We develop a stochastic model for the velocity gradients dynamics along a Lagrangian trajectory in isotropic and homogeneous turbulent flows. Comparing with different attempts proposed in the literature, the present model, at the cost of introducing a free parameter known in turbulence phenomenology as the intermittency coefficient, gives a realistic picture of velocity gradient statistics at any Reynolds number. To achieve this level of accuracy, we use as a first modelling step a regularized self-stretching term in the framework of the Recent Fluid Deformation (RFD) approximation that was shown to give a realistic picture of small scales statistics of turbulence only up to moderate Reynolds numbers. As a second step, we constrain the dynamics, in the spirit of Girimaji & Pope (1990), in order to impose a peculiar statistical structure to the dissipation seen by the Lagrangian particle. This probabilistic closure uses as a building block a random field that fulfils the statistical description of the intermittency, i.e. multifractal, phenomenon. To do so, we define and generalize to a statistically stationary framework a proposition made by Schmitt (2003). These considerations lead us to propose a non-linear and non-Markovian closed dynamics for the elements of the velocity gradient tensor. We numerically integrate this dynamics and observe that a stationary regime is indeed reached, in which (i) the gradients variance is proportional to the Reynolds number, (ii) gradients are typically correlated over the (small) Kolmogorov time scale and gradients norms over the (large) integral time scale (iii) the joint probability distribution function of the two non vanishing invariants Q and R reproduces the characteristic teardrop shape, (iv) vorticity gets preferentially aligned with the intermediate eigendirection of the deformation tensor and (v) gradients are strongly non-Gaussian and intermittent, a behaviour that we quantify by appropriate high order moments. Additionally, we examine the problem of rotation rate statistics of (axisymmetric) anisotropic particles as observed in Direct Numerical Simulations. Although our realistic picture of velocity gradient fluctuations leads to better results when compared to the former RFD approximation, it is still unable to provide an accurate description for the rotation rate variance of oblate spheroids.

The Recent Fluid Deformation Closure (RFDC) model of lagrangian turbulence is recast in pathintegral language within the framework of the Martin-Siggia-Rose functional formalism. In order to derive analytical expressions for the velocity-gradient probability distribution functions (vgPDFs), we carry out noise renormalization in the low-frequency regime and find approximate extrema for the Martin-Siggia-Rose effective action. We verify, with the help of Monte Carlo simulations, that the vgPDFs so obtained yield a close description of the single-point statistical features implied by the original RFDC stochastic differential equations.

No abstract

We perform a detailed analytical study of the Recent Fluid Deformation (RFD) model for the onset of Lagrangian intermittency, within the context of the Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD) path integral formalism. The model is based, as a key point, upon local closures for the pressure Hessian and the viscous dissipation terms in the stochastic dynamical equations for the velocity gradient tensor. We carry out a power counting hierarchical classification of the several perturbative contributions associated to fluctuations around the instanton-evaluated MSRJD action, along the lines of the cumulant expansion. The most relevant Feynman diagrams are then integrated out into the renormalized effective action, for the computation of velocity gradient probability distribution functions (vgPDFs). While the subleading perturbative corrections do not affect the global shape of the vgPDFs in an appreciable qualitative way, it turns out that they have a significant role in the accurate description of their non-Gaussian cores. arXiv:1711.06339v3 [physics.flu-dyn]

The standard Black-Scholes theory of option pricing is extended to cope with underlying return fluctuations described by general probability distributions. A Langevin process and its related Fokker-Planck equation are devised to model the market stochastic dynamics, allowing us to write and formally solve the generalized Black-Scholes equation implied by dynamical hedging. A systematic expansion around a non-perturbative starting point is then implemented, recovering the Matacz's conjectured option pricing expression. We perform an application of our formalism to the real stock market and find clear evidence that while past financial time series can be used to evaluate option prices before the expiry date with reasonable accuracy, the stochastic character of volatility is an essential ingredient that should necessarily be taken into account in analytical option price modeling.There has been a great interest in the study of the stochastic dynamics of financial markets through ideas and techniques borrowed from the statistical physics context. A set of well-established phenomenological results, universally valid across global markets, yields the motivating ground for the search of relevant models [1,2,3].A flurry of activity, in particular, has been related to the problem of option price valuation. Options are contracts which assure to its owner the right to negotiate (i.e, to sell or to buy) for an agreed value, an arbitrary financial asset (stocks of some company, for instance) at a future date. The writer of the option contract, on the other hand, is assumed to comply with the option owner's decision on the expiry date. Options are a crucial element in the modern markets, since they can be used, as convincingly shown by Black and Scholes [4], to reduce portfolio risk.The Black-Scholes theory of option pricing is a closed analytical formulation where risk vanishes via the procedure of dynamical (also called "delta") hedging, applied to what one might call "log-normal efficient markets". Real markets, however, exhibit strong deviations of lognormality in the statistical fluctuations of stock index returns, and are only approximately efficient.Our aim in this letter is to introduce a theoretical framework for option pricing which is general enough to account for important features of real markets. We also perform empirical tests, taking the London market as the arena where theory and facts can be compared. More concretely, we report in this work observational results concerning options of european style, based on the FTSE 100 index, denoted from now on as S t [5].To start with, we note, in fact, that the returns of the FTSE 100 index do not follow log-normal statistics for small time horizons. We have considered a financial time series of 242993 minutes (roughly, two years) ending on 17th november, 2005. The probability distribution function (pdf) ρ(ω) of the returns given by ω ≡ ln(S t /S t−1 ), taken at one minute intervals is shown in Fig.1. We verify that the Student t-distribution with three degrees of freedom...