Markov Processes and Parabolic Functions

Transcription

1 Markov Processes and Parabolic Functions Arindam Sengupta Department of Mathematics Indian Institute of Technology Guwahati Guwahati , Assam INDIA Abstract: This paper investigates the extent of richness of the class of parabolic functions in time and space for a given stochastic process M = (M t : t 0) that guarantees its Markov property. We describe a sufficient condition for this in terms of denseness in the various Lebesgue spaces for the law of the process at every given time and obtain certain consequences for some special Markov processes. Key-words: Time-space harmonic function, polynomially harmonisable process, conditional determinacy. 1 Introduction Time-space harmonic functions for a stochastic process may be defined as functions of two variables which yield martingales when evaluated at an arbitrary point of time and the value of the process at that time, as this time point varies. They play a significant role in revealing many important properties of processes, particularly, certain Markov processes that we shall study in this paper. The class of all time-space harmonic functions for a process forms a real vector space. When the underlying process is real-valued, a subspace which has been extensively studied, consists of all time-space harmonic polynomials (see Definition 1). In [4], [8], [9], [7] and [6], much of the interaction between properties of time-space harmonic polynomials, and properties of the underlying process was brought out. One thus has the natural definition of a polynomially harmonisable, or p-harmonisable, process (Definition 2). One of the first important properties of a polynomially harmonisable process to be established, under mildly restrictive additional assumptions, is the Markov property. Specifically, Markov property of a p-harmonisable process was proved imposing a condition on the existence of moment generating function of each random variable in the process, in Corollary of [8]. For an exact statement see Proposition 1. The discrete-time variant of this result can be found in Corollary 2.1 of [4]. In a sense, p-harmonisability really represents one notion of richness of the class of all time-space harmonic polynomials. It thus seems a natural question to ask when Markov property can be deduced simply from some notion of richness of the superclass of time-space harmonic functions, even when the subclass need not be rich in any way, or even be non-empty. Clearly, a result in this direction would be far more general, encompassing processes which need not be p-harmonisable, removing our dependence on the somewhat special nature of that property. Even processes which arise from p-harmonisable ones, by the application of one-to-one functions on the state space, for example, are not covered by results such as the ones mentioned above; while obviously retaining Markov property. Thus it is clearly desirable to extend these results to apply in the general 1

2 setting. This leads us to ask the question : how rich does the class of time-space harmonic functions have to be in order for the underlying process to be Markovian? This note will provide a couple of examples when the answer to this question may be found. The most well-known examples, however, really arise in the context of time-space harmonic polynomials. We are, therefore, also motivated to apply our findings to this setting and succeed in weakening the hypotheses in the results mentioned above somewhat to attain more generality. In this setting, one is naturally interested in actual procedures to obtain time-space harmonic polynomials for specific polynomially harmonisable Markov processes. Particularly for timehom-ogeneous processes, there are several methods, a few of which we describe. In further special cases, these methods are particularly simple to use, as we illustrate with examples in the sequel. 