A Gaussian calculation

A Discrete Construction for Gaussian Markov Processes

Abstract

In the Lévy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process and to define the Wiener process as the almost sure path-wise limit of when tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes which can be written with and being continuous functions. We build the finite-dimensional process so that it gives an exact representation of the conditional expectation of with respect to the filtration generated by for . Moreover, we prove that the process converges in distribution toward .

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A Discrete Construction for Gaussian Markov Processes

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Gaussian Process \kwdMarkov Process \kwdDiscrete Representation

1 Introduction

Given some probability space, it is often challenging to establish results about continuous adapted stochastic processes. As a matter of fact, the mere existence of such processes can prove lengthy and technical: the direct approach to build continuous Markov processes consists in evaluating the desired finite-dimensional distributions of the process, and then constructing the measure associated with the process on an appropriate measurable space, so that this measure consistently yields the expected finite-dimensional distributions (13).
In that respect, it is advantageous to have a discrete construction of a continuous stochastic process. For more general purpose, a discrete representation of a continuous process proves very useful as well. Assuming some mode of probability convergence, at stake is to write a process as a convergent series of random functions

 Xt=∞∑n=0fn(t)⋅ξn=limN→∞N∑n=0fn(t)⋅ξn,

where is a deterministic function and is a given random variable.
The Lévy construction of Brownian motion –later referred as Wiener process– provides us with a first example of discrete representation for a continuous stochastic process. Noticing the simple form of the probability density of a Brownian bridge, it is based on completing sample paths by interpolation according to the conditional probabilities of the Wiener process (10). More especially, the coefficients are Gaussian independent and the elements , called Schauder elements, are obtained by time-dependent integration of the Haar elements. This latter point is of relevance since, for being a Hilbert system, the introduction of the Haar basis greatly simplify the demonstration of the existence of the Wiener process (3).
From another perspective, fundamental among discrete representations is the Karhunuen-Loève decomposition. Instead of yielding a convenient construction scheme, it represents a stochastic process by expanding it on a basis of orthogonal functions (9); (11). The definition of the basis elements depends only on the second-order statistics of the considered process and the coefficients are pairwise uncorrelated random variables. Incidentally, such a decomposition is especially suited to study Gaussian processes because the coefficients of the representation then become Gaussian independent. For these reasons, the Karhunen-Loéve decomposition is of primary importance in exploratory data analysis, leading to methods referred as “principal component analysis”, “Hotelling transform”  (6) or “proper orthogonal decomposition” (12) according to the field of application. In particular, it was directly applied to the study of stationary Gaussian Markov processes in the theory of random noise in radio receivers (7).

In view of this, we propose a construction of Gaussian Markov processes using a Haar-like basis of functions. The class of processes we consider is general enough to encompass commonly studied centered Gaussian Markov processes that satisfy minimal properties of continuity. We stress the connection with the Haar basis because our basis of decomposition is the exact analog of the Haar-derived Schauder functions used in the Lévy construction of the Wiener process. As opposed to the Karhunene-Loève decomposition, our basis is not made of orthogonal functions but the elements are such that the random coefficients are always independent and Gaussian with law , i.e. with zero mean and unitary variance.
The almost sure path-wise convergence of our decomposition toward a well-defined continuous process is quite straightforward. Most of the work lies in proving that the candidate process provides us with an exact representation of a Gaussian Markov process and in demonstrating that our decomposition converges in distribution toward this representation. Validating the decomposition essentially consists in proving the weak convergence of all the finite-dimensional measures induced by our construction on the Wiener space: it requires the introduction of an auxiliary orthonormal system of functions in view of using the Parseval relation. To furthermore establish the convergence in distribution of the representation, we only need demonstrating the tightness of this family of induced measures.

