arXiv subjects

Ludovic Goudenège

Publications and source records attributed to Ludovic Goudenège.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative approximation of the Vlasov(-Fokker-Planck)-Navier-Stokes system by stochastic particle systems

This paper is concerned with a fluid-particle system given by the incompressible Navier-Stokes equations coupled with the Vlasov(-Fokker-Planck) equation through a drag force. Such a model arises naturally in the study of aerosols, sprays, and more generally two-phase flows. In dimensions $d\in \{2,3\}$, we establish a rate of convergence for a system of $N$ interacting stochastic particles coupled with a fluid, towards the Vlasov(-Fokker-Planck)-Navier-Stokes system, as $N\to \infty$. The case of particles with a noise that vanishes as $N\to \infty$ is considered and leads specifically to the Vlasov-Navier-Stokes system. More precisely, we prove that the empirical measure associated with the particle system converges to the Vlasov(-Fokker-Planck) component, while the fluid velocity converges to the Navier-Stokes component of the coupled system. The proofs combine stochastic calculus and PDE techniques to establish energy estimates and commutator estimates for both the discrete and continuous systems.

math.PR↗

arXiv2025

Robust Pricing of Equity-Indexed Annuities under Uncertain Volatility and Stochastic Interest Rate

In this paper, we propose a novel methodology for pricing equity-indexed annuities featuring cliquet-style payoff structures and early surrender risk, using advanced financial modeling techniques. Specifically, the market is modeled by an equity index that follows an uncertain volatility framework, while the dynamics of the interest rate are captured by the Hull-White model. Due to the inherent complexity of the market dynamics under consideration, we develop a numerical algorithm that employs a tree-based framework to discretize both the interest rate and the underlying equity index, enhanced with local volatility optimization. The proposed algorithm is compared with a machine learning-based algorithm. Extensive numerical experiments demonstrate its high effectiveness. Furthermore, the numerical framework is employed to analyze key features of the insurance contract, including the delineation of the optimal exercise region when early surrender risk is incorporated.

q-fin.PR↗

arXiv2024

Numerical approximation of SDEs with fractional noise and distributional drift

We study the numerical approximation of SDEs with singular drifts (including distributions) driven by a fractional Brownian motion. Under the Catellier-Gubinelli condition that imposes the regularity of the drift to be strictly greater than $1-1/(2H)$, we obtain an explicit rate of convergence of a tamed Euler scheme towards the SDE, extending results for bounded drifts. Beyond this regime, when the regularity of the drift is $1-1/(2H)$, we derive a non-explicit rate. As a byproduct, strong well-posedness for these equations is recovered. Proofs use new regularising properties of discrete-time fBm and a new critical Grönwall-type lemma. We present examples and simulations.

math.PR↗

arXiv2024

Numerical approximation of the stochastic heat equation with a distributional reaction term

We study the numerical approximation of the stochastic heat equation with a distributional reaction term. Under a condition on the Besov regularity of the reaction term, it was proven recently that a strong solution exists and is unique in the pathwise sense, in a class of Hölder continuous processes. For a suitable choice of sequence $(b^k)_{k\in \mathbb{N}}$ approximating $b$, we prove that the error between the solution $u$ of the SPDE with reaction term $b$ and its tamed Euler finite-difference scheme with mollified drift $b^k$, converges to $0$ in $L^m(Ω)$ with a rate that depends on the Besov regularity of $b$. In particular, one can consider two interesting cases: first, even when $b$ is only a (finite) measure, a rate of convergence is obtained. On the other hand, when $b$ is a bounded measurable function, the (almost) optimal rate of convergence $(\frac{1}{2}-\varepsilon)$-in space and $(\frac{1}{4}-\varepsilon)$-in time is achieved. Stochastic sewing techniques are used in the proofs, in particular to deduce new regularising properties of the discrete Ornstein-Uhlenbeck process.

math.PR↗

arXiv2024

Enhancing Valuation of Variable Annuities in Lévy Models with Stochastic Interest Rate

