Alberto Lanconelli
We derive an Itô's-type formula for the one dimensional stochastic heat equation driven by a space-time white noise. The proof is based on elementary properties of the $\mathcal{S}$-transform and on the explicit representation of the solution process. We also discuss the relationship with other versions of this Itô's-type formula existing in literature.
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An Itô type formula for the additive stochastic heat equation
本文利用正则结构理论，为一维时空白噪声驱动的周期随机热方程解建立 Itô 型公式，并用重构算子刻画 $φ(u)$ 及其分布作用。
One of the variants to proof the generalized Ito-Wentzell's formula is introduced and examined in this paper. The relationship between different representations of the generalized Ito-Wentzell's formula/ is considered.
Generalization of the Itô-Wentzell formula for the generalized Itô's SDE (Itô's GSDE) system with a non-centered measure is constructed on the basis of the stochastic kernel of integral transformation. The Itô's GSDE system for the kernel the solution of which is the kernel of the integral invariant, is formed. This invariant is connected with the solution of the Itô's GSDE system non-centered measure. The concept of a stochastic first integral of the Itô's GSDE system with non-centered measure is introduced and conditions that when being performed the random function is the first integral of …
Itô's Formula for the Rearranged Stochastic Heat Equation
本文为重排随机热方程建立了一个方便版本的伊藤公式，识别出其解在 Lions 光滑函数上的生成元，并证明反射项与 Lions 导数正交。
Itô's Formula for Itô processes defined with respect to a cylindrical-martingale valued measure
基于作者近期发展的随机积分理论，证明了希尔伯特空间值Itô过程的Itô公式，并推导了Burkholder不等式及鞅值测度的Itô公式。
We present the product formula for complex multiple Wiener-Ito integrals. As applications, we show the Ustunel-Zakai independent criterion, the Nourdin-Rosinski asymptotic moment-independent criterion and joint convergence criterion for complex multiple Wiener-Ito integrals.
Using the Malliavin calculus with respect to Gaussian processes and the multiple stochastic integrals we derive Itô's and Tanaka's formulas for the $d$-dimensional bifractional Brownian motion.
We consider two approaches for obtain of the generalized Ito-Wentzell formula: the first way uses the generalized Ito's formula; the second one is based on a concept of kernel functions for integral invariants.
We clearly characterize the relation between real and complex Wiener-Ito integrals. Given a complex multiple Wiener-Ito integral, we get explicit expressions for two kernels of its real and imaginary parts. Conversely, consider a two-dimensional real Wiener-Ito integral, we obtain the representation formula by a finite sum of complex Wiener-Ito integrals. The main tools are a recursion technique and Malliavin derivative operators. We build a bridge between real and complex Wiener-Ito integrals.
This paper is complete proof of one method for obtaining the generalized Ito-Wentzell formula, its basic idea was announced earlier in a pre-print (arXiv:1309.3038v1). This proof sets the approach which uses the Ito formula and the stochastic approximation theory.
We prove Itô's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form.
This is a survey note of the author's observations on the discrete-time analogues of Itô formulas.
We use Yosida approximation to find an Itô formula for mild solutions $\left\{X^x(t), t\geq 0\right\}$ of SPDEs with Gaussian and non-Gaussian coloured noise, the non Gaussian noise being defined through compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in $C^{1,2}([0,T]\times H)$, as in the case considered for SDEs in [9]. Using this Itô formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such Itô formula to an Itô formula for mild sol…
We introduce a Skorokhod type integral and prove an Ito formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Ito formula unifies and extends the classical one for general (i.e., possibly discontinuous) Gaussian martingales in the sense of Ito integration and the one for stochastically continuous Gaussian non-martingales in the Skorokhod sense, which was first derived in Alos et al. (Ann. Probab. 29, 2001).
A well-known Itô formula for finite dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite dimensional Itô formula for continuous semimartingales proved by Krylov to a class of $L_p$-valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.
An Itô-Wentzell formula for the fractional Brownian motion
本文证明了分数布朗运动的Itô-Wentzell公式，并以此推导出由该过程驱动的一类随机微分方程的存在性与唯一性结果。
We prove the Ito-Tanaka formula and the existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue-Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of Ito-Tanaka formula for pricing and hedging of financial derivatives.
