From Diffusions to Semimartingales

This chapter is a quick review of the theory of semimartingales, these processes being those for which statistical methods are considered in this book.

A process is a collection X = (Xt) of random variables with values in the Euclidean space Rd for some integer d ≥ 1, and indexed on the half line R+ = [0, ∞), or a subinterval of R+, typically [0, T] for some real T > 0. The distinctive feature however is that all these variables are defined on the same probability space (Ω, F, P). Therefore, for any outcome ω [member of] Ω one can consider the path (or "trajectory"), which is the function t [??] Xt(ω), and X can also be considered as a single random variable taking its values in a suitable space of Rd-valued functions on R+ or on [0, T].

In many applications, including the modeling of financial data, the index t can be interpreted as time, and an important feature is the way the process evolves through time. Typically an observer knows what happens up to the current time t, that is, (s)he observes the path s [??] Xs(ω) for all s [member of] [0,t], and wants to infer what will happen later, after time t. In a mathematical framework, this amounts to associating the "history" of the process, usually called the filtration. This is the increasing family (Ft)t≥0 of σ-fields associated with X in the following way: for each t, Ft is the σ-field generated by the variables Xs for s [member of] [0,t] (more precise specifications will be given later). Therefore, of particular interest is the law of the "future" after time t, that is of the family (Xs : s > t), conditional on the σ-field Ft.

In a sense, this amounts to specifying the dynamics of the process, which again is a central question in financial modeling. If the process were not random, that would consist in specifying a differential equation governing the time evolution of our quantity of interest, or perhaps a non-autonomous differential equation where dXt = f (t, Xs : s ≤ t) dt for a function f depending on time and on the whole "past" of X before t. In a random setting, this is replaced by a "stochastic differential equation."

Historically, it took quite a long time to come up with a class of processes large enough to account for the needs in applications, and still amenable to some simple calculus rules. It started with the Brownian motion, or Wiener process, and then processes with independent and stationary increments, now called Lévy processes after P. Lévy, who introduced and thoroughly described them. Next, martingales were considered, mainly by J. L. Doob, whereas K. Itô introduced (after W. Feller and W. Doeblin) the stochastic differential equations driven by Brownian motions, and also by Poisson random measures. The class of semimartingales was finalized by P.-A. Meyer in the early 1970s only.

In many respects, this class of processes is the right one to consider by someone interested in the dynamics of the process in the sense sketched above. Indeed, this is the largest class with respect to which stochastic integration is possible if one wants to have something like the dominated (Lebesgue) convergence theorem. It allows for relatively simple rules for stochastic calculus. Moreover, in financial mathematics, it also turns out to be the right class to consider, because a process can model the price of an asset in a fair market where no arbitrage is possible only if it is a semimartingale. This certainly is a sufficient motivation for the fact that, in this book, we only consider this type of process for modeling prices, including exchange rates or indices, and interest rates.

Now, of course, quite a few books provide extensive coverage of semi-martingales, stochastic integration and stochastic calculus. Our aim in this chapter is not to duplicate any part of those books, and in particular not the proofs therein: the interested reader should consult one of them to get a complete mathematical picture of the subject, for example, Karatzas and Shreve (1991) or Revuz and Yor (1994) for continuous processes, and Dellacherie and Meyer (1982) or Jacod and Shiryaev (2003) for general ones. Our aim is simply to introduce semimartingales and the properties of those which are going to be of constant use in this book, in as simple a way as possible, starting with the most commonly known processes, which are the Brownian motion or Wiener process and the diffusions. We then introduce Lévy processes and Poisson random measures, and finally arrive at semimartingales, presented as a relatively natural extension of Lévy processes.

1.1 Diffusions

1.1.1 The Brownian Motion

The Brownian motion (or Wiener process), formalized by N. Wiener and P. Lévy, has in fact been used in finance even earlier, by T. N. Thiele and L. Bachelier, and for modeling the physical motion of a particle by A. Einstein. It is the simplest continuous-time analogue of a random walk.

Mathematically speaking, the one-dimensional Brownian motion can be specified as a process W = (Wt)t≥0, which is Gaussian (meaning that any finite family ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is a Gaussian random vector), centered (i.e. E(Wt) = 0 for all t), and with the covariance structure

E(Wt Ws) = t A s (1.1)

where the notation t [conjunction] s means min(t, s). These properties completely characterize the law of the process W, by Kolmogorov's Theorem, which allows for the definition of a stochastic process through its finite-dimensional distributions, under conditions known as consistency of the finite-dimensional distributions. And, using for example the Kolmogorov continuity criterion (since E(|Wt+s - Ws|4) = 3s2 for all nonnegative t and s), one can "realize" the Brownian motion on a suitable probability space (Ω, F, P) as a process having continuous paths, i.e. t [??] Wt([oemga]) is continuous and with W0(ω) = 0 for all outcomes ω. So we will take the view that a Brownian motion always starts at W0 = 0 and has continuous paths.

One of the many fundamental properties of Brownian motion is that it is a Lévy process, that is it starts from 0 and has independent and stationary increments: for all s, t ≥ 0 the variable Wt+s - Wt is independent of (Wr : r ≤ t), with a law which only depends on s: here, this law is the normal law N(0, s) (centered with variance s). This immediately follows from the above definition. However, a converse is also true: any Lévy process which is centered and continuous is of the form σW for some constant σ ≥ 0, where W is a Brownian motion.

