The Impact of Jumps and Leverage in Forecasting the Co-Volatility of Oil and Gold Futures

This paper investigates the impact of jumps in forecasting co-volatility in the presence of leverage effects for daily crude oil and gold futures. We use a modified version of the jump-robust covariance estimator of Koike (2016), such that the estimated matrix is positive definite. Using this approach, we can disentangle the estimates of the integrated co-volatility matrix and jump variations from the quadratic covariation matrix. Empirical results show that more than 80% of the co-volatility of the two futures contains jump variations and that they have significant impacts on future co-volatility but that the impact is negligible in forecasting weekly and monthly horizons.

The quadratic covariation for a weighted fractional Brownian motion

Let [Formula: see text] be a weighted fractional Brownian motion with indices [Formula: see text] and [Formula: see text] satisfying [Formula: see text] [Formula: see text] [Formula: see text]. In this paper, motivated by the asymptotic property [Formula: see text] for all [Formula: see text], we consider the generalized quadratic covariation [Formula: see text] defined by [Formula: see text] provided the limit exists uniformly in probability. We construct a Banach space [Formula: see text] of measurable functions such that the generalized quadratic covariation exists in [Formula: see text] and the generalized Bouleau–Yor identity [Formula: see text] holds for all [Formula: see text], where [Formula: see text] is the weighted local time of [Formula: see text] and [Formula: see text] is the Beta function.

The exponential Lie series for continuous semimartingales

We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logarithm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct explicit proof of the corresponding Chen–Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves.

Variational principles for stochastic fluid dynamics

This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent Itô representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Itô transformation. This term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics; namely, the Euler–Boussinesq and quasi-geostropic approximations.