A Discrete Construction for Gaussian Markov Processes

We first define the class of Gaussian Markov processes. Let us consider on some probability space $\left(\Omega ,F,P\right)$ a real adapted process $X=\{X_t,F_t;0\le t<\infty \}$ which takes value in the set of real numbers. We stress that the index $t$ of the random variable $X_t$ runs in the continuous set $R^{+}$. For a given realization $\omega$ in $\Omega$, the collection of outcomes $t\mapsto X_t\left(\omega \right)$ is a sample path of the process $X$. We only consider processes $X$ for which the sample paths $t\mapsto X_t\left(\omega \right)$ are continuous. With these definitions, we are in a position to state the two properties characterizing a Gaussian Markov process.

1. We say that $X$ is a Gaussian process if, for any integers $k$ and positive reals $t_1<t_2<\cdots <t_k$, the random vector $(X_{t_1},X_{t_2},\cdots ,X_{t_k})$ has a joint normal distribution.

2. We say that $X$ is a Markov process if, for any $s,t\ge 0$ and $\Gamma \in B\left(R\right)$, with $B\left(R\right)$ the set of real Borelians,

   $$\text{Extra open brace or missing close brace}$$

   which states that the conditional probability distribution of future states $X_{t+s}$, given the present state and all past states $F_s$, depends only upon the present state $X_s$.

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 $W_t$ for which $W_0=0$ and the increments $W_t-W_s$ are independent of $F_s$ and normally distributed with law $N\left(0,t-s\right)$. 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

We designate $U_t^{\alpha }$ the Ornstein-Ulhenbeck process of parameter $\alpha$ starting at $0$ for $t=0$ and we give its integral expression

The process $U^{\alpha }$ 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).

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 $X$, there exist a real non-zero function $g$ and a real function $f$ in $L_{loc}^2\left(R^{+}\right)$ such that we have the integral representation of $X$

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

then, for any $t,s\ge 0$, the covariance of $X$ can be expressed in terms of functions $h$ and $g$

The Doob’s representation (2) indicates that $X$ is obtained from $W$ by a change of variable in time $t\mapsto h\left(t\right)$ and, at any time $t$, by a change of variable in space by a time-dependent factor $x\mapsto g\left(t\right)\cdot x$. The couple of functions $(f,g)$ that intervenes in the Doob’s representation of $X$ is not determined univocally. Yet, one can defined a canonical class of functions $(f,g)$ which are uniquely defined almost surely in $L_{loc}^2\left(R^{+}\right)$ 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 $g\left(t\right)=e^{\alpha t}$ and $f\left(t\right)=e^{-\alpha t}$, which happens to be its canonical representation.

Form now on, we will suppose that the zeros of the function $f$ pertain to a negligible ensemble in $[0,1]$, causing the function $h$ to be strictly increasing. We will further restrain ourselves to Gaussian Markov processes for which the functions $f$ and $g$ belong to the set of continuous functions on $[0,1]$ denoted $C(0,1)$. We remark that in such a case, the functions $t\mapsto {}_X\mu \left(t\right)$ and $t\mapsto {}_X\sigma \left(t\right)$ are continuous on $[0,1]$.
Bearing in mind the example of the Lévy construction for the Wiener process, we want to define a basis of continuous functions ${\Psi }_{n,k}$ in $C(0,1)$ with $0\le k<2^n$ to form the discrete process

where ${\xi }_{n,k}$ are independent Gaussian random variables of standard normal law $N(0,1)$. We want to chose ${\Psi }_{n,k}$ so that $t\mapsto X^N\left(\omega \right)$ converges almost surely toward $t\mapsto X\left(\omega \right)$ when $N$ tends to infinity. Given the continuous nature of the processes $X_N$, we require that the convergence is uniform and normal on $[0,1]$ to ensure the definition of a continuous limit process $\overline{X}={lim}_{N\to \infty }{X}^N$ on $\left[0,1\right]$. Moreover, we want to define ${\Psi }_{n,k}$ on supports $S_{n,k}$ of the form

As a consequence, the basis of functions ${\Psi }_{n,k}$ 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 $n$ of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number.

