Regularity of models associated with Markov jump processes

1 Introduction and main results

Jump Markov processes, have found application in Bayesian statistics, chemistry, economics, information theory, finance, physics, population dynamics, speech processing, signal processing, statistical mechanics, traffic modeling, thermodynamics, and many others [1]. Regularity plays a significant role in the classical asymptotic statistics for parametric statistical models for jump Markov processes; see [2,3,4] for recent developments. Asymptotic normality or Bernstein-von Mises-type theorems impose several regularity conditions so that their results hold rigorously. In this article, we focus on the regularity conditions of several statistical models associated with a jump Markov process X with values E being an arbitrary space state, endowed with a σ -field ℰ . Let Ω be the canonical space of piecewise constant functions ω : R + ⟶ E , right continuous for the discrete topology. Let X = ( X t ) t ⩾ 0 be the canonical process, ( ℱ t ) t ⩾ 0 the canonical filtration, and ℱ = ∨ t ≥ 0 ℱ t . Let T 0 = 0 and ( T n ) n ≥ 0 be the sequence of the jump times of X , which are almost surely increasing to ∞ . To each θ ∈ Θ ⊂ R d and x ∈ E , we associate

We assume that, under P x , θ , the process ( X t ) t ⩾ 0 is Markovian, starts from x ∈ E , is non-exploding, and admits the infinitesimal generator

The existence of the probabilities P x , θ is guaranteed by the boundedness of the function μ θ for instance. The following facts clarify our focus on the different statistical models that will be presented later on:

• under P x , θ , and conditionally to ℱ T n − 1 , the distribution of T n − T n − 1 is exponential with parameter μ θ ( X T n − 1 ) ;

• Q θ ( x , d y ) = P x , θ ( X T 1 ∈ d y ) is the transition probability of the embedded Markov chain ( X T n ) n ≥ 0 ;

• Q ¯ θ k ( x , d y , d t ) , k ∈ N , is the distribution of ( X T k , T k ) under P x , θ ;

• The associated sub-Markovian transition kernels ( P t θ ) t ≥ 0 satisfy the backward Kolmogorov equations:

  (1) ∂ ∂ t P t θ ( x , A ) = ∫ E ( P t θ ( y , A ) − P t θ ( x , A ) ) π θ ( x , d y ) , s , t ≥ 0 , x ∈ E , A ∈ ℰ .

  The Markov process X is simple if π θ ( x , d y ) is a Markov kernel, i.e., for every x ∈ E , π θ ( x , d y ) is a probability measure on ( E , ℰ ) . In this case, the transition functions are also Markov and satisfy the Chapman-Kolmogorov equation

  P s + t θ ( x , A ) = ∫ E P t θ ( y , A ) P s θ ( x , d x ) , x ∈ E , A ∈ ℰ ,

  and

  P x , θ ( X s + t ∈ A ∣ ℱ t ) = P s θ ( X t , A ) , P x , θ -almost surely,

  cf. [5] for more account.

• The multivariate point process associated with the process ( X t ) t ⩾ 0 is

  λ ( ⋅ , d t , d y ) = ∑ k ≥ 1 ε ( T k , X T k ) ( d t , d y ) ,

  and its compensator, under P x , θ , is

  ν θ ( ⋅ , d t , d y ) = π θ ( X t , d y ) 1 R + ( t ) d t .

The model E x is not a proper statistical model since ( π θ ( x , d y ) ) θ ∈ Θ is not a probability measure. Nevertheless, the extension of the notion of regularity to models associated with families of finite positive measures is also feasible and is described as follows. Let ( R θ ) θ ∈ Θ be a family of finite positive measures in ( E , ℰ ) . For θ , ξ ∈ Θ , we denote by Π θ , ξ a measure that dominates R 0 , R θ , and R ξ , and by z θ , ξ , z ξ , θ , and Z θ , ξ , be Radon-Nikodym derivatives, respectively, of R θ and R ξ according to Π θ , ξ and of R θ according to R ξ . The Lebesgue decomposition of R θ , with respect to R ξ , is given by the pair ( N θ , ξ , Z θ , ξ ) ,

We start by recalling the notion of “error functions” which was introduced in [6] as follows.

