Victor Marx
We introduce in this paper a strategy to prove gradient estimates for some infinite-dimensional diffusions on $L_2$-Wasserstein spaces. For a specific example of a diffusion on the $L_2$-Wasserstein space of the torus, we get a Bismut-Elworthy-Li formula up to a remainder term and deduce a gradient estimate with a rate of blow-up of order $\mathcal{O}(t^{-(2+ε)})$.
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The Bismut-Elworthy-Li formula for mean-field stochastic differential equations
本文将所谓的Bismut-Elworthy-Li公式推广到一类系数可能依赖于解的规律的随机微分方程。我们给出了一些例子，这个公式可以应用于金融和计算的希腊人的背景下，并提供了一个简单的，但相当说明性的模拟实验表明，使用的Bismut-Elworthy-Li公式，也被称为Malliavin方法，是更有效的相比，有限差分法。
A Bismut-Elworthy formula for BSDEs with degenerate noise
本文在噪声非退化性较弱的假设下推导了BSDEs的Bismut-Elworthy公式，考虑了无限维Hilbert空间中随机微分方程的转移半群的梯度估计，并探讨了非线性Bismut公式的应用，应用于随机波方程和阻尼随机波方程。
In this work, we will show the existence, uniqueness, and weak differentiability of the solution to semi-linear mean-field stochastic differential equations driven by fractional Brownian motion. We prove an extension of the Bismut-Elworthy-Li formula and show some applications in the sensitivity analysis of variance swaps and also the price of derivatives with respect to the initial point.
The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps
考虑具有漂移、扩散和跳跃项的跳跃型随机微分方程。研究了求解过程中密度的对数导数，在扩散项和跳跃项系数的一致椭圆条件下，得到了Bismut-Elworthy-Li型公式.我们的方法是基于柯尔莫哥洛夫向后方程，充分利用马尔可夫性质的过程。
In this paper, we show the existence of unique Malliavin differentiable solutions to SDE`s driven by a fractional Brownian motion with Hurst parameter H<1/2 and singular, unbounded drift vector fields, for which we also prove a stability result. Further, using the latter results, we propose a stock price model with rough and correlated volatility, which also allows for capturing regime switching effects. Finally, we also derive a Bismut-Elworthy-Li formula with respect to our stock price model for certain classes of vector fields.
Bismut-Elworthy-Li Formulae for Forward-Backward SDEs with Jumps and Applications
本文提出了一种新的粒子方法，用于建立带有跳跃的前向后向SDEs的Bismut-Elworthy-Li型导数公式，并应用该公式研究了非局部拟线性积分偏微分方程的解的存在性和唯一性。
A Bismut Elworthy formula for quadratic BSDEs
在马尔可夫框架下考虑了生成元关于$Z$二次增长的过程对$(Y,Z)$的倒向随机微分方程.在非退化的假设下，我们证明了一个类似的著名的Bismut-Elworty公式时，发电机有二次增长的$Z$。应用于求解半线性Kolmogorov方程的未知$v$具有二次增长的非线性项关于$\nabla v$和最终条件只有有界和连续，以及应用于随机最优控制问题的二次增长.
In this article we are interested in the differentiability property of the Markovian semi-group corresponding to the Bessel processes of nonnegative dimension. More precisely, for all $δ\geq 0$ and $T>0$, we compute the derivative of the function $x \mapsto P^δ_{T} F (x) $, where $(P^δ_{t})_{t \geq 0}$ is the transition semi-group associated to the $δ$ - dimensional Bessel process, and $F$ is any bounded Borel function on $\mathbb{R}_{+}$. The obtained expression shows a nice interplay between the transition semi-groups of the $δ$ - and the $(δ+ 2)$-dimensional Bessel processes. As a consequen…
In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter $H<1/2$. "Singular" here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion…
We consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes $(Y,Z)$, with generator with quadratic growth with respect to $Z$. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut-Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to $Z$. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semi…
A Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces
该研究针对巴拿赫空间中线性随机发展方程，证明了Bismut-Elworthy-Li公式，在特定平方函数条件下实现了半群导数的随机积分表示，为固定时间高斯转移测度的Cameron–Martin公式提供了具体控制实现。
The Bismut-Elworthy-Li formula for jump-diffusions and applications to Monte Carlo pricing in finance
我们扩展的Bismut-Elworthy-Li公式的非退化跳跃扩散和“支付”功能取决于在未来多个时间的过程。在Fournie等人[13]和Davis和Johansson [9]的精神下，这可以改进具有跳跃的随机波动率模型的Monte Carlo数值。为此，我们需要所谓的Malliavin权，并给出了在跳跃情况下有效的显式公式：（a）在非退化情况下，推广的BEL公式将可能的Malliavin权表示为具有显式被积函数的Ito积分;（b）在亚椭圆设置中，我们回顾Arnaudon和Thalmaier [1]的工作，并且还找到了显式权重，现在涉及Malliavin协方差阵，但仍然很容易实现。(This与Forster，Lutkebohmert和Teichmann最近的工作相反，其中权重被构造为预期Skorohod积分。我们给出了（b）中的一些金融例子，但要注意的是，大多数蒙特卡罗性能较差的实际情况，例如数字小集团合同，可以通过扩展的BEL公式来处理，因此根本不依赖于Malliavin演算。然后，我们讨论了一些近似，往往被忽略的文献中，需要证明使用的Malliavin权重的上下文中的标准跳跃扩散模型。最后，由于这一切都是为了提高数值，我们给出了一些数值结果，重点是在Heston模型下的跳团。
On existence and uniqueness properties for solutions of stochastic fixed point equations with gradient-dependent nonlinearities
本文将半线性Kolmogorov偏微分方程相关随机不动点方程解的存在性与唯一性结果推广至梯度依赖非线性情形，并解决了不可微终端条件下导致的奇异性挑战。
Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wass…
Derivative formula and gradient estimate for SDEs driven by $α$-stable processes
本文证明了由$α$稳定噪声驱动的随机微分方程的Bismut-Elworthy-Li型导数公式和梯度估计，其中$α\in(0,2)$ .作为应用，给出了由从属柱布朗运动驱动的随机偏微分方程的强Feller性质.
