440 citations · 905 across the 22 of their papers we have counts for
22 papers
Well-posedness and approximation of reflected McKean-Vlasov SDEs with applications
P. D. Hinds, A. Sharma, M. V. Tretyakov
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 p…
Sampling from mixture distributions based on regime-switching diffusions
M. V. Tretyakov
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 t…
Sampling and estimation on manifolds using the Langevin diffusion
Karthik Bharath, Alexander Lewis, Akash Sharma +1
Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure $\text{d}μ_ϕ\propto e^{-ϕ} \mathrm…
Neural variance reduction for stochastic differential equations
P. D. Hinds, M. V. Tretyakov
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 varia…
Deep learning of transition probability densities for stochastic asset models with applications in option pricing
Haozhe Su, M. V. Tretyakov, David P. Newton
Transition probability density functions (TPDFs) are fundamental to computational finance, including option pricing and hedging. Advancing recent work in deep learning, we develop…
Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions
B. Leimkuhler, A. Sharma, M. V. Tretyakov
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…
Random walk algorithm for the Dirichlet problem for parabolic integro-differential equation
G. Deligiannidis, S. Maurer, M. V. Tretyakov
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 p…
Pricing FX Options under Intermediate Currency
S. Maurer, T. E. Sharp, M. V. Tretyakov
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 op…
Approximation of deterministic and stochastic Navier-Stokes equations in vorticity-velocity formulation
G. N. Milstein, M. V. Tretyakov
We consider a time discretization of incompressible Navier-Stokes equations with spatial periodic boundary conditions in the vorticity-velocity formulation. The approximation is ba…
Mutation and selection in bacteria: modelling and calibration
C. D. Bayliss, C. Fallaize, R. Howitt +1
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…
Geometric integrator for Langevin systems with quaternion-based rotational degrees of freedom and hydrodynamic interactions
Ruslan L. Davidchack, Thomas E. Ouldridge, Michael V. Tretyakov
We introduce new Langevin-type equations describing the rotational and translational motion of rigid bodies interacting through conservative and non-conservative forces, and hydrod…
Wiener chaos vs stochastic collocation methods for linear advection-diffusion equations with multiplicative white noise
Zhongqiang Zhang, Michael V. Tretyakov, Boris Rozovskii +1
We compare Wiener chaos and stochastic collocation methods for linear advection-reaction-diffusion equations with multiplicative white noise. Both methods are constructed based on…
New Langevin and Gradient Thermostats for Rigid Body Dynamics
R. L. Davidchack, T. E. Ouldridge, M. V. Tretyakov
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 an…
A Block Circulant Embedding Method for Simulation of Stationary Gaussian Random Fields on Block-regular Grids
M. Park, M. V. Tretyakov
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.…
On the long-time integration of stochastic gradient systems
B. Leimkuhler, C. Matthews, M. V. Tretyakov
This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on propert…
Layer methods for Navier-Stokes equations with additive noise
G. N. Milstein, M. V. Tretyakov
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 construct…
A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications
M. V. Tretyakov, Z. Zhang
A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDE) which coefficients are allowed to grow polynomially at infinity a…
Application of simplest random walk algorithms for pricing barrier options
M. Krivko, M. V. Tretyakov
We demonstrate effectiveness of the first-order algorithm from [Milstein, Tretyakov. Theory Prob. Appl. 47 (2002), 53-68] in application to barrier option pricing. The algorithm us…
Numerical integration of Heath-Jarrow-Morton model of interest rates
M. Krivko, M. V. Tretyakov
We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity ti…
Stable and fast semi-implicit integration of the stochastic Landau-Lifshitz equation
J. H. Mentink, M. V. Tretyakov, A. Fasolino +2
We propose new semi-implicit numerical methods for the integration of the stochastic Landau-Lifshitz equation with built-in angular momentum conservation. The performance of the pr…
Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations
Jonathan C. Mattingly, Andrew M. Stuart, M. V. Tretyakov
Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then use…
Langevin Thermostat for Rigid Body Dynamics
Ruslan L. Davidchack, Richard Handel, M. V. Tretyakov
We present a new method for isothermal rigid body simulations using the quaternion representation and Langevin dynamics. It can be combined with the traditional Langevin or gradien…