n = 0which admits the general solution(5.18) h n ( t ) = z n e ¯ λ n ( T − t ) where the z n are constants. The latter equation for h n ( t ) with the explicit form of V ( x ) in (5.13)and the expression of ¯ L in (5.16) yields the expression (1.38) for the Laplace transform.The coefficients z n are given by the final time condition (5.9):(5.19) q ( x ) x (cid:89) k =1 ¯ D ( x ) D ( x ) = N (cid:88) n =0 z n ¯ Q n ( x ) . Finally, multiplying on both sides by ¯ Q m ( x ) and the weight ¯ w ( x ), before summing over Λ N , givesthe final result (1.39) by orthogonality of the polynomials and concludes the proof of Theorem7.5.2. The Meixner process.
The Meixner polynomials provide a discrete lattice approximationto the Laguerre polynomials.
Definition 34.
The Meixner polynomials are defined as follows in case x is integer: (5.20) M n ( x ; β, c ) = F (cid:32) − n, − xβ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − c (cid:33) , n = 0 , , , . . . The Meixner polynomials satisfy an orthogonality relation with respect to the discrete measuresupported on Z + . Namely,(5.21) ∞ (cid:88) x =0 M m ( x ; β, c ) M n ( x ; β, c ) w ( x ) = c − n n !( β ) n (1 − c ) β δ nm . where the weight is(5.22) w ( x ) = ( β ) x x ! c x . The Meixner polynomials are solutions to the eigenvalue problem (1.27) with generator(5.23) L = σ x ∆ + ( a − bx ) ∇ , for x ∈ Z + , with a, b > λ n = − bn . The latter can be recast in the form (1.34)using the functions B ( x ) = − σ c ( x + β ) D ( x ) = − σ x, (5.24)for a = σ β and b = σ (1 − c ). The parameters are conditioned to β > < c < L . It is a discrete version of the CIR process.The following statement is a corollary to Theorem 7. Corollary 35.
Assume that L and ˜ L are reducible in the sense of Definition 4 to the Meixnerpolynomials. Assume also that the function φ ( x ) is given by φ ( x ) = (cid:37)x + ζ , (5.25) (cid:37) = σ (cid:0) c ( e ϕ −
1) + ( e − ϕ − (cid:1) , ζ = σ βc ( e ϕ − with the real parameter ϕ < − ln c . Then the transitional probability density for the Meixnerprocess is as follows: p T − t ( x, y ) = (1 − c ) β (1 − e ( T − t )( c − ) x + y (1 − ce ( T − t )( c − ) x + y + β ( β ) y c y y ! · F (cid:32) − x, − yβ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e ( T − t )( c − (1 − c ) c (1 − e ( T − t )( c − ) (cid:33) . (5.26) For q ( x ) = exp (cid:0) ωφ ( x ) (cid:1) , the Laplace transform is affine: (5.27) L T − t ( x, , ω ) = e m ( T − t ; ω ) x + n ( T − t ; ω ) . The functions of time m ( τ ; ω ) and n ( τ ; ω ) are as follows: m ( τ ; ω ) = log (cid:32) e ϕ − ce (cid:37)ω + ϕ − e (¯ c − e − ϕ τ (1 − e (cid:37)ω − ϕ )1 − ce (cid:37)ω + ϕ − ¯ ce (¯ c − e − ϕ τ (1 − e (cid:37)ω − ϕ ) (cid:33) n ( τ ; ω ) = − β log (cid:32) e σ ωc (1 − e ϕ ) − ce (cid:37)ω + ϕ − ¯ ce (¯ c − e − ϕ τ (1 − e (cid:37)ω − ϕ )1 − ¯ c (cid:33) . where ¯ c = ce ϕ .Proof. The transitional probability density follows from equation (1.35) in the discrete classifica-tion theorem. The definition of φ ( x ) suggests that we set ¯ B ( x ) = B ( x ) e ϕ and ¯ D ( x ) = D ( x ) e − ϕ in order to satisfy condition (1.37) in Theorem 7. The generator defined by ¯ B ( x ) and ¯ D ( x ) haseigenfunctions M n ( x ; β, ¯ c ) with eigenvalues ¯ λ n = σ e − ϕ n (¯ c − z n = ( β ) n ¯ c n (1 − ¯ c ) β n ! ∞ (cid:88) x =0 e ωφ ( x ) − ϕx M n ( x ; β, ¯ c ) ( β ) x ¯ c x x != e ωζ (cid:18) − ¯ c − ¯ ce (cid:37)ω − ϕ (cid:19) β ( β ) n ¯ c n n ! (cid:18) − e (cid:37)ω − ϕ − ¯ ce (cid:37)ω − ϕ (cid:19) n . APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 29