Note that if the regularity of the model ( E , ℰ , ( R θ ) θ ∈ Θ ) holds, then V is necessarily ( R 0 ) -square integrable. Furthermore, if ( R θ ) θ ∈ Θ is a family of probability measures, then E R 0 ( V ) = 0 . The Hellinger integral of order 1 2 between the measures R θ and R ξ , , is defined by

and is independent of the dominating measure Π θ , ξ . The regularity of the model ( E , ℰ , ( R θ ) θ ∈ Θ ) is equivalent to one of these two assertions:

1. There exist an error function f 1 and a random vector V (the same as before), such that

   (7) z θ , 0 − z 0 , θ − 1 2 z 0 , θ V ⋅ θ L 2 ( Π θ , 0 ) ≤ ∣ θ ∣ f 1 ( ∣ θ ∣ ) .

2. There exists an error function f 2 and a matrix I = [ I i j ] 1 ≤ i , j ≤ d , such that

   H 0 , 0 + H θ , ξ − H 0 , θ − H 0 , ξ − 1 4 θ ⋅ I ⋅ ξ ≤ ∣ θ ∣ ∣ ξ ∣ f 2 ( ∣ θ ∣ ∨ ∣ ξ ∣ ) .

   The matrix I is positive definite and is called the Fisher information matrix of the model at θ = 0 . It is linked to the vector V by

   I i j = R 0 ( V i V j ) ,

   cf. [6,7].

Let ( Ω , ℱ ) be a sample space endowed with a filtration ( ℱ t ) t ⩾ 0 , and a family of probability measures ( P θ ) θ ∈ Θ coinciding on ℱ 0 . The regularity of the statistical filtered model

mimics the one in Definition 2 and is expressed in terms of likelihood processes [8,7]. For a clear presentation, we need to introduce the likelihood process of P θ with respect to P ξ , θ , ξ ∈ Θ , defined in Jacod and Shiryaev’s book [9], by

The process Z t θ , ξ is a positive ( P ξ , ℱ t ) -supermartingale and is a martingale if

For any probability measure K θ , ξ , locally dominating P θ and P ξ , the ( K θ , ξ , ℱ t ) -martingales

and the stopping times

provide this version of Z θ , ξ :

As for non-filtered models, we have the following definition.

As in Definition 2 and according to [7, point 3.12], the regularity of the model is equivalent to the existence of a positive definite d × d matrix J T and of an error function f T , such that

where

is the Hellinger integral of order 1 2 , at time T , and which is independent of the choice of the dominating probability measure. The Fisher information matrix of the model is then

It is worth noting that if the regularity at a time T implies the regularity at any stopping time S ≤ T . In particular, if the regularity holds along a sequence S p , p ∈ N , increasing to infinity, then there exists a local martingale ( V t ) t ≥ 0 , locally square-integrable, null at zero, such that if T ≤ S p for some p , then V T is a version of the random variable in (10). In particular, if (9) and (10) are satisfied for all t ≥ 0 , then ( V t ) t ≥ 0 is a square-integrable martingale, null at 0, cf. [7, Corollary 3.16].

We are now able to introduce the notion of local regularity, which is less restrictive than the preceding one.

Note that if the model is regular along a localizing sequence, then it is necessarily locally regular. By Theorem [7, Theorem 4.6], the process ( V t ) t ≥ 0 is a locally square-integrable ( P 0 , ℱ t )-local martingale and the Fisher information process ( I t ) t ≥ 0 , at θ = 0 , is defined as the predicable quadratic covariation of ( V t ) t ≥ 0 :

The local regularity does guarantee the integrability of I ; however, it is the minimal condition we require to obtain the property of local asymptotic normality (LAN) for statistical models. In this case, the Fisher information quantities provide the lower bound of the variance of any estimator of the unknown parameters intervening in the models, see [10,11,12] for instance.

According to [7, Theorem 6.2], the local regularity is equivalent to the two following conditions:

and for all t ≥ 0 ,

where

• ( Var { . } t ) t ≥ 0 is the variation process of { . } .

• ( h t θ , ξ ) t ≥ 0 is a version of the Hellinger process of order 1 2 between P θ and P ξ , i.e., is a predictable nondecreasing process, null at zero, such that

  (17) z θ z ξ + [ z − θ z − ξ ] ∘ h θ , ξ is a ( K θ , ξ , ℱ t ) -martingale.

