Search arXiv⌕ Search

arXiv subjects

Neelima

Publications and source records attributed to Neelima.

8 recordsLinked to original sources
arXiv2025

Tamed Euler approximation for fully superlinear growth McKean-Vlasov SDE and their particle systems: sharp rates for strong propagation of chaos, convergence and ergodicity

We study McKean--Vlasov Stochastic Differential Equations (MV-SDEs) whose drift and diffusion coefficients are of superlinear growth in \textit{all} their variables thus also superlinear in the measure component (the meaning is specified in the body of the paper). We address the finite and infinite time horizon case. Our contribution is fourfold. (a) We establish well-posedness for this class of equations and the corresponding interacting particle system. (b) We prove two propagation of chaos results with explicit $L^2$-convergence rates: the first, is a general one where the rate degrades as the system's dimension $d$ increases; the second, attains the sharp rate $N^{-1/2}$ (in particle number $N$) uniformly over the dimension $d$ at the cost of a Vlasov kernel structure that is general and of superlinear growth for the measure dependency -- the latter's proof fully avoids the Kantorovich-Rubinstein duality argument. (c) Unlike existing works -- based on semi-implicit schemes or truncated Euler schemes -- we propose a fully explicit tamed Euler scheme that has reduced computational cost (comparatively). The explicit scheme is shown to converge in strong $L^p$-sense with rate $1/2$ (in timestep). (d) Lastly, we establish exponential ergodicity properties and long-time behavior for the MV-SDE, the corresponding interacting particle system, and the tamed scheme. The latter result is, to the best of our knowledge, fully novel.

math.PR↗
arXiv2022

An explicit Milstein-type scheme for interacting particle systems and McKean--Vlasov SDEs with common noise and non-differentiable drift coefficients

We propose an explicit drift-randomised Milstein scheme for both McKean--Vlasov stochastic differential equations and associated high-dimensional interacting particle systems with common noise. By using a drift-randomisation step in space and measure, we establish the scheme's strong convergence rate of $1$ under reduced regularity assumptions on the drift coefficient: no classical (Euclidean) derivatives in space or measure derivatives (e.g., Lions/Fr\'echet) are required. The main result is established by enriching the concepts of bistability and consistency of numerical schemes used previously for standard SDE. We introduce certain Spijker-type norms (and associated Banach spaces) to deal with the interaction of particles present in the stochastic systems being analysed. A discussion of the scheme's complexity is provided.

math.PR↗
arXiv2020

Well-posedness and tamed Euler schemes for McKean-Vlasov equations driven by L\'evy noise

We prove the well-posedness of solutions to McKean-Vlasov stochastic differential equations driven by L\'evy noise under mild assumptions where, in particular, the L\'evy measure is not required to be finite. The drift, diffusion and jump coefficients are allowed to be random, can grow super-linearly in the state variable, and all may depend on the marginal law of the solution process. We provide a propagation of chaos result under more relaxed conditions than those existing in the literature, and consistent with our well-posedness result. We propose a tamed Euler scheme for the associated interacting particle system and prove that the rate of its strong convergence is arbitrarily close to $1/2$. As a by-product, we also obtain the corresponding results on well-posedness, propagation of chaos and strong convergence of the tamed Euler scheme for McKean-Vlasov stochastic delay differential equations (SDDE) and McKean-Vlasov stochastic differential equations with Markovian switching (SDEwMS), both driven by L\'evy noise. Furthermore, our results on tamed Euler schemes are new even for ordinary SDEs driven by L\'evy noise and with super-linearly growing coefficients.

math.PR↗
arXiv2020

Well-posedness and tamed schemes for McKean-Vlasov Equations with Common Noise

In this paper, we first establish well-posedness of McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs) with common noise, possibly with coefficients having super-linear growth in the state variable. Second, we present stable time-stepping schemes for this class of McKean-Vlasov SDEs. Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for an interacting particle system associated with the McKean-Vlasov equation. We prove stability and strong convergence of order $1/2$ and $1$, respectively. To obtain our main results, we employ techniques from calculus on the Wasserstein space. The proof for the strong convergence of the tamed Milstein scheme only requires the coefficients to be once continuously differentiable in the state and measure component. To demonstrate our theoretical findings, we present several numerical examples, including mean-field versions of the stochastic $3/2$ volatility model and the stochastic double well dynamics with multiplicative noise.

math.PR↗
arXiv2020

On Explicit Milstein-type Scheme for Mckean-Vlasov Stochastic Differential Equations with Super-linear Drift Coefficient

We develop an explicit Milstein-type scheme for McKean-Vlasov stochastic differential equations using the notion of derivative with respect to measure introduced by Lions and discussed in \cite{cardaliaguet2013}. The drift coefficient is allowed to grow super-linearly in the space variable. Further, both drift and diffusion coefficients are assumed to be only once differentiable in variables corresponding to space and measure. The rate of strong convergence is shown to be equal to $1.0$ without using It\^o's formula for functions depending on measure. The challenges arising due to the dependence of coefficients on measure are tackled and our findings are consistent with the analogous results for stochastic differential equations.

math.PR↗
arXiv2017

$L^p$-estimates and regularity for SPDEs with monotone semilinearity

Semilinear stochastic partial differential equations on bounded domains $\mathscr{D}$ are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen--Cahn and Ginzburg--Landau equations. The first main result of this article are $L^p$-estimates for such equations. The $L^p$-estimates are subsequently employed in obtaining higher regularity. This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such equations. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space $H^2(\mathscr{D}')$ and $\ell^2$-integrable with values in $H^3(\mathscr{D}')$, for any compact $\mathscr{D}' \subset \mathscr{D}$. Using results from $L^p$-theory of SPDEs obtained by Kim~\cite{kim04} we get analogous results in weighted Sobolev spaces on the whole $\mathscr{D}$. Finally it is shown that the solution is H\"older continuous in time of order $\frac{1}{2} - \frac{2}{q}$ as a process with values in a weighted $L^q$-space, where $q$ arises from the integrability assumptions imposed on the initial condition and forcing terms.

math.PR↗
arXiv2016

Coercivity condition for higher order moments for nonlinear SPDEs and existence of solution under local monotonicity

Higher order moment estimates for solutions to nonlinear SPDEs governed by locally-monotone operators are obtained under appropriate coercivity condition. These are then used to extend known existence and uniqueness results for nonlinear SPDEs under local monotonicity conditions to allow derivatives in the operator acting on the solution under the stochastic integral.

math.PR↗