arXiv subjects

M. V. Tretyakov

Publications and source records attributed to M. V. Tretyakov.

At least 19 recordsLinked to original sources

arXiv2025

Numerical integrators for confined Langevin dynamics

We derive and analyze numerical methods for underdamped (kinetic) Langevin dynamics in a domain with elastic reflection at the boundary. First-order approximations are based on an Euler-type scheme incorporating collision-handling at the boundary. To achieve second order, composition schemes are derived based on decomposition of the generator into collisional drift, impulse, and stochastic momentum evolution. In a deterministic setting, this approach would typically lead to first-order approximation, even in symmetric compositions, but we find that the stochastic method can provide second-order weak approximation with a single gradient evaluation, both at finite times and in the ergodic limit. We provide analysis of this observation, as well as numerical demonstration, and we compare and contrast the performance of different variants of the integration method using model problems.

math.NA↗

arXiv2025

Sampling from mixture distributions based on regime-switching diffusions

It is proposed to use stochastic differential equations with state-dependent switching rates (SDEwS) for sampling from finite mixture distributions. An Euler scheme with constant time step for SDEwS is considered. It is shown that the scheme converges with order one in weak sense and also in the ergodic limit. Numerical experiments illustrate the use of SDEwS for sampling from mixture distributions and confirm the theoretical results.

math.NA↗

arXiv2024

Well-posedness and approximation of reflected McKean-Vlasov SDEs with applications

In this paper, we establish well-posedness of reflected McKean-Vlasov SDEs and their particle approximations in smooth non-convex domains. We prove convergence of the interacting particle system to the corresponding mean-field limit with the optimal rate of convergence. We motivate this study with applications to sampling and optimization in constrained domains by considering reflected mean-field Langevin SDEs and two reflected consensus-based optimization (CBO) models, respectively. We utilize reflection coupling to study long-time behaviour of reflected mean-field SDEs and also investigate convergence of the reflected CBO models to the global minimum of a constrained optimization problem. We numerically test reflected CBO models on benchmark constrained optimization problems and an inverse problem.

math.PR↗

arXiv2023

Deep learning of transition probability densities for stochastic asset models with applications in option pricing

Transition probability density functions (TPDFs) are fundamental to computational finance, including option pricing and hedging. Advancing recent work in deep learning, we develop novel neural TPDF generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions. The generators are ultra-fast, very accurate and can be trained for any asset model described by stochastic differential equations. These are "single solve", so they do not require retraining when parameters of the stochastic model are changed (e.g. recalibration of volatility). Once trained, the neural TDPF generators can be transferred to less powerful computers where they can be used for e.g. option pricing at speeds as fast as if the TPDF were known in a closed form. We illustrate the computational efficiency of the proposed neural approximations of TPDFs by inserting them into numerical option pricing methods. We demonstrate a wide range of applications including the Black-Scholes-Merton model, the standard Heston model, the SABR model, and jump-diffusion models. These numerical experiments confirm the ultra-fast speed and high accuracy of the developed neural TPDF generators.

q-fin.CP↗

arXiv2023

Neural variance reduction for stochastic differential equations

Variance reduction techniques are of crucial importance for the efficiency of Monte Carlo simulations in finance applications. We propose the use of neural SDEs, with control variates parameterized by neural networks, in order to learn approximately optimal control variates and hence reduce variance as trajectories of the SDEs are being simulated. We consider SDEs driven by Brownian motion and, more generally, by Lévy processes including those with infinite activity. For the latter case, we prove optimality conditions for the variance reduction. Several numerical examples from option pricing are presented.

math.NA↗

arXiv2022

Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions

A simple-to-implement weak-sense numerical method to approximate reflected stochastic differential equations (RSDEs) is proposed and analysed. It is proved that the method has the first order of weak convergence. Together with the Monte Carlo technique, it can be used to numerically solve linear parabolic and elliptic PDEs with Robin boundary condition. One of the key results of this paper is the use of the proposed method for computing ergodic limits, i.e. expectations with respect to the invariant law of RSDEs, both inside a domain in $\mathbb{R}^{d}$ and on its boundary. This allows to efficiently sample from distributions with compact support. Both time-averaging and ensemble-averaging estimators are considered and analysed. A number of extensions are considered including a second-order weak approximation, the case of arbitrary oblique direction of reflection, and a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition. The presented theoretical results are supported by several numerical experiments.

