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Dai Taguchi

Publications and source records attributed to Dai Taguchi.

At least 19 recordsLinked to original sources
arXiv2026

Strong rate of convergence for the Euler--Maruyama scheme of SDEs with unbounded H\"older continuous drift coefficient

In this paper, we provide the strong rate of convergence for the Euler--Maruyama scheme for multi-dimensional stochastic differential equations with uniformly locally (unbounded) H\"older continuous drift and multiplicative noise. Our technique is based on It\^o--Tanaka trick (Zvonkin transformation) for unbounded drift. Moreover, in order to apply the stochastic sewing lemma, we use the heat kernel estimate for the density function of the Euler--Maruyama scheme.

math.PR↗
arXiv2024

A generalized coupling approach for the weak approximation of stochastic functional differential equations

In this paper, we study functional type weak approximation of weak solutions of stochastic functional differential equations by means of the Euler--Maruyama scheme. Under mild assumptions on the coefficients, we provide a quantitative error estimate for the weak approximation in terms of the L\'evy--Prokhorov metric of probability laws on the path space. The weak error estimate obtained in this paper is sharp in the topological and quantitative senses in some special cases. We apply our main result to ten concrete examples appearing in a wide range of science and obtain a weak error estimate for each model. The proof of the main result is based on the so-called generalized coupling of probability measures.

math.PR↗
arXiv2024

Strong solution and approximation of time-dependent radial Dunkl processes with multiplicative noise

We study the strong existence and uniqueness of solutions within a Weyl chamber for a class of time-dependent particle systems driven by multiplicative noise. This class includes well-known processes in physics and mathematical finance. We propose a method to prove the existence of negative moments for the solutions. This result allows us to analyze two numerical schemes for approximating the solutions. The first scheme is a $\theta$-Euler--Maruyama scheme, which ensures that the approximated solution remains within the Weyl chamber. The second scheme is a truncated $\theta$-Euler--Maruyama scheme, which produces values in $\mathbb{R}^{d}$ instead of the Weyl chamber $\mathbb{W}$, offering improved computational efficiency.

math.PR↗
arXiv2024

Numerical schemes for radial Dunkl processes

We consider the numerical approximation for a class of radial Dunkl processes corresponding to arbitrary (reduced) root systems in $\mathbb{R}^{d}$. This class contains some well-known processes such as Bessel processes, Dyson's Brownian motions, and Wishart processes. We propose some semi--implicit and truncated Euler--Maruyama schemes for radial Dunkl processes, and study their rate of convergence with respect to the $L^{p}$-sup norm.

math.PR↗
arXiv2021

Semi-implicit Euler--Maruyama scheme for polynomial diffusions on the unit ball

In this article, we consider numerical schemes for polynomial diffusions on the unit ball, which are solutions of stochastic differential equations with a diffusion coefficient of the form $\sqrt{1-|x|^{2}}$. We introduce a semi-implicit Euler--Maruyama scheme with the projection onto the unit ball and provide the $L^{2}$-rate of convergence. The main idea to consider the numerical scheme is the transformation argument introduced by Swart for proving the pathwise uniqueness for some stochastic differential equation with a non-Lipschitz diffusion coefficient.

math.PR↗
arXiv2021

Approximations for adapted M-solutions of Type-II backward stochastic Volterra integral equations

In this paper, we study a class of Type-II backward stochastic Volterra integral equations (BSVIEs). For the adapted M-solutions, we obtain two approximation results, namely, a BSDE approximation and a numerical approximation. The BSDE approximation means that the solution of a finite system of backward stochastic differential equations (BSDEs) converges to the adapted M-solution of the original equation. As a consequence of the BSDE approximation, we obtain an estimate for the $L^2$-time regularity of the adapted M-solutions of Type-II BSVIEs. For the numerical approximation, we provide a backward Euler--Maruyama scheme, and show that the scheme converges in the strong $L^2$-sense with the convergence speed of order $1/2$. These results hold true without any differentiability conditions for the coefficients.

math.PR↗
arXiv2020

$L^{q}$-error estimates for approximation of irregular functionals of random vectors

Avikainen showed that, for any $p,q \in [1,\infty)$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $\mathbb{E}[|f(X)-f(\widehat{X})|^{q}] \leq C(p,q) \mathbb{E}[|X-\widehat{X}|^{p}]^{\frac{1}{p+1}}$, where $X$ is a one-dimensional random variable with a bounded density, and $\widehat{X}$ is an arbitrary random variable. In this article, we will provide multi-dimensional versions of this estimate for functions of bounded variation in $\mathbb{R}^{d}$, Orlicz--Sobolev spaces, Sobolev spaces with variable exponents, and fractional Sobolev spaces. The main idea of our arguments is to use the Hardy--Littlewood maximal estimates and pointwise characterizations of these function spaces. We apply our main results to analyze the numerical approximation for some irregular functionals of the solution of stochastic differential equations.