We prove that if the Ricci tensor $\mathrm{Ric}$ of a geodesically complete Riemannian manifold $M$, endowed with the Riemannian distance $\mathsf{d}$ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous function $k\colon M\to\mathbb{R}$ whose negative part $k^-$ satisfies, for every $t>0$, the exponential integrability condition \begin{equation*} \sup_{x\in M} \mathbb{E}\big[\mathrm{e}^{\int_0^t k^-(\mathsf{b}_r^x)/2\,\mathrm{d} r}\,1_{\{t < ζ^x\}}\big] < \infty, \end{equation*} then the lifetime $ζ^x$ of Brownian motion $\mathsf{b}^x$ on $M$ starting in any $x\in …
Bismut Formula and Gradient Estimates for Dirichlet Semigroups with Application to Singular Killed DDSDEs
本文通过建立Dirichlet半群在规则域上的局部Bismut公式，推导出具有奇异漂移的 Killed SDEs 的梯度估计，并应用该结果界定了Killed DDSDEs 解的总变差距离。
Quasi-invariant theorem on the Gaussian path space
本文引入一类高斯过程并证明关于高斯维纳测度的拟不变定理，涵盖分数布朗运动，进而建立分部积分公式、Bismut-Elworthy-Li公式及对数Sobolev不等式，并应用于金融数学。
Gradient estimates for SDEs Driven by Multiplicative Lévy Noise
本文首次给出了一类乘性Lévy噪声驱动的随机微分方程所对应的半群的梯度估计.特别是，估计是尖锐的$α$稳定型噪声。为了得到这些估计，一个新的导数公式的Bismut-Elworthy-Li的类型建立的半群通过使用Malliavin演算和有限跳近似的论点。
Log-Harnack Inequality and Bismut Formula for McKean-Vlasov SDEs with Singularities in all Variables
本文在漂移项于时间-空间可积且分布连续性弱于 Dini 的条件下，为所有变量均具有奇异性的 McKean-Vlasov 随机微分方程建立了 log-Harnack 不等式和 Bismut 公式，显著推广了现有结果。
Derivative Formulae and Gradient Estimates for Stochastic Hamiltonian Systems with Jumps
本文针对Lévy过程驱动的随机哈密顿系统，利用带跳跃的Malliavin微积分中的lent particle方法，建立了显式的Bismut--Elworthy--Li型导数公式和梯度估计，将退化系统的结果推广到非局部跳跃情形，填补了文献空白。
Gradient estimates for semigroups associated with stochastic differential equations driven by cylindrical Lévy processes
本文利用Bismut-Elworthy-Li公式，为柱状Lévy过程驱动的随机微分方程导出均匀梯度估计，并应用于Stein方法及Euler-Maruyama格式，获得Wasserstein距离下对步长线性依赖的最优逼近误差。
We prove gradient estimates for transition Markov semigroups $(P_t)$ associated to SDEs driven by multiplicative Brownian noise having possibly unbounded $C^1$-coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for $D_x P_tφ$ of the form \[ {\sqrt{t}} \, |D_x P_t φ(x) | \le c \, (1+ |x|^k) \, \| φ\|_{\infty} \] $t \in (0,1]$, $φ\in C_b ({\mathbb R}^d)$, $x \in {\mathbb R}^d.$ To prove the result we use two main tools. First, we consider a Fe…
Score-Based Diffusion Models in Infinite Dimensions: A Malliavin Calculus Perspective
本文从马尔可夫计算视角研究无穷维分数模型，推导了无穷维过渡测度的对数导数表达式，适用于希尔伯特空间的几何结构和一般迹类算子。
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