The Laplace transform, given by (1.38), is as follows: L T − t ( x, , ω ) = (cid:18) − ¯ c − ¯ ce (cid:37)ω − ϕ (cid:19) β e ωζ e ϕx · ∞ (cid:88) n =0 ( β ) n n ! (cid:18) ¯ ce σ (¯ c − e − ϕ ( T − t ) − e (cid:37)ω − ϕ − ¯ ce (cid:37)ω − ϕ (cid:19) n M n ( x ; β, ¯ c )= (cid:32) e σ ωc (1 − e ϕ ) − ce (cid:37)ω + ϕ − ¯ c (1 − e (cid:37)ω − ϕ ) e σ (¯ c − e − ϕ ( T − t ) − ¯ c (cid:33) − β · (cid:32) e ϕ − ce (cid:37)ω + ϕ − (1 − e (cid:37)ω − ϕ ) e σ (¯ c − e − ϕ ( T − t ) − ce (cid:37)ω + ϕ − ¯ c (1 − e (cid:37)ω − ϕ ) e σ (¯ c − e − ϕ ( T − t ) (cid:33) x . Note that the re-summation formula used to find the last two results is the generating function forthe Meixner polynomials which can be found in ( ? ). Also notice that M n ( x ; β, ¯ c ) = M x ( n ; β, ¯ c )by definition. (cid:3) The Racah Process.Definition 36.
The Racah polynomials R n ( λ ( x )) := R n ( λ ( x ); α, β, γ, δ ) are defined as follows: (5.28) R n ( λ ( x ); α, β, γ, δ ) = F (cid:18) − n, n + α + β + 1 , − x, x + γ + δ + 1 α + 1 , β + δ + 1 , γ + 1 (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) , n = 0 , , , . . . , N where λ ( x ) = x ( x + γ + δ + 1) and either α = − N − or β + δ = − N − or γ = − N − . The Racah polynomials satisfy an orthogonality relation with respect to the discrete measuresupported on the set Λ N . Namely,(5.29) (cid:88) x ∈ Λ N R m ( λ ( x )) R n ( λ ( x )) w ( x ) = d n δ nm where the weight is(5.30) w ( x ) := w ( x ; α, β, γ, δ ) = ( α + 1) x ( β + δ + 1) x ( γ + 1) x ( γ + δ + 1) x (( γ + δ + 3) / x ( − α + γ + δ + 1) x ( − β + γ + 1) x (( γ + δ + 1) / x ( δ + 1) x x !and the normalization factor is(5.31) d n = M ( n + α + β + 1) n ( α + β − γ + 1) n ( α − δ + 1) n ( β + 1) n n !( α + β + 2) n ( α + 1) n ( β + δ + 1) n ( γ + 1) n with M = ( − β ) N ( γ + δ + 2) N ( − β + γ + 1) N ( δ + 1) N if α = − N − − α + δ ) N ( γ + δ + 2) N ( − α + γ + δ + 1) N ( δ + 1) N if β + δ = − N − α + β + 2) N ( − δ ) N ( α − δ + 1) N ( β + 1) N if γ = − N − . The Racah polynomials are solutions to the eigenvalue problem (1.27) with generator (1.34)given by the functions B ( x ) = σ x + α + 1)( x + β + δ + 1)( x + γ + 1)( x + γ + δ + 1)(2 x + γ + δ + 1)(2 x + γ + δ + 2) ,D ( x ) = σ x ( x − α + γ + δ )( x − β + γ )( x + δ )(2 x + γ + δ )(2 x + γ + δ + 1) , (5.32)and eigenvalues λ n = − σ n ( n + α + β + 1). The Markov property is insured if B ( x ) ≤ D ( x ) ≤ ∀ x ∈ Λ N . The process generated by the latter generator is called the Racah process . Also define the corresponding functions¯ B ( x ) = σ x + ¯ α + 1)( x + ¯ β + ¯ δ + 1)( x + ¯ γ + 1)( x + ¯ γ + ¯ δ + 1)(2 x + ¯ γ + ¯ δ + 1)(2 x + ¯ γ + ¯ δ + 2) , ¯ D ( x ) = σ x ( x − ¯ α + ¯ γ + ¯ δ )( x − ¯ β + ¯ γ )( x + ¯ δ )(2 x + ¯ γ + ¯ δ )(2 x + ¯ γ + ¯ δ + 1)(5.33)for ¯ α, ¯ β, ¯ γ, ¯ δ ∈ R . Definition 37.