The discrete construction we present displays both analytical and numerical interests for further applications.
Analytically-wise, even if it does not exhibit the same orthogonal properties as the Karhunene-Loève decomposition, our representation can prove as advantageous to establish analytical results about Gaussian Markov processes. It is especially noticeable when computing quantities such as the characteristic functional of random processes (5); (2) as shown in annex. This is just an example of how, equipped with a discrete representation, one can expect to make demonstration of properties about continuous Gaussian Markov processes more tractable. If the measures induced by our decomposition on the classical Wiener space converge weakly toward the measure of a Gaussian Markov process, we put forward that the convergence of our decomposition is almost sure path-wise toward the representation of a Gaussian Markov process. This result contrasts with the convergence in mean of the Karhunene-Loève decomposition.
From another point of view, three Haar-like properties make our decomposition particularly suitable for certain numerical computations: all basis elements have compact support on an open interval with dyadic rational endpoints; these intervals are nested and become smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number. Thus the expansion in our basis, when evaluated at a dyadic rational, terminates in a finite number of steps. These properties suggest an exact schema to simulate sample paths of a Gaussian Markov process in an iterative “top-down” fashion. Assuming conditional knowledge of a sample path on the dyadic points of , one can decide to further the simulation of this sample path at any time in by drawing a point according to the conditional law of knowing , which is simply expressed in the framework of our construction. It can be used to great advantage in numerical computations such as dychotomic search algorithms for first passage times: considering a continuous boundary, we shall present elsewhere a fast Monte-Carlo algorithm that simulates sample-paths with increasing accuracy only in time regions where a first passage is likely to occur.

2 Main Result

Beforehand, we emphasize that the analytical and numerical advantages granted by the use of our decomposition come at the price of generality, being only suited for Gaussian Markov processes with minimal properties of continuity. We also remark that if the Karhunen-Loève decomposition is widely used in data analysis, our decomposition mainly provides us with a discrete construction scheme for Gaussian Markov processes.

Proposition.

Let be a real adapted process on some probability space which takes value in the set of real numbers and let with and be Gaussian random variables of law .
If there exist some non-zero continuous functions and such that

 Xt=g(t)⋅∫t0f(t)dWt,with0≤t≤1,

then there exists a basis of continuous functions for and such that the random variable

 XNt=N∑n=0∑0≤k<2n−1Ψn,k(t)⋅ξn,k

follows the same law as the conditional expectation of with respect to the filtration generated by for . The functions thus defined have support in and admit simple analytical expressions in terms of functions and .
Moreover, the path-wise limit defines almost surely a continuous process which is an exact representation of and we have

 XN\lx@stackrelD⟶X,

meaning that the finite-dimensional process converges in distribution toward when tends to infinity.

Remark.

The function can possibly be zero on a negligible set in in the previous proposition.

To prove this proposition, the paper is organized as follows. We first review some background about Gaussian Markov processes and their Doob representations as . Then we develop the rationale of our construction by focusing on the conditional expectations of the process with respect to the filtration generated by for . In the fifth section, we propose a basis of expansion to form the finite-dimensional candidate processes and justify the limit process as the almost sure path-wise convergent process . In the sixth section, we introduce the auxiliary Hilbert system and prove an important intermediate result. In the last section, we show that the finite-dimensional processes converge in distribution toward and that is an exact representation of .

3 Background on Gaussian Markov Processes

3.1 Basic Definitions

We first define the class of Gaussian Markov processes. Let us consider on some probability space a real adapted process which takes value in the set of real numbers. We stress that the index of the random variable runs in the continuous set . For a given realization in , the collection of outcomes is a sample path of the process . We only consider processes for which the sample paths are continuous. With these definitions, we are in a position to state the two properties characterizing a Gaussian Markov process.

1. We say that is a Gaussian process if, for any integers and positive reals , the random vector has a joint normal distribution.

2. We say that is a Markov process if, for any and , with the set of real Borelians,

 Extra open brace or missing close brace

which states that the conditional probability distribution of future states , given the present state and all past states , depends only upon the present state .