math.NA↗

arXiv2022

Consensus based optimization via jump-diffusion stochastic differential equations

We introduce a new consensus based optimization (CBO) method where interacting particle system is driven by jump-diffusion stochastic differential equations. We study well-posedness of the particle system as well as of its mean-field limit. The major contributions of this paper are proofs of convergence of the interacting particle system towards the mean-field limit and convergence of a discretized particle system towards the continuous-time dynamics in the mean-square sense. We also prove convergence of the mean-field jump-diffusion SDEs towards global minimizer for a large class of objective functions. We demonstrate improved performance of the proposed CBO method over earlier CBO methods in numerical simulations on benchmark objective functions.

math.PR↗

arXiv2021

Pricing FX Options under Intermediate Currency

We suggest an intermediate currency approach that allows us to price options on all FX markets simultaneously under the same risk-neutral measure which ensures consistency of FX option prices across all markets. In particular, it is sufficient to calibrate a model to the volatility smile on the domestic market as, due to the consistency of pricing formulas, the model automatically reproduces the correct smile for the inverse pair (the foreign market). We first consider the case of two currencies and then the multi-currency setting. We illustrate the intermediate currency approach by applying it to the Heston and SABR stochastic volatility models, to the model in which exchange rates are described by an extended skewed normal distribution, and also to the model-free approach of option pricing

q-fin.MF↗

arXiv2020

Random walk algorithm for the Dirichlet problem for parabolic integro-differential equation

We consider stochastic differential equations driven by a general Lévy processes (SDEs) with infinite activity and the related, via the Feynman-Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). We approximate the solution of PIDE using a numerical method for the SDEs. The method is based on three ingredients: (i) we approximate small jumps by a diffusion; (ii) we use restricted jump-adaptive time-stepping; and (iii) between the jumps we exploit a weak Euler approximation. We prove weak convergence of the considered algorithm and present an in-depth analysis of how its error and computational cost depend on the jump activity level. Results of some numerical experiments, including pricing of barrier basket currency options, are presented.

math.NA↗

arXiv2018

Mutation and selection in bacteria: modelling and calibration

Temporal evolution of a clonal bacterial population is modelled taking into account reversible mutation and selection mechanisms. For the mutation model, an efficient algorithm is proposed to verify whether experimental data can be explained by this model. The selection-mutation model has unobservable fitness parameters and, to estimate them, we use an Approximate Bayesian Computation (ABC) algorithm. The algorithms are illustrated using in vitro data for phase variable genes of Campylobacter jejuni.

q-bio.PE↗

arXiv2018

Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation

We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is based on freezing the velocity on time subintervals resulting in linear parabolic equations for vorticity. Probabilistic representations for solutions of these linear equations are given. At each time step, the velocity is expressed via vorticity using a formula corresponding to the Biot--Savart-type law. We show that the approximation is divergent free and of first order. The results are extended to two-dimensional stochastic Navier-Stokes equations with additive noise, where, in particular, we prove the first mean-square convergence order of the vorticity approximation.

math.NA↗

arXiv2017

Bayesian inversion in resin transfer molding

We study the Bayesian inverse problem of inferring the permeability of a porous medium within the context of a moving boundary framework motivated by Resin Transfer Molding (RTM), one of the most commonly used processes for manufacturing fiber-reinforced composite materials. During the injection of resin in RTM, our aim is to update our probabilistic knowledge of the per- meability of the material by inverting pressure measurements as well as observations of the resin moving domain. We consider both one-dimensional and two-dimensional forward models for RTM. Based on the analytical solution for the one-dimensional case, we prove existence of the sequence of posteriors that arise from a sequential Bayesian formulation within the infinite-dimensional framework. For the numerical characterisation of the Bayesian posteriors in the one-dimensional case, we investigate the application of a fully-Bayesian Sequential Monte Carlo method (SMC) for high-dimensional inverse problems. By means of SMC we construct a benchmark against which we compare performance of a novel regularizing ensemble Kalman algorithm (REnKA) that we propose to approximate the posteriors in a computationally efficient manner under practical scenarios. We investigate the robustness of the proposed REnKA with respect to tuneable param- eters and computational cost, and display advantages of REnKA compared with SMC with a small number of particles. We further apply REnKA to investigate, in both the one-dimensional and two-dimensional settings, practical aspects relevant to RTM which include the effect of pressure sensors configuration and the observational noise level in the uncertainty in the log-permeability quantified via the sequence of Bayesian posteriors.