Now, we need two extensions of the previous notion. The first one is straightforward: a d-dimensional Brownian motion is an Rd-valued process W = (Wt)t≥0 with components Wit for i = 1, ..., d (we keep the same notation W as in the one-dimensional case), such that each component process Wi = (Wit)t≥0 is a one-dimensional Brownian motion, and all components are independent processes. Equivalently, W is a centered continuous Gaussian process with W0 = 0, and with the following covariance structure:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

A d-dimensional Brownian motion retains the nice property of being a Lévy process.

The second extension is slightly more subtle, and involves the concept of a general filtered probability space, denoted by (Ω, F, (Ft)t≥0, P). Here (Ω, F, P) is a probability space, equipped with a filtration (Ft)t≥0: this is an increasing family of sub-σ-fields Ft of F (that is, Ft [subset] Fs [subset] F when t ≤ s), which is right-continuous (that is Ft = [intersection]s>tFs). The right-continuity condition appears for technical reasons, but is in fact an essential requirement. Again, Ft can be viewed as the amount of information available to an observer up to (and including) time t.

We say that a process X is adapted to a filtration (Ft), or (Ft)- adapted, if each variable Xt is Ft-measurable. The filtration generated by a process X is the smallest filtration with respect to which X is adapted. It is denoted as (FXt), and can be expressed as follows:

FXt = [intersection]s>t σ(Xr : r [member of] [0, s])

(this is right-continuous by construction).

We suppose that the reader is familiar with the notion of a martingale (a real process M = (Mt)t≥0 is a martingale on the filtered space if it is adapted, if each variable Mt is integrable and if E(Mt+s | Ft) = Mt for s, t ≥ 0), and also with the notion of a stopping time: a [0, ∞]-valued random variable τ is a stopping time if it is possible to tell, for any t ≥ 0, whether the event that τ has occurred before or at time t is true or not, on the basis of the information contained in Ft; formally, this amounts to saying that the event {τ ≤ t} belongs to Ft, for all t ≥ 0. Likewise, Fτ denotes the σ-field of all sets A [member of] F such that A [intersection] {τ ≤ t} [member of] Ft for all t ≥ 0, and it represents the information available up to (and including) time τ.

A process X is called progressively measurable if for any t the function (ω, s) [??] Xs(ω) is Ft [cross product] B([0, t])- measurable on Ω × [0,t]; here, B([0,t]) denotes the Borel σ-field of the interval [0,t], that is the σ-field generated by the open sets of [0,t]. Then, given a stopping time τ and a progressively measurable process Xt, the variable Xτ 1{τ<∞} is Fτ-measurable; moreover one can define the new process Xτ[conjunction]t, or X stopped at τ, and this process is again adapted. We note the following useful property: if M is a martingale and τ1 and τ2 are two stopping times such that 0 ≤ τ1 ≤ τ2 ≤ T a.s. (almost surely), then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Coming back to Brownian motion, we say that a d-dimensional process W = (Wi)1≤i≤d is a Brownian motion on the filtered space ([OMEGA F, (Ft)t≥0, P), or an (Ft) -Brownian motion, if it satisfies the following three conditions:

1. It has continuous paths, with W0 = 0.

2. It is adapted to the filtration (Ft).

3. For all s, t ≥ 0 the variable Wt+s - Wt is independent of the σ- field Ft, with centered Gaussian law N(0, sId) (Id is the d × d identity matrix).

It turns out that a Brownian motion in the previously stated restricted sense is an (FWt)-Brownian motion. This property is almost obvious: it would be obvious if FWt were σ(Wr : r [member of] [0, t]), and its extension comes from a so-called 0 - 1 law which asserts that, if an event is in FWs for all s > t and is also independent of FWv for all v < t, then its probability can only equal 0 or 1.

Another characterization of the Brownian motion, particularly well suited to stochastic calculus, is the Lévy Characterization Theorem. Namely, a continuous (Ft)-adapted process W with W0 = 0 is an (Ft)-Brownian motion if and only if it satisfies

the processes Wit and WitWij - δijt are (Ft) – martingales (1.3)

where δij = 1 if i = j and 0 otherwise denotes the Kronecker symbol. The necessary part is elementary; the sufficient part is more difficult to prove.

Finally, we mention two well known and important properties of the paths of a one-dimensional Brownian motion:

• Lévy modulus of continuity: almost all paths satisfy, for any interval I of positive length,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

• Law of iterated logarithm: for each finite stopping time T, almost all paths satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

By symmetry, the lim inf in the law of iterated logarithm is equal to -√2. Despite the appearances, these two results are not contradictory, because of the different position of the qualifier "almost all." These facts imply that, for any ρ < 1/2, almost all paths are locally Hölder with index ρ, but nowhere Hölder with index 1/2, and a fortiori nowhere differentiable.

1.1.2 Stochastic Integrals

A second fundamental concept is stochastic integration. The paths of a one-dimensional Brownian motion W being continuous but nowhere differentiable, a priori the "differential" dWt makes no sense, and neither does the expression ∫t0 HsdWs. However, suppose that H is a "simple," or piecewise constant, process of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.6)

where 0 = t0< t1< ··· and tn -> [inifinity] as n -> ∞. Then it is natural to set t

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.7)

This would be the usual integral if t [??] Wt were the distribution function of a (signed) measure on [0, t], which it is of course not. Nevertheless it turns out that, due to the martingale properties (1.3) of W, this "integral" can be extended to all processes H having the following properties:

H is progressively measurable, and not too big, in the sense that ∫t0H2sds< ∞ for all t. (1.8)