A Gaussian Markov process is a stochastic process that satisfies both Gaussian and Markov properties.
The Wiener process and the Ornstein-Uhlenbeck process are two well-known examples of Gaussian Markov processes. The Wiener process is defined as the only continuous process for which and the increments are independent of and normally distributed with law . These requirements naturally place the Wiener process in the class of Gaussian Markov process. The Ornstein-Ulhenbeck process can be defined as a solution of the stochastic differential equation of the form

 dXt=αXtdt+dWtwithα∈R,

We designate the Ornstein-Ulhenbeck process of parameter starting at for and we give its integral expression

 Uαt=∫t0eα(u−t)dWu. (1)

The process naturally appears as a Gaussian Markov process as well: it is Gaussian for integrating independent Gaussian contributions and Markov for being solution of a first-order stochastic differential equation. It is known that both processes can be described as discrete processes with an appropriate basis of random functions (14).

3.2 The Doob Representation

The discrete construction of the Wiener process and the Ornstein-Uhlenbeck process is likely to be generalized to a wider class of Gaussian Markov processes because any element of this class can be represented in terms of the Wiener process. By Doob’s theorem (4), for any Gaussian Markov process , there exist a real non-zero function and a real function in such that we have the integral representation of

 Xt=g(t)⋅∫t0f(t)dWt. (2)

where is the standard Wiener process. If we introduce the non-decreasing function defined as

 h(t)=∫t0f2(u)du,

then, for any , the covariance of can be expressed in terms of functions and

 E(Xt⋅Xs)=g(t)g(s)⋅h(min(t,s)). (3)

The Doob’s representation (2) indicates that is obtained from by a change of variable in time and, at any time , by a change of variable in space by a time-dependent factor . The couple of functions that intervenes in the Doob’s representation of is not determined univocally. Yet, one can defined a canonical class of functions which are uniquely defined almost surely in if we omit their signs (5). Incidentally, we can compare the integral formulation (2) with expression (1) of the Ornstein-Uhlenbeck process: we remark that the representation of this Gaussian Markov process is provided by setting and , which happens to be its canonical representation.

3.3 Analytical Results

The discrete construction of Gaussian Markov processes will rely on two analytical results that we detail in the following.
First, the Doob’s representation allows us to give an analytical expression for the transition kernel of a general Gaussian Markov process. As the Doob’s representation is a simple change of variables, it is easy to transform the expression of the Wiener transition kernel to establish

 p(Xt=x|Xt0=x0)=1g(t)√2π(h(t)−h(t0))⋅exp⎛⎜ ⎜ ⎜⎝−(xg(t)−x0g(t0))22(h(t)−h(t0))⎞⎟ ⎟ ⎟⎠.

We have to mention that this expression is only valid if , otherwise is deterministic and .
Second, we can use this result to evaluate with , the probability density of knowing its values and at two framing times and . Because is a Markov process, a sample path which originates from and joins through is just the junction of two independent paths: a path originating in going to and a path originating from going to . Therefore, after normalization by the absolute probability for a path to go from to , we have the probability density

 q(Xty=y|Xtx=x,Xtz=z)=p(Xty=y|Xtx=x)⋅p(Xtz=z|Xty=y)p(Xtz=z|Xtx=x).

Thanks to the previous expression, we can compute the distribution of knowing and , which is expected to be a normal law because we only consider Gaussian processes. For a general Gaussian Markov process , we refer to that probability law as , with mean value and variance . We show in annex that these parameters satisfy

 Xμ(ty)=g(ty)g(tx)⋅h(tz)−h(ty)h(tz)−h(tx)⋅x+g(ty)g(tz)⋅h(ty)−h(tx)h(tz)−h(tx)⋅z, (4)
 Xσ(ty)2=g2(ty)⋅(h(ty)−h(tx))(h(tz)−h(ty))h(tz)−h(tx). (5)

Once more, we have to mention that these expressions are only valid if . Wether considering a Wiener process or an Ornstein-Uhlenbeck process, the evaluation of (4) and (5) with the corresponding expression of and leads to the already known results (14).