The set of parameters { ¯ α, ¯ β, ¯ γ, ¯ δ } will be called acceptable with respect to { α, β, γ, δ } if it satisfies condition (1.37) in Theorem 7, i.e. if ¯ B ( x −
1) ¯ D ( x ) = B ( x − D ( x ) , and if ¯ B ( x ) ≤ and ¯ D ( x ) ≤ , ∀ x ∈ Λ N . The following statement is a corollary to Theorem 7.
Corollary 38.
Assume that L and ˜ L are reducible in the sense of Definition 4 to Racah poly-nomials. Assume also that for an acceptable set of parameters { ¯ α, ¯ β, ¯ γ, ¯ δ } : (5.34) φ ( x ) = B ( x ) + D ( x ) − ¯ B ( x ) − ¯ D ( x ) . Then the transitional probability density for the Racah process is given by (5.35) p T − t ( x, y ) = N (cid:88) n =0 e − σ n ( n + α + β +1)( T − t ) d n R n ( λ ( x )) R n ( λ ( y )) w ( y ) . while the Laplace transform is given by the following convergent series: (5.36) L T − t ( x,
1) = x (cid:89) k =1 D ( k )¯ D ( k ) N (cid:88) n =0 e − σ n ( n +¯ α + ¯ β +1)( T − t ) z n R n (¯ λ ( x ); ¯ α, ¯ β, ¯ γ, ¯ δ ) . The coefficients z n are as follows: (5.37) z n = 1¯ d n (cid:88) x ∈ Λ N x (cid:89) k =1 ¯ D ( k ) D ( k ) q ( x ) R n (¯ λ ( x ); ¯ α, ¯ β, ¯ γ, ¯ δ ) ¯ w ( x ) where ¯ d n = d n (¯ α, ¯ β, ¯ γ, ¯ δ ) , ¯ w ( x ) = w ( x ; ¯ α, ¯ β, ¯ γ, ¯ δ ) and ¯ λ ( x ) = λ ( x ; ¯ γ, ¯ δ ) .Proof. The restrictions imposed on the set of parameters { ¯ α, ¯ β, ¯ γ, ¯ δ } ensures that condition (1.37)in Theorem 7 is satisfied. This is a necessary condition. The rest of the corollary is a directapplication of Theorem 7. (cid:3) The Dual Hahn Process.Definition 39.
The dual Hahn polynomials R n ( λ ( x )) := R n ( λ ( x ); γ, δ, N ) are defined as follows: (5.38) R n ( λ ( x ); γ, δ, N ) = F (cid:18) − n, − x, x + γ + δ + 1 γ + 1 , − N (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) , n = 0 , , , . . . , N where λ ( x ) = x ( x + γ + δ + 1) . For γ > − δ > − γ < − N and δ < − N , the dual Hahn polynomials satisfy anorthogonality relation with respect to the discrete measure supported on the set Λ N . Namely,(5.39) (cid:88) x ∈ Λ N R m ( λ ( x )) R n ( λ ( x )) w ( x ) = d n δ nm where the weight is(5.40) w ( x ) := w ( x ; γ, δ, N ) = (2 x + γ + δ + 1)( γ + 1) x ( − N ) x N !( − x ( x + γ + δ + 1) N +1 ( δ + 1) x x !and the normalization factor is(5.41) d n = 1 (cid:18) γ + nn (cid:19) (cid:18) δ + N − nN − n (cid:19) . APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 31
The dual Hahn polynomials are solutions to the eigenvalue problem (1.27) with generator (1.34)given by the functions B ( x ) = − σ x + γ + 1)( x + γ + δ + 1)( N − x )(2 x + γ + δ + 1)(2 x + γ + δ + 2) ,D ( x ) = − σ x ( x + γ + δ + N + 1)( x + δ )(2 x + γ + δ )(2 x + γ + δ + 1) , (5.42)and eigenvalues λ n = − σ n . The discrete Markov process generated by the latter generator iscalled the dual Hahn process .The following statement is another corollary to Theorem 7. Corollary 40.