Now remains to propose an analytical expression for ${\Psi }_{n,k}$. If we denote $D_N$ the set of reals $\{k{2}^{-N}|0\le k\le 2^N\}$, the key point is to consider $Z_t^N=E(X_t|\left\{X_s\right\},s\in D_N)$ the conditional expectation of the random variable $X_t$ given $X_s$ with $s$ pertaining to the set of dyadic points $D_N$. The collection of random variables $Z_t^N$ defined on $\Omega$ specify a continuous random process $Z^N$ on $\Omega$. We notice that, if $t_x=k{2}^{-N}$ and $t_z=(k+1)2^{-N}$ with $0\le k<2^{-N}$ are the two successive points of $D_N$ framing t, the random variable $Z_t^N$ is only conditioned by $X_{t_x}$ and $X_{t_z}$:

Using expression (4), we can express the sample paths $t\mapsto Z_t^N\left(\omega \right)$ as a function of $t$ on $[t_x,t_z]$: for a given $\omega$ in the sample space $\Omega$, we write

where the conditional dependency upon parameters $X_{t_x}\left(\omega \right)=x$ and $X_{t_z}\left(\omega \right)=z$ is implicit in ${}_X{\mu }^{N,k}$. The random process $Z_t^N$ appears then as a parametric function of ${\left\{X_s\right\}}_{s\in D_N}$: for any $\omega$ in the sample space $\Omega$, the sample path $t\mapsto X_t\left(\omega \right)$ determines a set of value ${\left\{x_s\right\}}_{s\in D_N}={\left\{X_s\right(\omega \left)\right\}}_{s\in D_N}$ and by extension a sample path $t\mapsto Z_t^N\left(\omega \right)$ for the process $Z^N$.
Now two points are worth noticing: first, $t\mapsto X_t\left(\omega \right)$ and $t\mapsto Z_t^N\left(\omega \right)$ are continuous sample paths that coincide on the set $D_N$; second, we have $\{0,1\}=D_0\subset D_1\subset \cdots \subset D_N$ a growing sequence of sets with limit ensemble $D$ the set of dyadic points in $\left[0,1\right]$, which is dense in $[0,1]$. Then, provided the path-wise convergence is almost surely normal and uniform on $[0,1]$, the limit process of $Z^N$ when $N$ tends to infinity should be continuous and the processes ${lim}_{N\to \infty }{Z}^N$ and $X$ should be indistinguishable on $\Omega$.

We recall that we carry out the case for which the function $f$ has a negligible set of zeros in $[0,1]$, which directly follows from the previous section. Before specifying the candidate basis elements ${\Psi }_{n,k}$, we introduce the following short notations to simplify the writing of their expressions

Then for $n>0$ and $0\le k<2^{n-1}$, the explicit formulation of the basis of functions ${\Psi }_{n,k}$ reads

where we use the constants $L_{n,k}$ and $R_{n,k}$ that are defined by the relations

For $N=0$, the basis element ${\Psi }_{0,0}$ needs to satisfy the relation

which completely defines the analytical expression of ${\Psi }_{0,0}$ as follows

As expected, we directly ascertain the continuity of the ${\Psi }_{n,k}$ by continuity of $f$ and $g$.
We should briefly discuss the form of the functions ${\Psi }_{n,k}$. In the case for which $g\left(t\right)=1$ and $h\left(t\right)=t$, we find the usual expression of ${\Psi }_{n,k}$ for the Lévy construction of a Wiener process: the elements of the basis are the triangular wedged-functions obtained by integration of $H_{n,k}$ the standard Haar functions. In the general case of a Gaussian Markov process, the expression of ${\Psi }_{n,k}$ can be derived from the Wiener process basis elements by three operations: a change of variable in time $dt\mapsto h^{\prime }\left(t\right)dt=f^2\left(t\right)dt$, a time-dependent change of variable in space $x\mapsto g\left(t\right)\cdot x$ and a multiplication by the coefficients $L_{n,k}$ and $R_{n,k}$. The effect of this multiplication by $L_{n,k}$ and $R_{n,k}$ 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 $S_{n,k}$. Let us consider a given segment $I_{0,0}=[l_{0,0},r_{0,0}[$ and construct by recurrence such a tree. We suppose that we have the following partition

For each $k$ such that $0\le k<2^{n-1}$, we draw a point $m_{n,k}$ in $S_{n,k}$. Then, we have

and by construction, we posit $m_{n,k}=l_{n+1,2k+1}=r_{n+1,2k}$. Iterating the process for increasing $n$, we build a tree of nested compact supports $S_{n,k}=\overline{I_{n,k}}$.
The definition (6) enables us to explicit elements ${\Psi }_{n,k}$ that are adapted to any such tree. The so-defined functions ${\Psi }_{n,k}$ will appear to be valid basis elements to build a discrete representation of $X$ under the only requirement that