4 The Rationale of the Construction

4.1 Form of the Discrete Representation

Form now on, we will suppose that the zeros of the function pertain to a negligible ensemble in , causing the function to be strictly increasing. We will further restrain ourselves to Gaussian Markov processes for which the functions and belong to the set of continuous functions on denoted . We remark that in such a case, the functions and are continuous on .
Bearing in mind the example of the Lévy construction for the Wiener process, we want to define a basis of continuous functions in with to form the discrete process

 XNt=N∑n=0∑0≤k<2n−1Ψn,k(t)⋅ξn,k,

where are independent Gaussian random variables of standard normal law . We want to chose so that converges almost surely toward when tends to infinity. Given the continuous nature of the processes , we require that the convergence is uniform and normal on to ensure the definition of a continuous limit process on . Moreover, we want to define on supports of the form

 Sn,k=[k⋅2−n+1,(k+1)2−n+1]

As a consequence, the basis of functions will have the following properties : all basis elements have compact support on an open interval with dyadic endpoints; these intervals are nested and becomes smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number.

4.2 Conditional Averages of the Process

Now remains to propose an analytical expression for . If we denote the set of reals , the key point is to consider the conditional expectation of the random variable given with pertaining to the set of dyadic points . The collection of random variables defined on specify a continuous random process on . We notice that, if and with are the two successive points of framing t, the random variable is only conditioned by and :

 ZNt=E(Xt|{Xs},s∈DN)=E(Xt|Xtx,Xtz).

Using expression (4), we can express the sample paths as a function of on : for a given in the sample space , we write

 ZNt(ω)=Xμtx,tz(t,x,z)\lx@stackreldef=XμN,k(t),

where the conditional dependency upon parameters and is implicit in . The random process appears then as a parametric function of : for any in the sample space , the sample path determines a set of value and by extension a sample path for the process .
Now two points are worth noticing: first, and are continuous sample paths that coincide on the set ; second, we have a growing sequence of sets with limit ensemble the set of dyadic points in , which is dense in . Then, provided the path-wise convergence is almost surely normal and uniform on , the limit process of when tends to infinity should be continuous and the processes and should be indistinguishable on .

4.3 Identification of Conditional Averages and Partial Sums

Identifying the process with the partial sums provides us with a rationale to build the functions .
We first need to consider the random variable on the support of the function . The Markov property of the process entails

 ZN+1t = E(Xt|Xtx,Xty,Xtz) = {E(Xt|Xtx,Xty)if \> tx≤t≤ty,E(Xt|Xty,Xtz)if \> ty≤t≤tz.

Hence, for a given the estimation of the sample paths is now dependent upon with the midpoint of and

 ZN+1t(ω)={Xμtx,ty(t,x,y)if \> tx≤t≤tyXμty,tz(t,y,z)if % \> ty≤t≤tz\lx@stackreldef=XνN,k(t,y).

We can identify the conditional process and the partial sums . Then, for any in , writing the sample path as a function of , we have

 ΨN+1,k(t)⋅ξN+1,k(ω)=XN+1t(ω)−XNt(ω)=ZN+1t(ω)−ZNt(ω)=XνN,k(t,y)−XμN,k(t).

Assuming conditional knowledge on , the quantity becomes deterministic and the outcome of the random variable is only dependent upon the values of the process on . More precisely, on the support , the outcome of is determined through the function by the outcome of given and .
The distribution of given and follows the law and we denote a Gaussian variable distributed according to such a law. With this notation, we are in a position to propose the following criterion to compute the function : the element is the only positive function with support included in such that the random variable has the same law as . Direct calculations confirms that the previous relation provides us with a consistent paradigm to define the functions . Incidentally, we have an interpretation for the statistical contribution of the components : if one has previous knowledge of on , the function represents the uncertainty about that is discarded by the knowledge of its value on .

5 The Candidate Discrete Process

5.1 The Basis of Functions

We recall that we carry out the case for which the function has a negligible set of zeros in , which directly follows from the previous section. Before specifying the candidate basis elements , we introduce the following short notations to simplify the writing of their expressions

 ln,k=(2k)2−n,mn,k=(2k+1)2−n,rn,k=2(k+1)2−n.