2 Basic definitions and the main theorem Suppose M := {M t : t 0} is a stochastic process defined on some probability space (Ω, F, P) taking values in an arbitrary measurable space (E, E), to be referred to as the state space. The natural filtration (F t ) t 0 of M is defined as F t = σ({m s : 0 s t}). A time-space harmonic function is a two-variable function which, evaluated at t and M t, leads to an (F t )-martingale. To be precise, let V := {f : [0, ) E R : f(t, M t ) is an (F t )-martingale} be the vector space whose elements we call timespace harmonic functions for M. For f V, its first argument is called the time variable, while the second is referred to as the space variable. The point at issue here is to describe the right notion of richness that would apply. A natural notion of richness could be denseness in some Lebesgue space, which has to be interpreted suitably in our context. Of course, we want denseness of the space of functions in one of the variables, namely, the space variable, while keeping the time variable fixed. Thus, for t 0, consider the following vector space of real valued functions on E : V t := {f(t, ) : f V }. From the definition of V, it is clear that V t L 1 (E, E, L(M t )) t 0. It should be pointed out that the Markov property that we shall prove will not guarantee timehomogeneity, though in applications, this case appears frequently. Throughout, for a random variable X, L(X) denotes its distribution. Theorem 1 If V t is dense in L 1 (E, E, L(M t )) for each t 0, then M is Markov. Proof : We show that for every n 2, 0 t 1 < t 2 <... < t n and every bounded measurable f : E R, E [ ] f(m tn ) M t1, M t2,..., M tn 1 = E [ ] f(m tn ) M tn 1 almost surely. For this, it is enough to show that the left hand side of the above equation is equal almost surely to a function of M tn 1. In fact, the above equation holds for every f L 1 (E, E, L(M tn )). To show this, suppose f L 1 (E, E, L(M tn )). By applying denseness, choose a sequence {f m V tn : m 1} such that f m f as m. This means, by change of variable, that f m (M tn ) f(m tn ) in L 1 ((Ω, F, P)). Now, the well-known contraction property in L 1 of conditional expectations implies that E [ ] f m (M tn ) M t1, M t2,..., M tn 1 E [ f(m tn ) M t1, M t2,..., M tn 1 ] 1 = E[f m (M tn ) f(m tn ) M t1, M t2,..., M tn 1 ] 1 f m (M tn ) f(m tn ) 1 0 as m. 2

3 This means that E [ ] f m (M tn ) M t1, M t2,..., M tn 1 converges to E [ ] f(m tn ) M t1, M t2,..., M tn 1 in L 1 ((Ω, F, P)) as m. But by the harmonicity of f m, for each m, each member of this sequence is a.s. equal to a function of M tn 1 ; viz. the random variable g m (t n 1, M tn 1 ), where g m V is such that f m ( ) = g m (t n, ). Using the almost sure convergence of a subsequence, the limit must also equal a function of M tn 1 almost surely. Remark 1 If p 1, then any subset of L p that is dense in L p is also dense in L q for every 1 q p, and hence in L 1 too. Thus, if one can check, for every t, the inclusion and denseness of V t in L pt (L(M t )) for some p t 1, Theorem 1 would apply, and result in M being Markov. 3 Consequences Applying Theorem 1 to the special case when the state space E is the real line R and M is p- harmonisable, we can generalise Corollary of [8]. Before stating the consequent result, we need to recall the definition of a p-harmonisable process as one admitting, for each k 1, a polynomial P k V of degree k in the space variable. First, we state Definition 1 A time-space harmonic polynomial for real-valued process M is a polynomial P in two variables such that P V. The set of all time-space harmonic polynomials for a process is denoted P(M), and for every k 1, we define the subset P k (M) as P k (M) := {P P(M) : deg(p (t, ) = k}. Definition 2 A real-valued process M is called polynomially harmonisable, or p-harmonisable for short, if for every k 1, P k (M). Simply this definition, however, is not sufficient for us to obtain much further by way of any result; and the mildest condition under which Markov property might at all be expected can be put as complete p-harmonisability : Definition 3 We call a p-harmonisable process M completely polynomially harmonisable if for any t 0 and k 1, there exists a polynomial P k,(t) V such that the degree of P k,(t) (t, ) is exactly k. A special case of a completely polynomially harmonisable process is what is called a restricted polynomially harmonisable process, where for each k 1, there exists a polynomial P k P k (M) whose leading coefficient in the space variable is a constant; that is, the coefficient of x k in P k (t, x) is free of t. Then, for each k 1, this P k serves as a choice for P k,(t) for all t 0. As is easy to see, complete p-harmonisability implies that for every t 0, P is contained in V t, where P stands for the set of all polynomials in one variable with real coefficients. For given any P P with P (0) = 0, it can be written as a (finite) linear combination from the set {P k,(t) (t, ) : k 1}, each of whose elements is in V t. In [8] (Corollary 3.2.1), Markov property for a completely p-harmonisable process M was obtained under the additional restriction that for each t 0, M t have moment generating function defined in a both-sided neighbourhood of 0. We restate the relevant result as Proposition 1 before proceeding further : Proposition 1 If for a completely p- harmonisable process M, the moment generating function (mgf) of M t exists in a neighbourhood of 0 for every t 0, then M is Markov. We may recall here the following Definition 4 A probability measure on (R, B) is called determinate if it has moments of all orders and they determine the measure uniquely. A realvalued random variable is called determinate if its distribution is determinate. 3

