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    Horvath B, Jacquier A, Lacombe C,

    Asymptotic behaviour of randomised fractional volatility models

    , Journal of Applied Probability, ISSN: 0021-9002

    We study the asymptotic behaviour of a class of small-noise diffusions drivenby fractional Brownian motion, with random starting points. Different scalingsallow for different asymptotic properties of the process (small-time and tailbehaviours in particular). In order to do so, we extend some results on samplepath large deviations for such diffusions. As an application, we show how theseresults characterise the small-time and tail estimates of the impliedvolatility for rough volatility models, recently proposed in mathematicalfinance.

    Jacquier A, Pakkanen MS, Stone H, 2019,

    Pathwise large deviations for the rough Bergomi model

    , Journal of Applied Probability, Vol: 55, Pages: 1078-1092, ISSN: 0021-9002

    We study the small-time behaviour of the rough Bergomi model, introduced byBayer, Friz and Gatheral (2016), and prove a large deviations principle for arescaled version of the normalised log stock price process, which then allowsus to characterise the small-time behaviour of the implied volatility.

    Cass T, Lim N, 2019,

    A Stratonovich-Skorohod integral formula for Gaussian rough paths

    , Annals of Probability, Vol: 47, Pages: 1-60, ISSN: 0091-1798

    Given a Gaussian process X, its canonical geometric rough path lift X, and a solution Y to the rough differential equation (RDE) dYt=V(Yt)∘dXt, we present a closed-form correction formula for ∫Y∘dX−∫YdX, that is, the difference between the rough and Skorohod integrals of Y with respect to X. When X is standard Brownian motion, we recover the classical Stratonovich-to-Itô conversion formula, which we generalize to Gaussian rough paths with finite p-variation, p<3, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with H>13. To prove the formula, we first show that the Riemann-sum approximants of the Skorohod integral converge in L2(Ω) by using a novel characterization of the Cameron–Martin norm in terms of higher-dimensional Young–Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a rebalancing of terms.

    Crisan D, Míguez J, 2018,

    Nested particle filters for online parameter estimation in discrete-time state-space Markov models

    , Bernoulli, Vol: 24, Pages: 2429-2460, ISSN: 1350-7265

    © 2018 ISI/BS. We address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system using a sequential Monte Carlo method. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability measure of the static parameters and the dynamic state variables of the system of interest, in a vein similar to the recent “sequential Monte Carlo square” (SMC2) algorithm. However, unlike the SMC2 scheme, the proposed technique operates in a purely recursive manner. In particular, the computational complexity of the recursive steps of the method introduced herein is constant over time. We analyse the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the approximation errors vanish asymptotically in Lp (p ≥ 1) with convergence rate proportional to 1N + 1M , where N is the number of Monte Carlo samples in the parameter space and N × M is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC2 algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.

    Crisan D, Flandoli F, Holm DD, 2018,

    Solution properties of a 3D stochastic euler fluid equation

    , Journal of Nonlinear Science, ISSN: 0938-8974

    We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.

    Engel M, Lamb JSW, Rasmussen M,

    Bifurcation Analysis of a Stochastically Driven Limit Cycle

    We establish the existence of a bifurcation from an attractive randomequilibrium to shear-induced chaos for a stochastically driven limit cycle,indicated by a change of sign of the first Lyapunov exponent. This addresses anopen problem posed by Kevin Lin and Lai-Sang Young, extending results byQiudong Wang and Lai-Sang Young on periodically kicked limit cycles to thestochastic context.

    Dutordoir V, Salimbeni H, Deisenroth M, Hensman Jet al., 2018,

    Gaussian Process Conditional Density Estimation

    Conditional Density Estimation (CDE) models deal with estimating conditionaldistributions. The conditions imposed on the distribution are the inputs of themodel. CDE is a challenging task as there is a fundamental trade-off betweenmodel complexity, representational capacity and overfitting. In this work, wepropose to extend the model's input with latent variables and use Gaussianprocesses (GP) to map this augmented input onto samples from the conditionaldistribution. Our Bayesian approach allows for the modeling of small datasets,but we also provide the machinery for it to be applied to big data usingstochastic variational inference. Our approach can be used to model densitieseven in sparse data regions, and allows for sharing learned structure betweenconditions. We illustrate the effectiveness and wide-reaching applicability ofour model on a variety of real-world problems, such as spatio-temporal densityestimation of taxi drop-offs, non-Gaussian noise modeling, and few-shotlearning on omniglot images.

    Wilson J, Hutter F, Deisenroth MP,

    Maximizing acquisition functions for Bayesian optimization

    , Advances in Neural Information Processing Systems (NIPS) 2018, Publisher: Massachusetts Institute of Technology Press, ISSN: 1049-5258

    Bayesian optimization is a sample-efficient approach to global optimization that relies on theoretically motivated value heuristics (acquisition functions) to guide its search process. Fully maximizing acquisition functions produces the Bayes' decision rule, but this ideal is difficult to achieve since these functions are frequently non-trivial to optimize. This statement is especially true when evaluating queries in parallel, where acquisition functions are routinely non-convex, high-dimensional, and intractable. We first show that acquisition functions estimated via Monte Carlo integration are consistently amenable to gradient-based optimization. Subsequently, we identify a common family of acquisition functions, including EI and UCB, whose characteristics not only facilitate but justify use of greedy approaches for their maximization.

    Gulisashvili A, Horvath B, Jacquier A, 2018,

    Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

    , Quantitative Finance, Vol: 18, Pages: 1753-1765, ISSN: 1469-7688

    We study the mass at the origin in the uncorrelated SABR stochasticvolatility model, and derive several tractable expressions, in particular whentime becomes small or large. As an application--in fact the original motivationfor this paper--we derive small-strike expansions for the implied volatilitywhen the maturity becomes short or large. These formulae, by definitionarbitrage free, allow us to quantify the impact of the mass at zero on existingimplied volatility approximations, and in particular how correct/erroneousthese approximations become.

    Noven RC, Veraart AED, Gandy A, 2018,

    A latent trawl process model for extreme values

    , JOURNAL OF ENERGY MARKETS, Vol: 11, Pages: 1-24, ISSN: 1756-3607

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