We develop the functional Itô/path-dependent calculus with respect to fractional Brownian motion with Hurst parameter $H> \frac{1}{2}$. Firstly, two types of integrals are studied. The first type is Stratonovich integral, and the second type is Wick-Itô integral. Then we establish the functional Itô formulas for fractional Brownian motion, which extend the functional Itô formulas in Dupire (2009) and Cont-Fournié (2013) to the case of non-semimartingale. Finally, as an application, we deal with a class of fractional backward stochastic differential equations (BSDEs). A relation between fractio…
We provide an Itô's formula for stochastic dynamical equation on general time scales. Based on this Itô's formula we give a closed form expression for stochastic exponential on general time scales. We then demonstrate a Girsanov's change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (q-time scale).
We derive an Ito-formula for the Dawson-Watanabe superprocess, a well-known class of measure-valued processes, extending the classical Ito-formula with respect to two aspects. Firstly, we extend the state-space of the underlying process $(X(t))_{t\in [0,T]}$ to an infinite-dimensional one - the space of finite measure. Secondly, we extend the formula to functions $F(t,X_t)$ depending on the entire paths $X_t=(X(s\wedge t))_{s \in [0,T]}$ up to times $t$. This later extension is usually called functional Ito-formula. Finally we remark on the application to predictable representation for marting…
We derive a generalised Itō formula for stochastic processes which are constructed by a convolution of a deterministic kernel with a centred Lévy process. This formula has a unifying character in the sense that it contains the classical Itō formula for Lévy processes as well as recent change-of-variable formulas for Gaussian processes such as fractional Brownian motion as special cases. Our result also covers fractional Lévy processes (with Mandelbrot-Van Ness kernel) and a wide class of related processes for which such a generalised Itō formula has not yet been available in the literature.
In this paper we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Itô-type formula for the process X.
This article shows an Itô-Wentzell type formula adapted to rough paths with $α$-Hölder regularity $α\in (\frac{1}{3},\frac{1}{2}]$. We improve previous results of R. Castrequini and P. Catuogno for the Young integral and C. Keller and J. Zhang for rough paths.
On the Itô-Alekseev-Gröbner formula for stochastic differential equations
本文提出一种新的Itô-Alekseev-Gröbner公式，用于计算随机微分方程解与Itô过程的测试函数之差，推广了经典Alekseev-Gröbner公式和Itô公式，为推导随机方程扰动与逼近的强逼近速率提供了有力工具。
We show an It\^ o's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in Itô's s formula as an integral over space and time with respect to local time.
We give a short proof of Itô's formula for stochastic Hilbert-space valued processes in the setting $V\subset H\subset V^{*}$ based on the possibility to lift the stochastic differentials, which are originally in $V^{*}$, into $H$. Using this result we also prove the maximum principle for second-order SPDEs in arbitrary domains.
An Optimal Functional Itô's Formula For Lévy Processes
本文针对Lévy过程提出了一个最优的局部时-空泛函Itô公式，适用于任意有界连续泛函，且不依赖Hölder连续性条件。
The mild Ito formula proposed in Theorem 1 in [Da Prato, G., Jentzen, A., \& Röckner, M., A mild Ito formula for SPDEs, arXiv:1009.3526 (2012), To appear in the Trans.\ Amer.\ Math.\ Soc.] has turned out to be a useful instrument to study solutions and numerical approximations of stochastic partial differential equations (SPDEs) which are formulated as stochastic evolution equations (SEEs) on Hilbert spaces. In this article we generalize this mild Itô formula so that it is applicable to solutions and numerical approximations of SPDEs which are formulated as SEEs on UMD (unconditional martingal…
We extend the Itô-Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to $k$-form-valued stochastic processes. The result is the Kunita-Itô-Wentzell (KIW) formula for $k$-forms. We also establish a correspondence between the KIW formula for $k$-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic ideal fluid dynamics. This geometric structure includes Eulerian and Lagrangian variational principles, Lie--Poisson Hamiltonian formulations and natural analogues of the Kelvin cir…
Ito's formula for flows of conditional measures on semimartingales
本文在一般半鞅的共同滤波下建立了条件概率测度流的Ito公式，通过构建条件独立副本并证明随机积分的等价性，推广了现有关于Ito过程和确定性测度流的研究成果。
In this article we study existence of pathwise stochastic integrals with respect to a general class of $n$-dimensional Gaussian processes and a wide class of adapted integrands. More precisely, we study integrands which are functions that are of locally bounded variation with respect to all variables. Moreover, multidimensional Itô formula is derived.