4 Since it is well-known that a sufficient condition for determinacy of a random variable is the existence of its mgf in a both-sided neighbourhood of the origin, it may not be difficult to guess that determinacy plays some role in the proof of Proposition 1. What is really crucial there is that the existence of mgf implies the determinacy not just of the original random variable, but also of almost each regular conditional distribution given any sub-σ-field. Thus, any condition which ensures this, or what we may call almost sure conditional determinacy, would also yield Markov property for a completely p-harmonisable process. However, for Markov property to hold, this is not necessary, as we see in the sequel. As has been known for a long time, determinacy of a random variable is enough to guarantee that polynomials be dense in the L 1 for its distribution (see [1], [2] and references therein for more information regarding the connection between polynomial density and determinacy). In fact, polynomial density in L 1 (µ) for a measure µ on (R, B) is equivalent to µ being an extreme point of the convex set M(µ) of measures with the same moment sequence as µ. We are thus led to the following corollary to Theorem 1, generalising Proposition 1: Corollary 1 If for a completely p-harmonisable process M = {M t : t 0}, the distribution L t of M t is determinate, or more generally, an extreme point of M(L t ) for every t 0, then M is Markov. One can easily derive the following discretetime versions of Theorem 1 and Corollary 1 with proofs running exactly along the same lines. Let M = (M n ) n 0 be a discrete-time process, (F n ) n 0 its natural filtration, V := {f : N {0} E R such that f(n, M n ) is an (F n )-martingale } be the class of time-space harmonic functions and V n := {f(n, ) : f V } for n 0. Theorem 2 If for every n 1, V n is dense in L 1 (E, E, L(M n )), then M is Markov. Corollary 2 If for a real-valued completely p- harmonisable discrete-time process M, the distribution L n of M n is either determinate, or an extreme point of M(L n ) for every n 1, then M is Markov. We may observe here that since for any random variable with all moments finite, polynomials form a subset of L p for every p 1, by virtue of Remark 1, the denseness of P in L pt (L(M t )) for some p t 1 for every t 0, would also imply that M is Markov. A natural question pertains to the extension of Corollaries 1 and 2 to the multivariate case, when the state space is R d for some d 2. But before stating the results, we need to give the appropriate definitions in this case. We define as the degree of a d-variable polynomial P as the multi-index k = (k 1, k 2,..., k d ) such that the degree of P (x 1,..., x i 1,, x i+1,..., x d ) equals k i for i i d and coeff(x k 1 1 xk 2 2 xk d d ) 0. As is clear, we may not be able to assign degrees to any arbitrary polynomial in this way. Accordingly, we shall call an R d -valued process p-harmonisable if for every multi-index k = (k 1, k 2,..., k d ) with k i 0, V admits a polynomial P with degree k. Completely and restricted p-harmonisable processes are defined in the obvious way. It is now clear that for completely p-harmonisable processes, the class of all polynomials is contained in V t for every t 0. One thus has the same result that polynomial denseness in L 1 (L(M t )) implies Markov property. As for the relation with determinacy, we refer to [5] where for a random vector also, determinacy was proved to imply denseness of polynomials in L 1. In turn, determinacy of each of its 1-dimensional marginals was shown to be sufficient for its determinacy. We state the continuous-time result here: Corollary 3 If for a d-dimensional completely p-harmonisable process M, the distribution of M t at any time t 0 is determinate, then M is Markov. In particular, the determinacy of the distribution of each of the components M i,t at any 4