Assume that L and ˜ L are reducible in the sense of Definition 4 to dual Hahnpolynomials. Let δ > γ > − or δ < γ < − N such that: (5.43) φ ( x ) = σ δ − γ ) (cid:20) x ( x + γ + δ + N + 1)(2 x + γ + δ )(2 x + γ + δ + 1) − ( x + γ + δ + 1)( N − x )(2 x + γ + δ + 1)(2 x + γ + δ + 2) (cid:21) . Then the transitional probability density for the dual Hahn process is given by (5.44) p T − t ( x, y ) = N (cid:88) n =0 e − σ n ( T − t ) d n R n ( λ ( x )) R n ( λ ( y )) w ( y ) . while the Laplace transform can be expressed as the following convergent series: (5.45) L T − t ( x,
1) = ( δ + 1) x ( γ + 1) x N (cid:88) n =0 e − σ n ( T − t ) z n R n ( λ ( x ); δ, γ, N ) . The coefficients z n are as follows: (5.46) z n = 1¯ d n (cid:88) x ∈ Λ N ( γ + 1) x ( δ + 1) x q ( x ) R n ( λ ( x ); δ, γ, N ) w ( x ; δ, γ, N ) where ¯ d n = d n ( δ, γ, N ) .Proof. For ¯ γ = δ and ¯ δ = γ , define the functions¯ B ( x ) = − σ x + ¯ γ + 1)( x + ¯ γ + ¯ δ + 1)( N − x )(2 x + ¯ γ + ¯ δ + 1)(2 x + ¯ γ + ¯ δ + 2) , ¯ D ( x ) = − σ x ( x + ¯ γ + ¯ δ + N + 1)( x + ¯ δ )(2 x + ¯ γ + ¯ δ )(2 x + ¯ γ + ¯ δ + 1) . (5.47)The corollary then follows from Theorem 7. (cid:3) Limit relations
Limit relations between orthogonal polynomials are well-known, see ( ? ) and ( ? ). In thissection, we show that the transitional probability densities and Laplace transforms obtainedfrom the various corollaries of Theorems 5 and 7 are similarly connected to each other. We startby rigorously stating the relation between the Jacobi process and its dual. Remark 41.
The dual Jacobi process is obtained from the Jacobi process with the change ofvariable transformation (6.1) Z ( x ) = x (2 − x ) applied to the underlying process X t . The same holds for its transitional probability density andLaplace transform. For the discrete X t process, the limit relation between the Racah process and the dual Hahnprocess is not as obvious. Remark 42.
The Racah process converges to the dual Hahn process in three different ways,corresponding to the three families of Racah polynomials: (1) α = − N − with the acceptable set of parameters {− N − , β + δ − γ, δ, γ } , conditionedto either (cid:26) β ≥ γ + Nδ > γ > − or (cid:26) β ≥ − δ − δ < γ < − N in the limit as β → ∞ . (2) β = − δ − N − with the acceptable set { α, − γ − N − , δ, γ } , conditioned to either (cid:26) α ≥ γ + δ + Nδ > γ > − or (cid:26) α ≥ − δ < γ < − N in the limit as α → ∞ . (3) γ = − N − with the acceptable set {− α + δ − N − , β, − N − , δ } , conditioned to either α > − β ≥ − δ > α + N + 1 or α < − Nβ ≥ − δ − δ < α + N + 1 , with first the mapping δ (cid:55)→ δ + α + N + 1 and then the limit β → ∞ . The dual Hahnparameters are in this third case ( α, δ ) .The result extends to the transitional probability densities and Laplace transforms. The next proposition states that the Meixner process converges to the CIR process in theaffine case.
Remark 43.
Under the transformations (cid:37) (cid:55)→ (cid:37) (1 − c ) and x (cid:55)→ x − c in the Meixner process,the limit c → yields the CIR process with parameter α = β − . The Laplace transform is affinein this case.Proof. The proof follows from the limit relationlim c → M n (cid:18) x − c ; α + 1 , c (cid:19) = L ( α ) n ( x ) L ( α ) n (0) . Both the transitional probability density and the Laplace transform in the Meixner case thenconverge to the affine CIR case, sincelim c → φ (cid:18) x − c (cid:19) = (cid:37) x + ζ. (cid:3) APLACE TRANSFORMS FOR INTEGRALS OF MARKOV PROCESSES 33
The last proposition demonstrates the connection between a discrete and a continuous un-derlying process X t by showing that the dual Jacobi process is actually a limiting case of theRacah process. Remark 44.