• ( A t θ ) t ≥ 0 is the predictable nondecreasing process intervening in the Doob-Meyer decomposition of the supermartingale ( Z t θ ) t ≥ 0 . Since P θ and P 0 coincide on ℱ 0 , then necessarily Z 0 θ = 1 and there exists a ( P 0 , ℱ t ) -local martingale ( M t θ ) t ≥ 0 such that

  Z θ = 1 + M θ − A θ .

The results that we obtain complete those of Höpfner et al. [6], who proved that if ( π θ ( x , d y ) ) θ ∈ Θ is regular and if the process X satisfies a condition of positive recurrence (resp. null recurrence), then the model ( P x , θ ) θ ∈ Θ localized around the parameter θ = 0 is or locally asymptotically normal or is locally asymptotically mixed normal. The main result is as follows.

Models (2)–(5) are described in depth in Section 2. We also provide a full scheme linking them by their regularity, see Theorems 6–8 and 10. The proofs are given in Section 3.

2 Additional regularity properties

Our notations and the calculus of the Hellinger integrals and of the likelihood processes are borrowed from Höpfner [13] and Höpfner et al. [6]. For x ∈ E and θ , ξ ∈ θ , the following measures will be used in the sequel.

1. Π x θ , ξ ( d y ) is a measure dominating π θ ( x , d y ) , π ξ ( x , d y ) , and π 0 ( x , d y ) . Thus, Π x θ , ξ ( d y ) also dominates Q θ ( x , d y ) , Q ξ ( x , d y ) , and Q 0 ( x , d y ) ;

2. Q x θ , ξ ( d y , d t ) is a transition probability on E × R + dominating Q ¯ θ ( x , d y , d t ) , Q ¯ ξ ( x , d y , d t ) , and Q ¯ 0 ( x , d y , d t ) ;

3. K x θ , ξ is a probability measure, locally dominating P x , θ , P x , ξ , and P x , 0 ;

4. Π x θ ( d y ) = Π x θ , 0 ( d y ) , Q x θ ( d y , d t ) = P x θ , 0 ( d y , d t ) , and K x θ = K x θ , 0 .

If we choose

and if K x θ , ξ is the probability under which the canonical process ( X t ) t ⩾ 0 , starts from x , and has the infinitesimal generator Π x θ , ξ ( d y ) , then we have

With the convention ∏ j = 1 0 = 1 , a version of the likelihood processes of P x , θ , with respect to K x θ , ξ and to ( ℱ t ) t ⩾ 0 , is given in [6] by

With the notations

a version of the likelihood processes of P x , θ , relative to P x , 0 and ( ℱ t ) t ⩾ 0 , is explicitly given by

The Hellinger integral of order 1 2 between π θ ( x , d y ) and π ξ ( x , d y ) is then

and the Hellinger integral of order 1 2 at time t between P x , θ and P x , ξ relative to the filtration ( ℱ t ) t ⩾ 0 , is expressed by

We also consider the quantities

which are used to define the Hellinger process ( h t θ , ξ ) t ≥ 0 , of order 1 2 , between P x , θ and P x , ξ , and relative to ( ℱ t ) t ⩾ 0 . It is expressed by

Finally, we define the function

where I ( x ) is the Fisher information matrix of the model E x at θ = 0 , whenever it is regular. Consequently, H ¯ 0 , θ ( x ) is expressed by

Observe that the function g in (20) is such that the function

is nondecreasing in u , satisfies ∣ g ( x , θ , ξ ) ∣ ≤ f ( x , ∣ θ ∣ ∨ ∣ ξ ∣ ) . Thus, f has the vocation to be an error function.

We can now state a first technical but intuitive result.

In the three following theorems, we complete our results by studying the regularity of the filtered model ℰ x , at fixed times t > 0 , or at the jump times T k , k ∈ N . In this direction, we obtain only partial results appealing to some additional conditions of integrability.

We conclude with our last result.

3 Proofs of the theorems

We will sometimes use the notion of isomorphism between two statistical models. Referring to Strasser’s book [14], we say that two models G = ( A , A , ( P θ ) θ ∈ Θ ) and ℋ = ( B , ℬ , ( Q θ ) θ ∈ Θ ) are isomorphic if they are randomized of each other. To illustrate this notion, assume for instance that G and ℋ are, respectively, dominated by P and Q . Then, the model ℋ is randomized from G , if there exists a Markovian operator M : L ∞ ( A , A , P θ ) → L ∞ ( B , ℬ , Q θ ) , such that