This paper extends the valuation and optimal surrender framework for variable annuities with guaranteed minimum benefits in a Lévy equity market environment by incorporating a stochastic interest rate described by the Hull-White model. This approach frames a more dynamic and realistic financial setting compared to previous literature. We exploit a robust valuation mechanism employing a hybrid numerical method that merges tree methods for interest rate modeling with finite difference techniques for the underlying asset price. This method is particularly effective for addressing the complexities of variable annuities, where periodic fees and mortality risks are significant factors. Our findings reveal the influence of stochastic interest rates on the strategic decision-making process concerning the surrender of these financial instruments. Through comprehensive numerical experiments, and by comparing our results with those obtained through the Longstaff-Schwartz Monte Carlo method, we illustrate how our refined model can guide insurers in designing contracts that equitably balance the interests of both parties. This is particularly relevant in discouraging premature surrenders while adapting to the realistic fluctuations of financial markets. Lastly, a comparative statics analysis with varying interest rate parameters underscores the impact of interest rates on the cost of the optimal surrender strategy, emphasizing the importance of accurately modeling stochastic interest rates.

q-fin.PR↗

arXiv2023

Backward Hedging for American Options with Transaction Costs

In this article, we introduce an algorithm called Backward Hedging, designed for hedging European and American options while considering transaction costs. The optimal strategy is determined by minimizing an appropriate loss function, which is based on either a risk measure or the mean squared error of the hedging strategy at maturity. The proposed algorithm moves backward in time, determining, for each time-step and different market states, the optimal hedging strategy that minimizes the loss function at the time the option is exercised, by assuming that the strategy used in the future for hedging the liability is the one determined at the previous steps of the algorithm. The approach avoids machine learning and instead relies on classic optimization techniques, Monte Carlo simulations, and interpolations on a grid. Comparisons with the Deep Hedging algorithm in various numerical experiments showcase the efficiency and accuracy of the proposed method.

q-fin.CP↗

arXiv2023

A New Non-Linear Density Fluctuations Stochastic Partial Differential Equation With a Singular Coefficient of Relevance to Polymer Dynamics and Rheology: Discussions, Proofs of Solution Existence, Uniqueness, and a Conjecture

In this paper we consider an entirely new - previously unstudied to the best of our knowledge - type of density fluctuations stochastic partial differential equation with a singular coefficient involving the inverse of a probability density. The equation was recently introduced by Schieber \cite{jay3} while working on a new polymer molecular dynamics approach that pertains to the generally called polymer reptation (aka tube) theory. The corresponding probability density is the solution of an evolution equation (a stochastic transport) defined on a dynamical one-dimensional subspace. A peculiarity of the here studied equation is its very singular pattern, even though it exhibits a well-posed structure. As a first step towards furthering the understanding of this new clas

math-ph↗

arXiv2022

Convergence of the stochastic Navier-Stokes-$α$ solutions toward the stochastic Navier-Stokes solutions

Loosely speaking, the Navier-Stokes-$α$ model and the Navier-Stokes equations differ by a spatial filtration parametrized by a scale denoted $α$. Starting from a strong two-dimensional solution to the Navier-Stokes-$α$ model driven by a multiplicative noise, we demonstrate that it generates a strong solution to the stochastic Navier-Stokes equations under the condition $α$ goes to 0. The initially introduced probability space and the Wiener process are maintained throughout the investigation, thanks to a local monotonicity property that abolishes the use of Skorokhod's theorem. High spatial regularity a priori estimates for the fluid velocity vector field are carried out within periodic boundary conditions.

math.AP↗

arXiv2022

Likelihood-based non-Markovian models from molecular dynamics

We introduce a new method to accurately and efficiently estimate the effective dynamics of collective variables in molecular simulations. Such reduced dynamics play an essential role in the study of a broad class of processes, ranging from chemical reactions in solution to conformational changes in biomolecules or phase transitions in condensed matter systems. The standard Markovian approximation often breaks down due to the lack of a proper separation of time scales and memory effects must be taken into account. Using a parametrization based on hidden auxiliary variables, we obtain a generalized Langevin equation by maximizing the statistical likelihood of the observed trajectories. Both the memory kernel and random noise are correctly recovered by this procedure. This data-driven approach provides a reduced dynamical model for multidimensional collective variables, enabling the accurate sampling of their long-time dynamical properties at a computational cost drastically reduced with respect to all-atom numerical simulations. The present strategy, based on the reproduction of the dynamics of trajectories rather than the memory kernel or the velocity-autocorrelation function, conveniently provides other observables beyond these two, including e.g. stationary currents in non-equilibrium situations, or the distribution of first passage times between metastable states.