Consider the acceptable set of parameters { α, ¯ β, γ, δ } where ¯ β = − β − δ − N − γ = − N − . (6.2) Assume furthermore the inequalities: α > − β > β > − γ > δ. (6.3) Then, applying the transformation x (cid:55)→ xN to the Racah process such that x ∈ [0 , , yields thedual Jacobi process in the limit N → ∞ for the special case of ¯ α = α and ¯ β > | β | . This resultapplies to both the transitional probability density and the Laplace transform.Proof. The inequalities in the assumption insure that the process X t satisfies the Markov prop-erty, as both B ( x ) and D ( x ) are negative for x ∈ Λ N . Since on top of ¯ α = α, ¯ γ = γ, ¯ δ = δ , wehave − β + γ = ¯ β + ¯ δ and β + δ = − ¯ β + ¯ γ , it is immediate that { α, ¯ β, γ, δ } is acceptable. Thefunction φ ( x ) reduces to(6.4) φ ( x ) = σ β − β ) (cid:20) x ( x − α + γ + δ )( x + δ )(2 x + γ + δ )(2 x + γ + δ + 1) − ( x + α + 1)( x + γ + 1)( x + γ + δ + 1)(2 x + γ + δ + 1)(2 x + γ + δ + 2) (cid:21) or equivalently, φ ( x ) = σ β − β ) (cid:20) x ( x − N − α − ¯ β − β − x − N − ¯ β − β − x − N − ¯ β − β − x − N − ¯ β − β − − ( x + α + 1)( x − N )( x − N − ¯ β − β − x − N − ¯ β − β − x − N − ¯ β − β ) (cid:21) . (6.5) φ ( x ) is bounded from below, from the inequality ¯ β > β > − x (cid:55)→ N x , the function φ ( N x ) converges, in the limit N → ∞ , to its continuous counterpart of Corollary 33 for ϑ = 1.Moreover, the finite difference infinitesimal generator L in (1.34) converges to the dual Jacobidiffusion generator in (4.49), under the transformation x (cid:55)→ N x : − ¯ D ( N x )∆ h + (cid:0) ¯ D ( N x ) − ¯ B ( N x ) (cid:1) ∇ h + (6.6) −→ σ α + 1)(1 − x ) − (2 β + 1) x (2 − x )4(1 − x ) ∂∂x + σ x (2 − x )4 ∂ ∂x in the limit h = 1 /N → n !( α + 1) n , since F (cid:18) − n, n + α + β + 1 , − N x, N x − N − − β − ¯ βα + 1 , − β − N, − N (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) −→ F (cid:18) − n, n + α + β + 1 α + 1 (cid:12)(cid:12)(cid:12)(cid:12) Z ( x ) (cid:19) (6.7)as N → ∞ , with Z ( x ) = x (2 − x ). We also have as N → ∞ ,(6.8) Nx (cid:89) k =1 D ( k )¯ D ( k ) = Nx (cid:89) k =1 (cid:32) β − ¯ βN (1 − kN ) + ¯ β + 1 (cid:33) −→ exp (cid:18)(cid:90) x β − ¯ β − y dy (cid:19) = (1 − x ) ¯ β − β . Finally, from all the above limit relations, the Laplace transform for the integral of the Racahprocess, given by (5.36) with the appropriate assumptions, provides an extension with underlyingprocess X t on the lattice to the Laplace transform (4.53) for the integral of the dual Jacobi processin the particular case ¯ α = α , β > | ¯ β | and ¯ β ≥ − (cid:3) Conclusion
We have given a complete classification scheme for diffusion processes for which Laplacetransforms for integrals of stochastic processes and transitional probability densities can beexpressed as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. The known models such as the Ornstein-Uhlenbeck process, the CIR processand the geometric Brownian motion fit into this classification scheme. We have also presentedextensions to these models in the quadratic Ornstein-Uhlenbeck process and the Jacobi process.An extension of the framework towards finite-state Markov processes related to hypergeometricpolynomials in the discrete series of the Askey classification tree has been derived. Finally, wehave explicitly computed some limit relations between discrete and continuous processes.
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