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Projection of fBm on the Space of Martingales

Consider the fractional Brownian motion (fBm) with Hurst index H ? (0, 1). Its definition and properties will be considered in more detail in section 1.1; however, let us mention immediately that fBm is a Gaussian process and anyhow not a martingale or even a semimartingale for H ? . Hence, a natural question arises: what is the distance between fBm and the space of Gaussian martingales in an appropriate metric and how do we determine the projection of fBm on the space of Gaussian martingales? Why is it not reasonable to consider non-Gaussian martingales? In this chapter, we will answer this and other related questions. The chapter is organized as follows. In section 1.1, we give the main properties of fBm, including its integral representations. In section 1.2, we formulate the minimizing problem simplifying it at the same time. In section 1.3, we strictly propose a positive lower bound for the distance between fBm and the space of Gaussian martingales. Sections 1.4 and 1.5 are devoted to the general problem of minimization of the functional f on L2([0, 1]) that has the following form:

with arbitrary kernel z(t, s) satisfying condition

(A) for any t ? [0, 1] the kernel z(t, ·) ? L2([0, t]) and

We shall call the functional f the principal functional. It is proved in section 1.4 that the principal functional f is convex, continuous and unbounded on infinity, consequently the minimum is reached. Section 1.5 gives an example of the kernel z(t, s) where a minimizing function for the principal functional is not unique (moreover, being convex, the set of minimizing functions is infinite). Sections 1.6-1.8 are devoted to the problem of minimization of principal functional f with the kernel z corresponding to fBm, i.e. with the kernel z from [1.7]. It is proved in section 1.6 that in this case, the minimizing function for the principal functional is unique. In section 1.7 it is proved that the minimizing function has a special form, namely a probabilistic representation, and many properties of the minimizing function have been established. Since we have no explicit analytical representation of the minimizing function, in section 1.8 we provide the discrete-time counterpart of the minimization problem and give the results explaining how to calculate the minimizing function numerically via evaluation of the Chebyshev center, illustrating the numerics with a couple of plots.

1.1. fBm and its integral representations

In this section, we define fBm and collect some of its main properties. We refer to the books [BIA 08, MIS 08, MIS 18, NOU 12] for the detailed presentation of this topic.

Let (O, , P) be a complete probability space with a filtration satisfying the standard assumptions.

DEFINITION 1.1.- An fBm with associated Hurst index H ? (0, 1) is a Gaussian process , such that

1. 1) ,
2. 2) .

The following statements can be derived directly from the above definition.

1. 1) If H = , then an fBm is a standard Wiener process.
2. 2) An fBm is self-similar with the self-similarity parameter H, i.e. for any c > 0. Here, means that all finite-dimensional distributions of both processes coincide.
3. 3) An fBm has stationary increments that is implied by the form of its incremental covariance: [1.3]
4. 4) The increments of an fBm are independent only in the case H = 1/2. They are negatively correlated for H ? (0, 1/2) and positively correlated for H ? (1/2, 1).

Due to the Kolmogorov continuity theorem, property [1.3] implies that an fBm has a continuous modification. Moreover, this modification is