5 time t 0 and 1 i d, implies Markov property for M. In the multi-dimensional case too, determinacy can be relaxed, and replaced by polynomial density; and so, by extremality, both for the measure as well as one-dimensional marginals. 4 Some Constructions In this section, we describe a few methods of construction of a sequence of time-space harmonic polynomials for a given polynomially harmonisable Markov process. The first method derives from the work of Neveu [N] for discrete-time processes which arise as partial sums of independent and identically distributed (iid) random variables, naturally from a distribution with all moments finite. Writing the summands as a sequence {X n : n 1} of iid random variables, one assumes to begin with that the cumulant generating function (cgf) ϕ(α) = log E{exp(αX 1 )} is defined in a neighbourhood of 0. This is of course equivalent to assuming the existence of the moment generating function (mgf) in the same neighbourhood. One then expands the function η(α, t, x) = exp{αx tϕ(α)} as a power series in α in this neighbourhood: η(α, t, x) = k=0 P k (t, x) αk k!. (1) Now, P 0 (, ) 1, and for each k 1, P k (, ) turns out to be a time-space harmonic polynomial for the Markov process M = {M n n 1} defined as: M 0 = 0, and M n = n i=1 X j, n 1. Further, P k is of degree k in both its arguments. The harmonicity of P k follows from successively interchanging the orders of the two operations of k-th order partial differentiation at α = 0 and of taking conditional expectation, given M 0,..., M n 1, on the exponential martingale {η(α, n, M n )}. The coefficients involved in P k depend, as is natural to expect, on the various moments of X 1 of order upto k for each k 1. Some of the relevant details may be found in [4]. A natural continuous-parameter extension of this situation exists to processes with stationary independent increments; or in reality Lévy processes, if one allows the minor additional restriction of rcll (right-continuous with left limits everywhere) paths. The necessary details are worked out in [8]. Thus, considering Brownian motion, the Poisson process, Gamma process etc. for the underlying process M, one gets many familiar sequences like the two-variable Hermite, Charlier polynomials and so on. Even when the mgf does not exist in an open set containing 0, but the moments of X 1 are all finite, one can obtain the polynomials P k which are given by exactly the same relations from the moments as in the case when the mgf exists. Thus though the exponential martingale may not be defined as before, same formal calculations can be carried out. However, the harmonicity of these polynomials needs to be established through some algebra involving moments, using the linearity of ordinary and conditional expectations. This alternative approach easily extends to processes with summands (respectively, increments), in the discrete-parameter (respectively, continuous-para-meter) case, which are independent but not necessarily identically distributed (respectively, stationary). For the discrete case, a necessary and sufficient condition for the process of partial sums to possess the polynomial harmonisability property is given in [4] with proof. The continuousparameter counterpart is available in [8]. Naturally, Markov processes covered by this situation are not necessarily time-homogeneous in contrast to the case of partial sums of iid summands or Lévy processes. For another class of continuous-time homogeneous Markov processes, however, an idea similar to the expansion of the exponential martingale is applicable. This class of processes is those for which the infinitesimal generator admits eigenfunctions which possess analyticity in a neigh- 5

6 bourhood of the origin. Specifically, let M = {M t : t 0} be timehomogenous as a Markov process. Then its semigroup possesses an operator A, called its generator, defined on a suitable subspace of the set of measurable functions on the state space, with the property that for every function f in its domain, Z t := f(m t ) t 0 Af(M s)ds is a martingale. Consider, for λ in some open subset Γ of R, the equation Aφ λ + λφ λ = 0. (2) Under the conditions that we describe in the sequel, solving this equation can give rise to the polynomials we seek. Firstly, assuming solvability of (2) in φ λ, one can show that φ λ (M t ) e λt is a martingale. Next, we assume that for all x, φ λ (x) is analytic as a function of λ in Λ, and for every k 1, its partial derivative of order k with respect to λ at a point λ 0 Γ is a polynomial in x of degree k, say q k (x). Then, expanding φ λ (x) e λt = (λ λ 0 ) k k=0 q j (x) j t k j j!