Conditional Full Support of Gaussian Processes with Stationary Increments

Abstract

We investigate the conditional full support (CFS) property, introduced in Guasoni et al. (2008a), for Gaussian processes with stationary increments. We give integrability conditions on the spectral measure of such a process which ensure that the process has CFS or not. In particular, the general results imply that, for a process with spectral density f such that f(λ) ∼ c1λpe−c2λq for λ → ∞ (with necessarily p < 1 if q = 0), the CFS property holds if and only if q < 1.

No arbitrage and lead–lag relationships

“…Therefore, by Lemma 4.1 it is enough to show that Y ν has CFS with respect to F Y ν for each ν = 1, 2. Since Y ν is a continuous Gaussian process with stationary increments whose spectral density is 1 − |f |, the desired result follows from Theorem 2.1 of [17]. This completes the proof.…”

“…The proof is an easy extension of the original one and we omit it. The last one is a straightforward multivariate extension of Lemma 3.2 from [17] (see also Remark 2.4(iii) of [39]):…”

“…Let X = (X t ) t∈[0,T ] be a d-dimensional continuous F-adapted process. Then, X has CFS with respect to F if and only if it has CFS with respect to the usual augmentation of F.The third one is a straightforward multivariate extension of Lemma 3.1 from[17]. For a d-dimensionalprocess X = (X t ) t∈[0,T ] we write F X = (F X t ) t∈[0,T ] the natural filtration of X.Lemma 2.5.…”

Sticky processes, local and true martingales

“…Other than this, stochastic processes with the conditional full support (henceforth, CFS) property are also sticky. The CFS property (see Remark 2.3 below for its definition) was introduced in the paper Guasoni et al [13] and a large class of stochastic processes, including fractional Brownian motion (fBm), enjoys this property, see [4,9,15,21] for example.…”

On the conditional small ball property of multivariate Lévy-driven moving average processes

“…When X is continuous, the CSBP is equivalent to the conditional full support (CFS) property, originally introduced by Guasoni et al [25], in connection with no-arbitrage and superhedging results for asset pricing models with transaction costs. For recent results on the CFS property, see, e.g., [16,20,23,35,36].…”