Shota Tsumurai
It is well known that Malliavin calculus can be applied to a stochastic differential equation with Lipschitz continuous coefficients in order to clarify the existence and the smoothness of the solution. In this paper, we apply Malliavin calculus to the CEV-type Heston model whose diffusion coefficient is non-Lipschitz continuous and prove the Malliavin differentiability of the model.
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In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a G{â}teaux derivative in the directions of the Cameron-Martin space. Incidentally, we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces $D^{1,p}$ .
In this paper we study rough differential equations driven by Gaussian rough paths from the viewpoint of Malliavin calculus. Under mild assumptions on coefficient vector fields and underlying Gaussian processes, we prove that solutions at a fixed time is smooth in the sense of Malliavin calculus. Examples of Gaussian processes include fractional Brownian motion with Hurst parameter larger than $1/4$.
Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates
本文研究了由分数布朗运动驱动的随机微分方程Euler格式的Malliavin可微性，并推导出步长一致且逐路径的上界估计。
In this article, we consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term $σ$. We prove that under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If $σ$ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.
Malliavin differentiability of McKean-Vlasov SDEs with common noise
该研究在全局Lipschitz假设下证明了带公共噪声的McKean-Vlasov随机微分方程的Malliavin可微性，明确了条件分布关于公共噪声的Malliavin导数的意义，并推导了对应维纳空间上的分部积分公式。
In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.
Malliavin differentiability of McKean-Vlasov SDEs with locally Lipschitz coefficients
本文证明了在漂移项满足局部Lipschitz和单边Lipschitz条件、扩散系数一致Lipschitz的情况下，McKean-Vlasov随机微分方程具有Malliavin可微性，并进一步研究了该可微性从相互作用粒子系统到极限方程的传递。
We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a locally Lipschitz condition in the $Z$ and $U$ variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these re…
On Malliavin differentiability and absolute continuity of one-dimensional doubly perturbed diffusion processes
本文研究一维双重扰动扩散过程，在参数条件下证明其Malliavin可微性、绝对连续性及光滑密度的存在性，推广了先前工作，并采用Malliavin微积分方法。
Malliavin differentiability for the excursion measure of a Gaussian field
本文研究高斯场游程体积的 Malliavin 可微性，证明协方差核矩的多项式衰减可精确确定游程测度的 p 阶平方可积 Malliavin 导数，并给出平稳或各向同性情形下的充分条件。
We consider a one-dimensional Stochastic Differential Equation with reflection where we allow the drift to be merely bounded and measurable. It is already known that such equations have a unique strong solution. Recently, it has been shown that non-reflected SDE's with discontinuous drift possess more regularity than one could expect, namely they are Malliavin differentiable and weakly differentiable w.r.t. the initial value. In this paper we show that similar results hold for one-dimensional SDE's with reflection. We then apply the results to get a Bismut-Elworthy-Li formula for the correspon…
We consider Malliavin smoothness of random variables $f(X_1)$, where $X$ is a pure jump Lévy process and $f$ is either bounded and Hölder continuous or of bounded variation. We show that Malliavin differentiability and fractional differentiability of $f(X_1)$ depend both on the regularity of $f$ and the Blumenthal-Getoor index of the Lévy measure.
We investigate solutions of backward stochastic differential equations (BSDE) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Levy process. In this new type of equations, the generator can depend on the past values of a solution, by feeding them back into the dynamics with a time lag. For such time delayed BSDE, we prove existence and uniqueness of solutions provided we restrict on a sufficiently small time horizon or the generator possesses a sufficiently small Lipschitz constant. We study differentiability in the va…
We consider the class of non-linear stochastic partial differential equations studied in \cite{conusdalang}. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point $(t,x)\in[0,T]\times \Rd$ is differentiable in the Malliavin sense. For this, an extension of the integration theory in \cite{conusdalang} to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise inte…
We consider Malliavin calculus based on the Itô chaos decomposition of square integrable random variables on the Lévy space. We show that when a random variable satisfies a certain measurability condition, its differentiability and fractional differentiability can be determined by weighted Lebesgue spaces. The measurability condition is satisfied for all random variables if the underlying Lévy process is a compound Poisson process on a finite time interval.