(k j)! as power series in λ λ 0 and interchanging as before the orders of partial differentiation wrt λ and conditional expectation, we get that for every k 1, P k (t, x) := q j (x) j t k j j!(k j)! is time-space harmonic for M. The conditions required for this procedure to actually be applicable can be simplified in some cases, as for instance the case when M is a semistable Markov process. We recall that a process {M t : t 0} with M 0 = 0 is called semi-stable of index β > 0 if for every c > 0, the processes {M ct : t 0} and {c β M t : t 0} have the same distribution. For homogeneous (in time) Markov processes ((Ω, F, P), (F t ) t 0, (P x ) x 0 ) with the right halfline [0, ) as state space, the above definition of semi-stability implies (and in fact is equivalent to) the equality of the laws of (M ct ) under P x and of (c β M t ) under P x/cβ for all c > 0 and all x. This means, for the transition function p t (, ) of M, that p ct (x, A) = p t (x/c β, A/c β ). Properties of time-space harmonic polynomials for a polynomially harmonisable semi-stable Markov process are detailed in [8]. The simplification of our method for semistable Markov processs owes itself to semistability of the process manifesting in the fact that φ λ (x) = φ λ1 (λx/λ 1 ). So, by choosing a suitable λ 1 so that if only φ λ1 is analytic, we can obtain the required expansion. Given a polynomially harmonisable semi-stable Markov process, another method to obtain the same property of some other semi-stable Markov processes from this process, exploits what is known as the intertwining relation between them [3]. By an intertwining relation between two Markov semigroups (P t ) and (Q t ), or the corresponding processes, we mean the existence of an operator Λ such that ΛP t = Q t Λ t. In some cases, this operator Λ is given by the multiplicative kernel for a random variable Z, that is, Λf(x) = Ef(xZ). In such a case, if P (t, x) = k p j(t)x j is a time-space harmonic polynomial for the process corresponding to the semigroup (P t ), then Q (t, x) = ΛP (t, x) = p j (t)ez j x j is one for the process with semigroup (Q t ). This fact is easy to verify from the definition. To illustrate these methods with specific examples, we treat here first the square (M t ) of the Bessel process with dimension 2a, say, which forms, as a varies, the only family of semi-stable Markov processes with continuous paths as remarked in [3]. Its generator, at least on the space of twice continuously differentiable functions f, is given by Af = 2x f + 2a f. Actually, for our purpose, solving the equation (2) for any one particular value of a is good enough because time-space harmonic polynomials obtained for one value of a give rise to those for the other values as well, as we shall see later. We con- 6

7 sider the value a = 1 2, in which case the function φ λ (x) = cos( 2λx) can be easily seen to be a solution to (2). Now, the standard power series expansion cos θ = φ λ (x) = k=0 φ λ (x) e λt = ( 2λx) k (2k)! k=0 k=0 ( θ2 ) k /k! yields ; and in turn, λ k ( 2x) j t k j (2j)!(k j)!. By previous arguments, it follows that {P k (t, x) = k ( 2) j t k j (2j)!(k j)! xj : k 1} is a sequence of time-space harmonic polynomials for M. The above construction of time-space harmonic polynomials is merely an illustration of the method outlined, since our choice of α makes M into BES 2 (1), or simply, the square of one-dimensional Brownian motion. Thus one could have taken just the sequence P k (t, x) = H 2k (t, x), k 1, where {H k } are the Hermite polynomials. However, as indicated ealier, this method is applicable for more general Markov processes also, if the solution to (2) is wellbehaved in λ. We now consider a few examples of polynomially harmonisable processes arising out of intertwinings of Markov semigroups and calculate the resulting polynomials. Some examples of random variables which lead to semigroups intertwined with that of the square M of the Bessel process of dimension 2a, say, in the way that was described previouly are : Z = Z a,b β a,b has a beta distribution with parameters a and b, Z = 2Z a+b, where Z c γ c has a gamma distribution with parameter c. In the former case, we recover the semigroup of another Bessel process, this time of dimension 2b, and in the latter, a semigroup of a certain process detailed in [10] with increasing saw-teeth paths. The above procedure also allows one to obtain time-space harmonic polynomials for Azema s martingale (see e.g. [11]), M t = sign(b t ) t g t, t 0, where B is a Brownian motion and g t is the last hitting time of 0 by B before time t. Its semigroup is intertwined with that of Brownian