Then for and , the explicit formulation of the basis of functions reads

 Missing or unrecognized delimiter for \big (6)

where we use the constants and that are defined by the relations

 Ln,k= ⎷h(rn,k)−h(mn,k)(h(rn,k)−h(ln,k))(h(mn,k)−h(ln,k))
 Rn,k= ⎷h(mn,k)−h(ln,k)(h(rn,k)−h(ln,k))(h(rn,k)−h(mn,k))

For , the basis element needs to satisfy the relation

 Ψ0,0(t)⋅ξ0,0=E(Xt|{Xs},s∈D0={0,1}),

which completely defines the analytical expression of as follows

 Ψ0,0(t)=g(t)⋅(h(t)−h(l0,0))√h(r0,0)−h(l0,0).

As expected, we directly ascertain the continuity of the by continuity of and .
We should briefly discuss the form of the functions . In the case for which and , we find the usual expression of for the Lévy construction of a Wiener process: the elements of the basis are the triangular wedged-functions obtained by integration of the standard Haar functions. In the general case of a Gaussian Markov process, the expression of can be derived from the Wiener process basis elements by three operations: a change of variable in time , a time-dependent change of variable in space and a multiplication by the coefficients and . The effect of this multiplication by and will be explain in section 7.
Moreover, the paradigm of the construction makes no assumption about the form of the binary tree of nested compact supports . Let us consider a given segment and construct by recurrence such a tree. We suppose that we have the following partition

 Missing or unrecognized delimiter for \big

For each such that , we draw a point in . Then, we have

 In,k=[ln,k,mn,k[⋃[mn,k,rn,k[=In+1,2k⋃In+1,2k+1

and by construction, we posit . Iterating the process for increasing , we build a tree of nested compact supports .
The definition (6) enables us to explicit elements that are adapted to any such tree. The so-defined functions will appear to be valid basis elements to build a discrete representation of under the only requirement that

 limn→∞sup0≤k<2n−1|rn,k−ln,k|=0. (7)

5.2 The almost sure Normal and Uniform Convergence

We want to prove the validity of the discrete representation of a Gaussian Markov process with Doob’s representation (2) using the proposed basis of functions . Let us consider the partial sums defined on by

 XNt=N∑n=0∑0≤k<2n−1Ψn,k(t)⋅ξn,kfort∈S0,0=[0,1].

We need to study the path-wise convergence of the partial sums on to see in which sense we can consider as a proper stochastic process.
Again, we only consider Gaussian Markov processes for which the functions and belong to the set of continuous functions . If we designate the norms of and on by and , we can show that

 sup0≤k<2n−1sup0≤t≤1∣∣Ψn,k(t)∣∣ ≤ √(rn,k−mn,k)(mn,k−ln,k)rn,k−ln,k⋅∥g∥∞⋅∥f∥∞ (8) ≤ 2−n+12⋅∥g∥∞⋅∥f∥∞.

For -bounded Gaussian Markov processes, this inequality provides us with the same upper bound to the elements as in the case of a Wiener process times a constant . By the same Borel-Cantelli argument as for the Haar construction of the Wiener process (8), for almost every in , the sample path converges almost surely normally and uniformly in to a function when goes to infinity.
It is worth noticing that, since and are continuous functions, so are the basis functions . Then, for every in , the sample path is a continuous function in . As the convergence when tends to infinity is normal and uniform in t, the limit functions results to be in almost surely on . This allows us to define on a limit process with continuous paths.
Showing that is an admissible discrete representation of the Gaussian Markov process only amounts to demonstrate that, for any integers and positive reals , the random vector has a the same joint distribution as . As is defined as the path-wise almost sure limit of when tends to infinity, this result is implied by the convergence in distribution of the continuous processes toward their limit  (8); (1). We will therefore establish the convergence in distribution of our representation in section 8 and incidentally validate as an exact representation of . In that perspective, we devote the following section to set out the meaning of the convergence in distribution of our candidate process .