In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
Differentiability in infinite dimension and the Malliavin calculus
本文在无限维分析和Malliavin演算框架下，研究了P. Cannarsa、G. Da Prato及L. Gross提出的两种可微性概念。
In this work we prove Malliavin differentiability for the solution to an SDE with locally Lipschitz and semi-monotone drift. To this end we construct a sequence of SDEs with globally Lipschitz drifts. We show that the solutions of these SDEs converge to the solution of the original SDE and the p-moments of their Malliavin derivatives are uniformly bounded.
We give a criteria for a Malliavin differentiable function to be strongly H-differentiable.
Malliavin differentiability of solutions of hyperbolic stochastic partial differential equations with irregular drifts
本文研究带不规则漂移项的双曲型随机偏微分方程，通过逐路径唯一性和Yamada-Watanabe原理建立强唯一性，并证明在漂移全局有界时唯一强解具有Malliavin可微性。
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…
In this paper, we provide a strong formulation of the stochastic G{â}teaux differentiability in order to study the sharpness of a new characterization, introduced in [6], of the Malliavin-Sobolev spaces. We also give a new internal structure of these spaces in the sense of sets inclusion.
We examine existence and uniqueness of strong solutions of multi-dimensional mean-field stochastic differential equations with irregular drift coefficients. Furthermore, we establish Malliavin differentiability of the solution and show regularity properties such as Sobolev differentiability in the initial data as well as Hölder continuity in time and the initial data. Using the Malliavin and Sobolev differentiability we formulate a Bismut-Elworthy-Li type formula for mean-field stochastic differential equations, i.e. a probabilistic representation of the first order derivative of an expectatio…
Malliavian differentiablity and smoothness of density for SDES with locally Lipschitz coefficients
本文研究了具有超线性增长漂移的随机微分方程解的Malliavin可微性，证明了在单调多项式增长漂移条件下的任意阶可微性，并在Hörmander假设下证明了密度的无限次可微性。
In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Hölder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2 has a smooth density. To this purpose, we use Malliavin calculus based on the Frechet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.
We study Malliavin differentiability of solutions to sub-critical singular parabolic stochastic partial differential equations (SPDEs) and we prove the existence of densities for a class of singular SPDEs. Both of these results are implemented in the setting of regularity structures. For this we construct renormalized models in situations where some of the driving noises are replaced by deterministic Cameron-Martin functions, and we show Lipschitz continuity of these models with respect to the Cameron-Martin norm. In particular, in many interesting situations we obtain a convergence and stabil…
Locally Lipschitz Path Dependent FBSDEs with Unbounded Terminal Conditions in Brownian and L{é}vy Settings
本文研究Lévy过程驱动的路径依赖局部Lipschitz正倒向随机微分方程，通过截断迭代等三步法证明解的存在唯一性，还基于新的空间刻画得到Malliavin可微性结果。
We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth (and random coefficients), namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Ray Absolute Continuity and Stochastic Gâteaux Differentiability. This method enables one to take limits in probability rather than mean square which bypasses the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology from Nualart's 2006 work, Lemma 1.2.…
Malliavin calculus and densities for chaos-driven stochastic differential equations
本文研究由有限阶混沌过程驱动的随机微分方程，通过路径积分建立解的存在性和唯一性，并利用马尔蒂嫩微积分方法证明解的绝对连续性，应用于多维赫尔米特驱动方程。
In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H < 1/2. Furthermore, we prove the rather surprising result of the higher order Frechet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can be also applied to the study of…
In this paper, we consider quadratic forward-backward SDEs (QFBSDEs), for {which} the drift in the forward equation does not satisfy the standard globally Lipschitz condition and the driver of the backward system {possesses} nonlinearity of type $f(|y|)|z|^2,$ where $f$ is any locally integrable function. We prove both the Malliavin and classical derivative of the QFBSDE and provide representations of these processes. We study a numerical approximation of this system in the sense of \cite{ImkDosReis} in which the authors assume that the drift is Lipschitz and the driver of the BSDE is quadrati…