stat.AP↗

arXiv2016

Stochastic resin transfer molding process

We consider one-dimensional and two-dimensional models of the stochastic resin transfer molding process, which are formulated as random moving boundary problems. We study their properties, analytically in the one-dimensional case and numerically in the two-dimensional case. We show how variability of time to fill depends on correlation lengths and smoothness of a random permeability field.

physics.flu-dyn↗

arXiv2014

New Langevin and Gradient Thermostats for Rigid Body Dynamics

We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometric numerical integrators for the Langevin thermostat and one for the gradient thermostat. The numerical integrators reflect key properties of the thermostats themselves. Namely, they all preserve the unit length of quaternions, automatically, without the need of a projection onto the unit sphere. The Langevin integrators also ensure that the angular momenta remain within the tangent space of the quaternion coordinates. The Langevin integrators are quasi-symplectic and of weak order two. The numerical method for the gradient thermostat is of weak order one. Its construction exploits ideas of Lie-group type integrators for differential equations on manifolds. We numerically compare the discretization errors of the Langevin integrators, as well as the efficiency of the gradient integrator compared to the Langevin ones when used in the simulation of rigid TIP4P water model with smoothly truncated electrostatic interactions. We observe that the gradient integrator is computationally less efficient than the Langevin integrators. We also compare the relative accuracy of the Langevin integrators in evaluating various static quantities and give recommendations as to the choice of an appropriate integrator.

physics.comp-ph↗

arXiv2014

A Block Circulant Embedding Method for Simulation of Stationary Gaussian Random Fields on Block-regular Grids

We propose a new method for sampling from stationary Gaussian random field on a grid which is not regular but has a regular block structure which is often the case in applications. The introduced block circulant embedding method (BCEM) can outperform the classical circulant embedding method (CEM) which requires a regularization of the irregular grid before its application. Comparison of BCEM vs CEM is performed on some typical model problems.

stat.CO↗

arXiv2014

On the long-time integration of stochastic gradient systems

This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic (stepsize $h \to 0$) convergence behavior of the error of finite time averages. Recently it has been demonstrated, by study of Fokker-Planck operators, that a non-Markovian numerical method [Leimkuhler and Matthews, 2013; Leimkuhler et al., 2013] generates approximations in the long time limit with higher accuracy order (2nd order) than would be expected from its weak convergence analysis (finite-time averages are 1st order accurate). In this article we describe the transition from the transient to the steady-state regime of this numerical method by estimating the time-dependency of the coefficients in an asymptotic expansion for the weak error, demonstrating that the convergence to 2nd order is exponentially rapid in time. Moreover, we provide numerical tests of the theory, including comparisons of the efficiencies of the Euler-Maruyama method, the popular 2nd order Heun method, and the non-Markovian method.

math.NA↗

arXiv2013

Layer methods for Navier-Stokes equations with additive noise

We propose and study a number of layer methods for stochastic Navier-Stokes equations (SNSE) with spatial periodic boundary conditions and additive noise. The methods are constructed using conditional probabilistic representations of solutions to SNSE and exploiting ideas of the weak sense numerical integration of stochastic differential equations. We prove some convergence results for the proposed methods. Results of numerical experiments on two model problems are presented.

math.NA↗

arXiv2013

A Recursive Sparse Grid Collocation Method for Differential Equations with White Noise

We consider a sparse grid collocation method in conjunction with a time discretization of the differential equations for computing expectations of functionals of solutions to differential equations perturbed by time-dependent white noise. We first analyze the error of Smolyak's sparse grid collocation used to evaluate expectations of functionals of solutions to stochastic differential equations discretized by the Euler scheme. We show theoretically and numerically that this algorithm can have satisfactory accuracy for small magnitude of noise or small integration time, however it does not converge neither with decrease of the Euler scheme's time step size nor with increase of Smolyak's sparse grid level. Subsequently, we use this method as a building block for proposing a new algorithm by combining sparse grid collocation with a recursive procedure. This approach allows us to numerically integrate linear stochastic partial differential equations over longer times, which is illustrated in numerical tests on a stochastic advection-diffusion equation.

math.NA↗