math.PR↗
arXiv2020

A generalized Avikainen's estimate and its applications

Avikainen provided a sharp upper bound of the difference $\mathbb{E}[|g(X)-g(\widehat{X})|^{q}]$ by the moments of $|X-\widehat{X}|$ for any one-dimensional random variables $X$ with bounded density and $\widehat{X}$, and function of bounded variation $g$. In this article, we generalize this estimate to any one-dimensional random variable $X$ with H\"older continuous distribution function. As applications, we provide the rate of convergence for numerical schemes for solutions of one-dimensional stochastic differential equations (SDEs) driven by Brownian motion and symmetric $\alpha$-stable with $\alpha \in (1,2)$, fractional Brownian motion with drift and Hurst parameter $H \in (0,1/2)$, and stochastic heat equations (SHEs) with Dirichlet boundary conditions driven by space--time white noise, with irregular coefficients. We also consider a numerical scheme for maximum and integral type functionals of SDEs driven by Brownian motion with irregular coefficients and payoffs which are related to multilevel Monte Carlo method.

math.PR↗
arXiv2019

On the Euler--Maruyama scheme for degenerate stochastic differential equations with non-sticky condition

The aim of this paper is to study weak and strong convergence of the Euler--Maruyama scheme for a solution of one-dimensional degenerate stochastic differential equation $\mathrm{d} X_t=\sigma(X_t) \mathrm{d} W_t$ with non-sticky condition. For proving this, we first prove that the Euler--Maruyama scheme also satisfies non-sticky condition. As an example, we consider stochastic differential equation $\mathrm{d} X_t=|X_t|^{\alpha} \mathrm{d} W_t$, $\alpha \in (0,1/2)$ with non-sticky boundary condition and we give some remarks on CEV models in mathematical finance.

math.PR↗
arXiv2018

Probability density function of SDEs with unbounded and path--dependent drift coefficient

In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path--dependent, and diffusion coefficient is bounded, uniformly elliptic and H\"older continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama--Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super--linear growth condition), Gaussian two--sided bound and H\"older continuity (under sub--linear growth condition) of a probability density function of a solution of SDEs with path--dependent drift coefficient. As an application of explicit representation, we provide the rate of convergence for an Euler--Maruyama (type) approximation, and an unbiased simulation scheme.

math.PR↗
arXiv2018

Newton-Kantorovitch method for decoupled forward-backward stochastic differential equations

We present and prove a Newton-Kantorovitch method for solving decoupled forward-backward stochastic differential equations (FBSDEs) involving smooth coefficients with uniformly bounded derivatives. As Newton's method is required a suitable initial condition to converge, we show that such initial conditions are solutions of a linear backward stochastic differential equation. In addition, we show that converges linearly to the solution.

math.PR↗
arXiv2018

On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case

We propose a positivity preserving implicit Euler-Maruyama scheme for a jump-extended Cox-Ingersoll-Ross (CIR) process where the jumps are governed by a compensated spectrally positive $\alpha$-stable process for $\alpha \in (1,2)$. Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV (Constant Elasticity Variance) process, the model considered here has infinite activity jumps. We calculate, in this specific model, the strong rate of convergence and give some numerical illustrations. Jump extended models of this type were initially studied in the context of branching processes and was recently introduced to the financial mathematics literature to model sovereign interest rates, power and energy markets.

math.PR↗
arXiv2018

Malliavin Calculus for Non-colliding Particle Systems

In this paper, we use Malliavin calculus to show the existence and continuity of density functions of $d$-dimensional non-colliding particle systems such as hyperbolic particle systems and Dyson Brownian motion with smooth drift. For this purpose, we apply results proved by Florit and Nualart (1995) and Naganuma (2013) on locally non-degenerate Wiener functionals.

math.PR↗
arXiv2017

On the Euler-Maruyama scheme for spectrally one-sided L\'evy driven SDEs with H\"older continuous coefficients

We study in this article the strong rate of convergence of the Euler-Maruyama scheme and associated with the jump-type equation introduced in Li and Mytnik. We obtain the strong rate of convergence under similar assumptions for strong existence and pathwise uniqueness. Models of this type can be considered as a generalization of the CIR (Cox-Ingersoll-Ross) process with jumps.

math.PR↗
arXiv2017

Semi-implicit Euler-Maruyama approximation for non-colliding particle systems

We introduce a semi-implicit Euler-Maruyama approximation which preservers the non-colliding property for some class of non-colliding particle systems such as Dyson Brownian motions, Dyson-Ornstein-Uhlenbeck processes and Brownian particles systems with nearest neighbour repulsion, and study its rates of convergence in both $L^p$-norm and path-wise sense.

math.PR↗
arXiv2015

On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients

We study the strong rates of the Euler-Maruyama approximation for one dimensional stochastic differential equations whose drift coefficient may be neither continuous nor one-sided Lipschitz and diffusion coefficient is H\"older continuous. Especially, we show that the strong rate of the Euler-Maruyama approximation is 1/2 for a large class of equations whose drift is not continuous. We also provide the strong rate for equations whose drift is H\"older continuous and diffusion is nonconstant

math.PR↗
arXiv2015

Convergence Implications via Dual Flow Method

Given a one-dimensional stochastic differential equation, one can associate to this equation a stochastic flow on $[0,+\infty )$, which has an absorbing barrier at zero. Then one can define its dual stochastic flow. In \cite{AW}, Akahori and Watanabe showed that its one-point motion solves a corresponding stochastic differential equation of Skorokhod-type. In this paper, we consider a discrete-time stochastic-flow which approximates the original stochastic flow. We show that under some assumptions, one-point motions of its dual flow also approximates the corresponding reflecting diffusion. We investigate the relation between them in weak and strong approximation sense.

math.PR↗