arXiv subjects

Takuya Nakagawa

Publications and source records attributed to Takuya Nakagawa.

5 recordsLinked to original sources

arXiv2026

Existence of densities and atoms for the running maximum of time-inhomogeneous jump diffusions

We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous Lévy--Itô diffusions driven by a Brownian motion and an independent non-truncated pure-jump Lévy process. Using Bismut's directional Malliavin calculus on the Wiener--Poisson space together with the running-maximum criteria of Song--Xie and Nakagawa--Suzuki, we reduce the problem to constructing an admissible direction $Θ$. The key is to ensure that the directional derivative $D_ΘX_t$ is strictly positive for all $t\in(0,T]$. We give explicit directions in two regimes. In the uniformly elliptic case with time-inhomogeneous coefficients, a purely Brownian perturbation yields an explicit positive integral representation for $D_ΘX_t$, and hence $X^{\ast}_T$ admits a density without truncating the jump component. In a Brownian-degenerate pure-jump model with a bounded deterministic time-dependent jump weight $κ(t)$ that may vanish on subintervals, we prove absolute continuity under the minimal nondegeneracy-in-time condition $\int_0^t κ(s)^2\,ds>0$ for every $t>0$ and infinite activity of the Lévy measure. A weighted Poisson positivity lemma is the key new input. Finally, we show that silent initial intervals can create atoms and we derive an explicit atom--density decomposition.

math.PR↗

arXiv2026

Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process

We prove quantitative pathwise stability estimates for one-dimensional stochastic differential equations driven by a common Brownian motion and a common symmetric $α$-stable process, where $α\in(1,2)$. The comparison is made under a synchronous coupling and is measured by $\sup_{0\le t\le T}\mathbb E|X_t-\wt X_t|^{α-1}$. The perturbed drift and Brownian diffusion coefficients are spatially Lipschitz, while the perturbed stable jump coefficient may be Hölder continuous down to the critical exponent $1/α$. The estimate is expressed in terms of the initial error and three law-weighted coefficient errors: the drift error $B$, the Brownian diffusion error $A$, and the stable jump-coefficient error $S$. The Brownian component produces a second-order correction term in Itô's formula. This term is controlled by a second-derivative estimate for a Komatsu-type mollification of $|x|^{α-1}$. For stable jump coefficients with Hölder exponent $\wtη>1/α$, the estimate gives explicit power rates in $B,A,S$; at $\wtη=1/α$, it gives a logarithmic rate. The same method yields estimates with predictable forcing errors and time-uniform tail bounds via stopped quasi-martingales.

math.PR↗

arXiv2026

$L^{α-1}$ distance between two one-dimensional stochastic differential equations with drift terms driven by a symmetric $α$-stable process

This paper establishes a quantitative stability theory for one-dimensional stochastic differential equations (SDEs) with non-zero drift, driven by a symmetric $α$-stable process for $α\in(1,2)$. Our work generalizes the classical pathwise comparison method, pioneered by Komatsu for uniqueness problems, to address the stability of SDEs featuring both non-zero drift and, crucially, time-dependent coefficients. We provide the first explicit convergence rates for this broad class of SDEs. The main result is a Hölder-type estimate for the $L^{α-1}(Ω)$ distance between two solution paths, which quantifies the stability with respect to the initial values and coefficients. A key innovation of our approach is the measurement of the distance between coefficients. Instead of using a standard supremum norm, which would impose restrictive conditions, we introduce a weighted integral norm constructed from the transition probability density of the baseline solution. This technique, which generalizes the framework of Nakagawa \cite{Nakagawa}, is essential for handling time-dependent perturbations and effectively localizes the error analysis. The proof is based on a refined analysis of a mollified auxiliary function, for which we establish a new, sharper derivative estimate to control the drift terms. Finally, we apply these stability results to derive corresponding convergence rates in probability, providing an upper bound for the tail probability of the uniform distance between solution paths.

math.PR↗

arXiv2025

Existence of density functions for SDEs driven by pure-jump processes

We verify the existence of density functions of the running maximum of a stochastic differential equation (SDE) driven by a Brownian motion and a non-truncated stable process. This is proved by the existence of density functions of the running maximum of Wiener-Poisson functionals resulting from Bismut's approach to Malliavin calculus for jump processes.

math.PR↗

arXiv2022

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↗