denoted by E [z] := l/T J dt z(t). For the probability density function (pdf) of z, we use p(z). A synthetic regular (or homogeneous) time series (RTS), spaced by 8t, derived from the irregular time series z, is denoted by RTS[<5f; zj. For a standardized time series for z, we use the notation z. = (z - E [z])/tr[z] and o[z\2 = E[(z - E[z])2]. The letter x is used to represent the logarithmic middle price as defined by Equation 3.4.

3.3.2 Linear Operator and Kernels

We focus on operators with the following useful properties:

Linear operators, where £2[zi + zi\ = &[zi] + Q[z2]

Time-translation invariant operators, where £l\z(t - AO 1(0 = Si[z{t)](t ~ A/)

Causal operators, where £2[z(0 exclusively depends on information already know at time t. If £2[ ](0 depends on future events after t, it does not respect causality at time t and is noncausal.

An operator with all these three properties can be represented by a convolution with a kernel w(t):

Si[z](t) = f dt co(t -1) z(t) (3.29)

J-oo

/•oo

= / dtw(t) z(t - t) Jo

The causal kernel w{t) is defined only on the positive semiaxis t > 0 and should decay for t large enough. With this convention for the convolution, the weight given to past events corresponds to the value of the kernel for positive argument. The value of the kernel (t - t) is the weight of events in the past, at a time interval t - t from t. In this convolution, z(t) is a continuous function of time. Actual time series z are known only at the sampling time f, and should be interpolated between sampling points. As in Section 3.2.1, we can define different interpolation procedures for the value of z(t) between tj-\ and tj. Three are used in practice:

Previous value, z(t) = Zj-i

Next value, (0 = Zj

Linear interpolation, z(t) = zj-\ + {zj - Zj-\){t - tj-\)/(tj - tj-\)

The linear interpolation seems preferable as it leads to a continuous interpolated function. Moreover, linear interpolation defines the mean path of a random walk, given the start and end values. Unfortunately, it is non-causal, because in the interval between f, i and tj, the value at the end of.the interval zi is used. Only the previous-value interpolation is causal, as only the information known at f,-t is used in the interval between t,-\ and t-t. Any interpolation can be used for historical computations, but for the real-time situation, only the causal previous-value interpolation is defined. In practice, the interpolation scheme is almost

irrelevant for good macroscopic operators, (i.e., if the kernel has a range longer than the typical sampling rate).

The kernel w(t) can be extended to t e R, with co(t) = 0 for t < 0. This is useful for analytical computation, particularly when the order of integral evaluations has to be changed. If the operator Q is linear and time-translation invariant but noncausal, the same representation can be used except that the kernel may bc nonzero on the whole time axis.

We often use two broad families of operators that share general shapes and properties:

An average operator has a kernel which is nonnegative, co(t) > 0, and normalized to unity, /dtco(t) = 1. This implies that £2[Parameters; Const] =Const.

Derivative and difference operators have kernels that measure the difference between a value now and a value in the past (with a typical lag of ). Their kernels have a zero average, / dt w(t) - 0, such that [Parameters; Const] = 0.

The integral in Equation 3.29 can also be evaluated in scaled time. In this case, the kernel is no more invariant with respect to physical time translation (i.e., it depends on t and t), but it is invariant with respect to translation in business time. If the operator is an average or a derivative, the normalization property is preserved in scaled time.

The -th moment of a causal kernel is defined as

The range R and the width w of an operator £1 are defined, respectively, by the following relations:

For most operators £2[ ] depending on a time range , the formula is set up so that

(3.30)

(3.31)

[ [ ]] = .

Linear operators can be applied successively:

£lc[z\ = &2 adz] = n2niz := Sliindz]]

It is easy to show that the kernel of Qc is given by the convolution of the kernels of £2\ and

= a>\ * a>2 or (3.32)

dt"co\(t -t")co2{t" -t)

(3.33)

For causal operators,

2 t t

oc(t)= I dt coi(- - t)co2(t + -) forf>0 (3.34)

J-t/2 2 2

and a>c(t) = 0 for t < 0. Under convolution, range and width obey the following simple laws:

Rc = R\+ R2

w2c = w} + w\ (3.35)

(t2)c = (t2)\+{t2)2+2rxr2

3.3.3 Build-Up Time Interval

As our basic building blocks are EMA operators, most kernels have an exponential tail for large t. This implies that, when starting the evaluation of an operator at time T, a build-up time interval must be elapsed before the result of the evaluation is accurate enough (i.e., the influence of the initial error at T has sufficiently faded). This heuristic statement can be expressed by quantitative definitions. We assume that the process z(t) is known since time -T and is modeled before as an unknown random walk with no drift. Equation 3.29 for an operator Q needs to be modified in the following way:

[- ; ](0 = f dtco{t -t)z{t) . (3.36)

The "infinite" build-up corresponds to oo; z]{t). For - T < 0, the average build-up error e at t = 0 is given by

- 21

el = E[(Q[-T;z](0)-[-cx>;z)(0))z] = E

dt co(-t) z{t)

(3.37)

where E is the expectation operator. For a given build-up error e, this equation is the implicit definition of the build-up time interval T. In order to compute the expectation, we need to specify the considered space of random processes. We assume simple random walks with constant volatility a, namely

E[(z(t)-z(t + 8t))2] = a- (3.38)