motion by the multiplicative kernel for the random variable denoted m 1, arising as the terminal value of the process {m u : 0 u 1} called Brownian meander. m 1 has what is known as a Rayleigh distribution, with density x exp( x 2 /2), x 0. Chapter 15 of [11], in the context of Chaotic Representation Property, also presents an alternative, or direct, proof of the p-harmonisability of the process M, as also of each member of the class of Emery s martingales. In fact, Theorem 15.2 there makes it possible to directly apply Theorem 8 of [8]. The specific polynomials that we get from the method just outlined are: Azema s martingale : The semigroup of M t is intertwined with that of Brownian motion by the multiplicative kernel of the random variable m 1. We have, ( ) k Em1 k = 2 k/2 Γ 2 + 1, whence P k (t, x) = EH k (t, m 1 x) ( ) j = 2 j/2 Γ h k j (t)x j are time-space harmonic polynomials for M where H k (t, x) = k hk j (t)x j are the Hermite polynomials. BES 2 (2b) : M t has semigroup intertwined with that of BES 2 (1) by the random variable Z 1/2,b β 1/2,b. EZ k 1/2,b = (1/2) k (1/2 + b) k, so the time-space harmonic polynomials we get for Bessel process of dimension 2b are given by ( 2) j (1/2) j t k j (2j)!(b + 1/2) j (k j)! x j 7

8 The final example in this context is that of the process M whose semigroup is intertwined with that of the square of BES(1) by Z = 2Z b+1/2 γ b+1/2. Here, EZ k = Γ(k+b+1/2) Γ(k) P k (t, x) =, so that 5 Further remarks ( 2) j Γ(k + b + 1/2)t k j Γ(k)(2j)!(k j)! Remark 2 There is no reason to limit ourselves to real-valued functions, and the same results would hold good when the definitions of the classes V, V t and P are extended in the selfevident way to contain complex-valued functions. In defining P, the state-space still remains R, but now we allow polynomials with complex coefficients. We can thus have analogues of Theorem 1 and Corollary 1; in fact, the statements require no change at all. One may prove them both in a similar way; for Theorem 1, one has only to apply our proof and use two observations : first, that both the real and imaginary parts of a complex-valued martingale are real-valued martingales; and second, that denseness of a subspace S of the complex L p is equivalent to the denseness of the real parts and imaginary parts of functions in S in the real L p separately. To prove Corollary 1, one uses the fact that the determinacy of a distribution implies polynomial density in L 1 is true both in the real as well as complex cases. Remark 3 In this paper, we have not addressed the question : is determinacy of a random variable a sufficient condition for almost sure conditional determinacy, as we have defined? To rephrase the question, suppose we are given a probability space (Ω, F, P ), a sub-σ-field G of F and a real random variable X having all moments. Let Q(, ) denote a version of the regular conditional distribution of X given G. The question is whether, if the distribution P X 1 of X is determinate, it is true that for P -almost every ω, Q(ω, ) is determinate. While the answer to x j that question is this generality is not known, if X itself, or more generally, a k-to-1 function of X for some k 1, is measurable with respect to G, then the answer is yes, irrespective of whether X is determinate or not. For then, for almost every ω, the first two (or respectively, the first 2k) moments determine Q(ω, ) uniquely (see Lemma 3.1 of [9]). Acknowledgement. The author wishes to express his gratitude to Professors C. Berg for graciously making available reprints of his papers, and M. Yor and A. Goswami for helpful ideas. References [1] Christian Berg. Recent Results about Moment Problems. In: H. Heyer (ed.) Probability Measures on Groups and Related Structures XI (Proceedings, Oberwolfach, 1994). World Scientific, Singapore, [2] Christian Berg. Moment Problems and Polynomial Approximation. Ann. Fac. Sci. Toulouse Mathématiques, Numero Special Stieljes, pages 9 32, [3] P. Carmona, F. Petit and M. Yor. Sur les fonctionelles exponentielles de certain processus de Lévy. Stochastics and Stochastics Reports, 47, , [4] A. Goswami and Arindam Sengupta. Time-space Polynomial Martingales Generated by a Discrete-time Martingale. Journal of Theoretical Probability, 8, No. 2, , [5] L. C. Petersen. On The Relation Between The Multidimensional Moment Problem and The One-dimensional Moment Problem. Math. Scand., 51, , [6] W. Schoutens. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics, volume 146. Springer-Verlag,

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