Explore the vast range of titles available.

Rough volatility models are known to reproduce the behavior of historical volatility data while at the same time fitting the volatility surface remarkably well, with very few parameters. However, managing the risks of derivatives under rough volatility can be intricate since the dynamics involve fractional Brownian motion. We show in this paper that surprisingly enough, explicit hedging strategies can be obtained in the case of rough Heston models. The replicating portfolios contain the underlying asset and the forward variance curve, and lead to perfect hedging (at least theoretically). From a probabilistic point of view, our study enables us to disentangle the infinite-dimensional Markovian structure associated to rough volatility models.

Fractional and bifractional Brownian motions can be defined on a metric space if the associated metric or distance function is conditionally negative definite (or of negative type). This paper introduces several forms of scalar or vector bifractional Brownian motions on various metric spaces and presents their properties. A metric space of particular interest is the arccos-quasi-quadratic metric space over a subset of R d + 1 such as an ellipsoidal surface, an ellipsoid, or a simplex, whose metric is the composition of arccosine and quasi-quadratic functions. Such a metric is not only conditionally negative definite but also a measure definite kernel, and the metric space incorporates several important cases in a unified framework so that it enables us to study (bi, tri, quadri)fractional Brownian motions on various metric spaces in a unified manner. The vector fractional Brownian motion on the arccos-quasi-quadratic metric space enjoys an infinite series expansion in terms of spherical harmonics, and its covariance matrix function admits an ultraspherical polynomial expansion. We establish the property of strong local nondeterminism of fractional and bi(tri, quadri)fractional Brownian motions on the arccos-quasi-quadratic metric space.

In this paper the authors study mesoscopic fluctuations for Dyson's Brownian motion with \\beta =2. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations. The authors investigate the transition on the intermediate, i.e. mesoscopic, scales. The time scales that they consider are such that the system is already in microscopic equilibrium (sine-universality for the local correlations), but have not yet reached equilibrium at the macrosopic scale. The authors describe the transition to equilibrium on all mesoscopic scales by means of Central Limit Theorems for linear statistics with sufficiently smooth test functions. They consider two situations: deterministic initial points and randomly chosen initial points. In the random situation, they obtain a transition from the classical Central Limit Theorem for independent random variables to the one for the GUE.

Brownian motion of particles affects many branches of science. We report on the Brownian motion of micrometer-sized beads of glass held in air by an optical tweezer, over a wide range of pressures, and we measured the instantaneous velocity of a Brownian particle. Our results provide direct verification of the energy equipartition theorem for a Brownian particle. For short times, the ballistic regime of Brownian motion was observed, in contrast to the usual diffusive regime. We discuss the applications of these methods toward cooling the center-of-mass motion of a bead in vacuum to the quantum ground motional state.

High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work, we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (1) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (2) a merged formulation of the PDE and the 2BSDE problem, (3) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (4) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a 100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 100-dimensional G -Brownian motion.

The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than

We study the diffusive motion of a particle in a subharmonic potential of the form U ( x ) = | x | c (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise ξ α ( t ) with 0 < α ⩽ 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent α . While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 − 1/ α ) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Lévy flights in shallow external potentials.

Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.

Diffusion of nanoparticles in the cytoplasm of live cells has frequently been reported to exhibit an anomalous and even heterogeneous character, i.e. particles seem to switch gears during their journey. Here we show by means of a hidden Markov model that individual trajectories of quantum dots in the cytoplasm of living cultured cells feature a dichotomous switching between two distinct mobility states with an overall subdiffusive mode of motion of the fractional Brownian motion (FBM) type. Using the extracted features of experimental trajectories as input for simulations of different variants of a two-state FBM model, we show that the trajectory-intrinsic and the ensemble-wise heterogeneity in the experimental data is mostly due to variations in the (local) transport coefficients, with only minor contributions due to locally varying anomaly exponents. Altogether, our approach shows that diffusion heterogeneities can be faithfully extracted and quantified from fairly short trajectories obtained by single-particle tracking in highly complex media.

In this paper, we study dynamic backward problems, with the computation of conditional expectations as a special objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of Dupire (Quant. Finance 19 (2019) 721–729) to our more general framework. In particular, unlike in (Quant. Finance 19 (2019) 721–729) where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths. This new feature is due to the time inconsistency involved in this paper. We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing and hedging in a financial market with rough volatility is presented.