cond-mat.stat-mech↗

arXiv2021

Numerical and convergence analysis of the stochastic Lagrangian averaged Navier-Stokes equations

The primary emphasis of this work is the development of a finite element based space-time discretization for solving the stochastic Lagrangian averaged Navier-Stokes (LANS-$α$) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of R^d , d $\in$ {2, 3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale $α$, namely we first assume $α$ to be controlled by the step size of the space discretization so that it vanishes when passing to the limit, then we provide an alternative study when $α$ is fixed. A preparatory analysis of uniform estimates in both $α$ and discretization parameters is carried out. Starting out from the stochastic LANS-$α$ model, we achieve convergence toward the continuous strong solutions of the stochastic Navier-Stokes equations in 2D when $α$ vanishes at the limit. Additionally, convergence toward the continuous strong solutions of the stochastic LANS-$α$ model is accomplished if $α$ is fixed.

math.NA↗

arXiv2021

Moving average options: Machine Learning and Gauss-Hermite quadrature for a double non-Markovian problem

Evaluating moving average options is a tough computational challenge for the energy and commodity market as the payoff of the option depends on the prices of a certain underlying observed on a moving window so, when a long window is considered, the pricing problem becomes high dimensional. We present an efficient method for pricing Bermudan style moving average options, based on Gaussian Process Regression and Gauss-Hermite quadrature, thus named GPR-GHQ. Specifically, the proposed algorithm proceeds backward in time and, at each time-step, the continuation value is computed only in a few points by using Gauss-Hermite quadrature, and then it is learned through Gaussian Process Regression. We test the proposed approach in the Black-Scholes model, where the GPR-GHQ method is made even more efficient by exploiting the positive homogeneity of the continuation value, which allows one to reduce the problem size. Positive homogeneity is also exploited to develop a binomial Markov chain, which is able to deal efficiently with medium-long windows. Secondly, we test GPR-GHQ in the Clewlow-Strickland model, the reference framework for modeling prices of energy commodities. Finally, we consider a challenging problem which involves double non-Markovian feature, that is the rough-Bergomi model. In this case, the pricing problem is even harder since the whole history of the volatility process impacts the future distribution of the process. The manuscript includes a numerical investigation, which displays that GPR-GHQ is very accurate and it is able to handle options with a very long window, thus overcoming the problem of high dimensionality.

q-fin.CP↗

arXiv2021

Revisiting the framework for intermittency in Lagrangian stochastic models for turbulent flows: a way to an original and versatile numerical approach

The characterization of intermittency in turbulence has its roots in the K62 theory, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in attempt to reproduce it. The first contribution of this work is to propose a requirement list to be satisfied by models designed within the Lagrangian framework. Multifractal stochastic processes are a natural choice to retrieve multifractal properties of the dissipation. Among them, following the proposition of \cite{Mandelbrot1968}, we investigate the Gaussian Multiplicative Chaos formalism, which requires the construction of a log-correlated stochastic process $X_t$. The fractional Gaussian noise of Hurst parameter $H = 0$ is of great interest because it leads to a log-correlation for the logarithm of the process.Inspired by the approximation of fractional Brownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeck processes, our second contribution is to propose a new stochastic model: $X_t = \int_0^\infty Y_t^x k(x) d x$, where $Y_t^x$ is an Ornstein-Uhlenbeck process with speed of mean reversion $x$ and $k$ is a kernel. A regularization of $k(x)$ is required to ensure stationarity, finite variance and logarithmic auto-correlation. A variety of regularizations are conceivable, and we show that they lead to the aforementioned multifractal models.To simulate the process, we eventually design a new approach relying on a limited number of modes for approximating the integral through a quadrature $X_t^N = \sum_{i=1}^N ω_i Y_t^{x_i}$, using a conventional quadrature method. This method can retrieve the expected behavior with only one mode per decade, making this strategy versatile and computationally attractive for simulating such processes, while remaining within the proposed framework for a proper description of intermittency.