6 The Convergence in Distribution

6.1 The Finite-dimensional Measures

In this section, we specify the finite-dimensional probability measures induced by the processes . Beforehand, we introduce the notations

 [n,k]=2n−1+kwith{[0,0]=0[0,1]=2N

to allow us to list the midpoints of the tree of supports in the prefix order. By reindexing according to

 t[n,k]=mn,k=(2k+1)2−n,

we get an ordered sequence . Let us now consider the finite-dimensional space of admissible functions for

 CN=Vect({Ψn,k}0≤n≤N0≤k<2n−1)

When it is equipped with the norm, is a complete, separable metric space under the distance . We can provide the space with the -algebra generated by the cylinder sets , which are defined for any collection of Borel sets in by

 CB0,⋯,B2N={x∈CN|x(t[n,k])∈B[n,k],0≤n≤N,0≤k<2n−1}

The random process induces a natural measure on , such that

 ∀B∈B(CN),PN(B)=P(ω|XN(ω)∈B).

Since for any in we have , the induced measure is entirely determined on the cylinder sets of the form with . Keeping this in mind, we show in annex that admits a probability density : for any cylinder set of with we have

 PN(CB0,⋯,B2N)=∫B1⋯∫B2NpN(x1,⋯,x2N)dx1⋯dx2N

where is made explicit with the help of the transition kernel of defined in (3.3)

 pN(x1,⋯,x2N)=2N−1∏k=0p(xk+1,tk+1|xk,tk).

We want to specify in which sense the finite-dimensional probability measures defined on converge to a limit measure associated to .

6.2 The Weak Convergence

Here, we consider that the stochastic processes and take value in the Wiener space, that is the space of continuous functions . This allows us to characterize the -induced measure associated with on . Defining the -induced measures on as well, we then state the convergence of toward in terms of weak convergence of toward .
The process defined on some probability space have continuous sample paths and so does the Gaussian Markov process . Being a complete, separable metric space under the distance , the Wiener space is a natural space to define - and -induced measures. We consider and as a random variables with values in the measurable space , where is the -field generated by the cylinder sets of : as previously and naturally induce the probability measures and defined by

 μN(B)=P{ω∈Ω∣∣XN(ω)∈B}andμ(B)=P{ω∈Ω∣∣X(ω)∈B}

for any in .
More specially, the measure is called the Wiener measure of the Gaussian Markov process . Assuming a general cylinder set to be

 Ct1,⋯,tnB1,⋯,Bn={x∈C[0,1]∣∣x(tk)∈Bk,0

for any and any Borel sets in , is the unique probability measure such that for any set

 μ(Ct1,⋯,tnB1,⋯,Bn)=∫B1⋯∫Bnp(x1,t1|0,0)⋯p(xn,tn|xn−1,tn−1)dxt1⋯dxtn.

Through the induced measures and , we want to study the convergence of the process toward from a probabilistic point of view. In that respect, the most general form of convergence one might expect is the weak convergence. We recall that is weakly convergent to if and only if for any bounded continuous function of , we have

 limN→∞∫Cϕ(x)dμN(x)=∫Cϕ(x)dμ(x),

where is a short notation for .

6.3 The Convergence in Distribution

We can now translate on the Wiener space the much desirable property that the continuous processes converges in distribution toward the Gaussian Markov process , which we denote

 XN\lx@stackrelD⟶X.

We recall that, by definition, converges in distribution to if and only if for any bounded continuous function in we have

 limN→∞EN(ϕ(XN))=E(ϕ(X))

where and are the expectations with respect to and respectively. With the definitions made in the previous section, the convergence in distribution of the representation is rigorously equivalent to the weak convergence of the measures toward on the Wiener space .
We will show this point in section 8 following the usual two steps reasoning inspired by the Prohorov theorem (1). To show the weak convergence of the sequence of measure , it is enough to prove the two statements:

1. For every integer and reals , the finite-dimensional vector converges in distribution to when tends to infinity.

2. The family of induced measures is tight: for every , there exist a compact such that for every .

Before establishing these two criteria, we first need to compute the limit of the covariance of