Exponential bounds for the density of the law of the solution of a SDE with locally Lipschitz coefficients
该论文在一致Hörmander假设下，针对局部Lipschitz漂移且满足单调性条件的SDE，利用Malliavin可微性结果证明了其解的概率密度的光滑性与指数界。
We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get…
An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Ito's formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Levy models. Since…
In this paper we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to the coefficients, weak differentiability with respect to starting points, and the Malliavin differentiability with respect to sample paths. We also establish Bismut-Elworthy-Li's formula for the solutions. As an application, we use the stochastic Lagrangian representation of incompressible Navier-Stokes equations given by Constantin-Iyer \cite{Co-Iy} to prove the local well-posedness of NSEs in $\m…
In the pathwise stochastic calculus framework, the paper deals with the general study of equations driven by an additive Gaussian noise, with a drift function having an infinite limit at point zero. An ergodic theorem and the convergence of the implicit Euler scheme are proved. The Malliavin calculus is used to study the absolute continuity of the distribution of the solution. An estimation procedure of the parameters of the random component of the model is provided. The properties are transferred on a class of singular stochastic differential equations driven by a multiplicative noise. A frac…
Malliavin Calculus for the one-dimensional Stochastic Stefan Problem
本文将Malliavin演算应用于一维带反射的随机Stefan问题，证明了极大解的路径连续性，并在无反射情况下引入局部化论证，证明了Malliavin导数的存在性及解的绝对连续性。
We apply Malliavin calculus to the $Φ^4_3$ equation on the torus and prove existence of densities for the solution of the equation evaluated at regular enough test functions. We work in the framework of regularity structures and rely on Besov-type spaces of modelled distributions in order to prove Malliavin differentiability of the solution. Our result applies to a large family of Gaussian space-time noises including white noise, in particular the noise may be degenerate as long as it is sufficiently rough on small scales.
Option pricing in Sandwiched Volterra Volatility model
该研究提出由任意Hölder连续高斯Volterra过程驱动的夹层波动率新模型，分析其测度、可积性等性质，并基于Malliavin微积分实现不连续收益期权的定价。
In this paper, we are interested in the following singular stochastic differential equation (SDE) $${\rm d} X_t = b(t,X_t) {\rm d} t + {\rm d} B_{t},\ 0\leq t\leq T,\ X_0 = x \in \mathbb{R}^d,$$ where the drift coefficient $b:[0,T]\times \mathbb{R}^{d}\longrightarrow \mathbb{R}^{d}$ is Borel measurable, possibly unbounded and has spatial linear growth. The driving noise $B_{t}$ is a $d-$ dimensional Brownian motion. The main objective of the paper is to establish the existence and uniqueness of a strong solution and a Sobolev differentiable stochastic flow for the above SDE. Malliavin differen…
Central limit theorem for stochastic nonlinear wave equation with pure-jump Lévy white noise
本文研究了具有常初值和乘性噪声σ(u)˙L的随机非线性波方程的随机场解，通过伊藤微积分和马尔可夫微积分工具，建立了解的马尔可夫可微性，并获得了马尔可夫导数的精确矩界。由此得出解的空间遍历性，进而得到空间积分的大数定律结果，并进一步获得高斯波动的收敛速率。
Splitting AVF method for generalized Langevin equations: probability density function and geometric ergodicity
本文提出一种结构保持的分裂平均向量场方法，用于求解广义 Langevin 方程，该方法在保持指数可积性、Malliavin 可微性和ergodic性的同时，保证了概率密度函数的存在性和光滑性，并通过数值实验验证了几何ergodic性。
Absolute continuity of the solution to stochastic generalized Burgers-Huxley equation
本文研究乘性时空白噪声驱动的随机广义Burgers-Huxley方程，证明其整体可解性，并借助比较定理与Malliavin弱可微性得到解的分布关于Lebesgue测度的绝对连续性及密度存在性。
Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields
针对有界可测向量场下拟线性双曲型SPDE确定性形式不适定的问题，结合马利亚万微积分与推广的变分技术，首次证明其存在唯一马利亚万可微强解，还得到解对初值的索伯列夫可微性，推进了平面SDE领域研究。
In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to Itô's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory to …
Revisiting Stochastic Realization Theory using Functional Itô Calculus
本文利用 Dupire 的泛函 Itô 演算，为连续半鞅驱动的因果系统输出随机过程构造有限维状态空间实现，扩展了 Hijab 的思想，并与确定性系统的实现理论相联系。
Density Analysis for coupled forward-backward SDEs with non-Lipschitz drifts and Applications
本文研究了具有非Lipschitz漂移的耦合二次前向-后向SDEs解的密度存在性，提出基于弱解耦场和Itô-Tanaka技巧的框架，并应用于利率模型与不可交易资产的对冲问题。
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