physics.flu-dyn↗

arXiv2021

Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises

We study the asymptotic properties of the stochastic Cahn-Hilliard equation with the logarithmic free energy by establishing different dimension-free Harnack inequalities according to various kinds of noises. The main characteristics of this equation are the singularities of the logarithmic free energy at 1 and --1 and the conservation of the mass of the solution in its spatial variable. Both the space-time colored noise and the space-time white noise are considered. For the highly degenerate space-time colored noise, the asymptotic log-Harnack inequality is established under the so-called essentially elliptic conditions. And the Harnack inequality with power is established for non-degenerate space-time white noise.

math.AP↗

arXiv2020

$α$-Navier-Stokes equation perturbed by space-time noise of trace class

We consider a stochastic perturbation of the $α$-Navier-Stokes model. The stochastic perturbation is an additive space-time noise of trace class. Under a natural condition about the trace of operator $Q$ in front of the noise, we prove the existence and uniqueness of strong solution, continuous in time in classical spaces of $L^{2}$ functions with estimates of non-linear terms. It is based on a priori estimate of solutions of finite-dimensional systems, and tightness of the approximated solution. Moreover, by studying the derivative of the solution with respect to the initial data, we can prove exponential moment of the approximated solutions, which is enough to obtain Strong Feller property and irreducibility of the transition semigroup. This leads naturally to the existence and uniqueness of an invariant measure.

math.PR↗

arXiv2019

Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension

In this paper we propose an efficient method to compute the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction technique. Specifically, the options we consider are written on a basket of assets, each of them following a Black-Scholes dynamics. In the wake of Ludkovski's approach (2018), we implement here a backward dynamic programming algorithm which considers a finite number of uniformly distributed exercise dates. On these dates, the option value is computed as the maximum between the exercise value and the continuation value, which is obtained by means of Gaussian process regression technique and Monte Carlo simulations. Such a method performs well for low dimension baskets but it is not accurate for very high dimension baskets. In order to improve the dimension range, we employ the European option price as a control variate, which allows us to treat very large baskets and moreover to reduce the variance of price estimators. Numerical tests show that the proposed algorithm is fast and reliable, and it can handle also American options on very large baskets of assets, overcoming the problem of the curse of dimensionality.

q-fin.CP↗

arXiv2019

Gaussian Process Regression for Pricing Variable Annuities with Stochastic Volatility and Interest Rate

In this paper we investigate price and Greeks computation of a Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity (VA) when both stochastic volatility and stochastic interest rate are considered together in the Heston Hull-White model. We consider a numerical method the solves the dynamic control problem due to the computing of the optimal withdrawal. Moreover, in order to speed up the computation, we employ Gaussian Process Regression (GPR). Starting from observed prices previously computed for some known combinations of model parameters, it is possible to approximate the whole price function on a defined domain. The regression algorithm consists of algorithm training and evaluation. The first step is the most time demanding, but it needs to be performed only once, while the latter is very fast and it requires to be performed only when predicting the target function. The developed method, as well as for the calculation of prices and Greeks, can also be employed to compute the no-arbitrage fee, which is a common practice in the Variable Annuities sector. Numerical experiments show that the accuracy of the values estimated by GPR is high with very low computational cost. Finally, we stress out that the analysis is carried out for a GMWB annuity but it could be generalized to other insurance products.

q-fin.CP↗

arXiv2019

Machine Learning for Pricing American Options in High-Dimensional Markovian and non-Markovian models

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black-Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. Moreover, these two methods solve the backward dynamic programming problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite this model is only bidimensional, the whole history of the process impacts on the price, and handling all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.

q-fin.CP↗

arXiv2019

Stochastic phase field $α$-Navier-Stokes vesicle-fluid interaction model

We consider a stochastic perturbation of the phase field alpha-Navier-Stokes model with vesicle-fluid interaction. It consists in a system of nonlinear evolution partial differential equations modeling the fluid-structure interaction associated to the dynamics of an elastic vesicle immersed in a moving incompressible viscous fluid. This system of equations couples a phase-field equation -- for the interface between the fluid and the vesicle -- to the alpha-Navier-Stokes equation -- for the viscous fluid -- with an extra nonlinear interaction term, namely the bending energy. The stochastic perturbation is an additive space-time noise of trace class on each equation of the system. We prove the existence and uniqueness of solution in classical spaces of $L^{2}$ functions with estimates of non-linear terms and bending energy. It is based on a priori estimate about the regularity of solutions of finite dimensional systems, and tightness of the approximated solution.

math.PR↗