Nonparametric tests for pathwise properties of semimartingales

Abstract

We propose two nonparametric tests for investigating the pathwise properties of a signal modeled as the sum of a L\'{e}vy process and a Brownian semimartingale. Using a nonparametric threshold estimator for the continuous component of the quadratic variation, we design a test for the presence of a continuous martingale component in the process and a test for establishing whether the jumps have finite or infinite variation, based on observations on a discrete-time grid. We evaluate the performance of our tests using simulations of various stochastic models and use the tests to investigate the fine structure of the DM/USD exchange rate fluctuations and SPX futures prices. In both cases, our tests reveal the presence of a non-zero Brownian component and a finite variation jump component.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ293 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

The speed of convergence of the Threshold estimator of integrated variance

“…ν(R) = ∞. This work contributes to the existing literature [5,8] precisely in the case where the jumps have also infinite variation. The Blumenthal-Getoor (BG) index of L, defined as…”

“…However the last term is negligible since if ( i X 1 ) 2 > 4r h then (by (18) in [5]: for any fixed c > 0 a.s. for sufficiently small h we have sup i=1..n | i X 0 | < c √ r h ) i N ̸ = 0 and thus, by assuming wlog σ bounded on Ω × [0, T ] (through localization, similarly to in [8], Lemma 4.6) and because W and N are independent [7], we have (7) in [5] and the boundedness of σ , then for sufficiently small h, √ r h − | i X 0 | > c √ r h , c a constant less than 1, and then either…”

“…In [5,8] it has been shown thatÎ V h is efficient when the jumps have fV, provided r h = ch β with β sufficiently close to 1, the speed ofÎ V h − I V being √ 2h I Q, where I Q :=  T 0 σ 4 s ds is the integrated quarticity of X . [8] proves it when σ and L are Ito semimartingales and L has constant jump index α, while [5] proves it for any càdlàg σ and α-stable L. However in [5] it is also shown that when J has infinite variation (iV) thenÎ V h is not efficient, the efficiency rate still being √ h [1].…”

“…In [5,8] it has been shown thatÎ V h is efficient when the jumps have fV, provided r h = ch β with β sufficiently close to 1, the speed ofÎ V h − I V being √ 2h I Q, where I Q :=  T 0 σ 4 s ds is the integrated quarticity of X . [8] proves it when σ and L are Ito semimartingales and L has constant jump index α, while [5] proves it for any càdlàg σ and α-stable L. However in [5] it is also shown that when J has infinite variation (iV) thenÎ V h is not efficient, the efficiency rate still being √ h [1]. In [8] it is shown that for r h = ch β , c ∈ R, β ∈]0, 1[, then for any δ > α the speed ofÎ V h − I V is higher than r 1−δ/2 h , while in [5] it is shown that it is lower than √ h. It is just by virtue of the different speeds under fV jumps or iV jumps that it was possible in [5] to construct the two tests for α < 1 versus α ≥ 1 and for σ ≡ 0 versus σ ̸ ≡ 0.…”

How precise is the finite sample approximation of the asymptotic distribution of realised variation measures in the presence of jumps?

“…Jacod (2004), Todorov and Tauchen (2011) for some first investigations along those lines. However, the assumption that a Brownian component is always present does not seem to be that unrealistic in the light of recent findings by Cont and Mancini (2011) and Aït-Sahalia and Jacod (2010).…”