Several versions of Itô's formula have been obtained in the context of the functional stochastic calculus. Here, we revisit this topic in two ways. First, by defining a notion of derivative along a functional, we extend the setting of the (semimartingale) functional Itô's formula and corresponding calculus. Second, for Lévy processes, an optimal local-time based Itô's formula is obtained. Some quick applications are then given.
The overarching goal of this paper is to establish a set-valued Itô's formula. As an application, we obtain the existence and uniqueness of solutions for the general set-valued backward stochastic differential equation which gives an answer to an open question proposed by Ararat et al. (C. Ararat, J. Ma and W.Q. Wu, Set-valued backward stochastic differential equation, arXiv:2007.15073).
The $C^{0,1}$ Itô-Ventzell formula for weak Dirichlet processes
本文将Itô-Ventzell公式推广到$C^{0,1}$中连续弱Dirichlet过程的随机流，并应用于椭圆型SPDE表示、二次变差公式推导及金融数学假设的放宽。
We study a Edgeworth-type refinement of the central limit theorem for the discretizacion error of Itô integrals. Towards this end, we introduce a new approach, based on the anticipating Itô formula. This alternative technique allows us to compute explicitly the terms of the corresponding expansion formula.
In this paper, we use the Malliavin calculus techniques to obtain an anticipative version of the change of variable formula for Lévy processes. Here the coefficients are in the domain of the anihilation (gradient) operator in the "future sense", which includes the family of all adapted and square-integrable processes. This domain was introduced on the Wiener space by Alòs and Nualart.
The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron-Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar …
Motivated by questions arising in financial mathematics, Dupire introduced a notion of smoothness for functionals of paths (different from the usual Fréchet--Gatéaux derivatives) and arrived at a generalization of Itō's formula applicable to functionals which have a pathwise continuous dependence on the trajectories of the underlying process. We study nonlinear functionals which do not have such pathwise continuity and further work simultaneously under the family of continuous semimartingale measures on path-space. We do this without introducing a second component, as carried out by Cont--Four…
Existence, Regularity, and a Strong Itô Formula for the Isochronal Phase of SPDE
本文证明了一类发展方程稳定不变流形等时映射的存在性与正则性，并据此为白噪声驱动 SPDE 中的行波、螺旋波等模式建立了等时相位的强 Itô 公式。
A Geometric Extension of the Itô-Wentzell and Kunita's Formulas
本文将 Itô-Wentzell 公式推广到流形上的张量值随机过程，推导随机微分同胚流下拉回与前推的 KIW 公式，以服务随机流体动力学和噪声适定性研究。
Generalization of the Itô-Wentzel formula for the generalized Itô's SDEs (Itô's GSDEs) system with not centered measure is constructed. This construction is based on the basis of the stochastic kernel of integral transformation. The Itô's GSDEs system for the kernel of the integral invariant is constructed. The concept of a stochastic first integral of the Itô's GSDEs system with not centered measure is introduced. The conditions for the random function, that it's the first integral of the set Itô's GSDEs system, are defined.
The paper introduces a novel Ito's formula for time dependent tempered generalized functions. As an application, we study the heat equation when initial conditions are allowed to be a generalized tempered function. A new proof of the Ustunel- Ito's formula for tempered distributions is also provided.
This article introduces a certain class of stochastic processes, which we suggest to call mild Ito processes, and a new - somehow mild - Ito type formula for such processes. Examples of mild Ito processes are mild solutions of SPDEs and their numerical approximation processes.
Controlled fields, rough stochastic calculus, and Itô-Wentzell-Alekseev-Gröbner identities
本文提出了一种用于粗糙随机系统的时空受控场微积分，推导了粗糙随机伊藤-文兹尔公式，并基于前人工作建立了相关恒等式。
SPDEs driven by standard symmetric $α$-stable cylindrical Lévy processes: existence, Lyapunov functionals and Itô formula
本文研究希尔伯特空间中标准对称$\alpha$-稳定柱形噪声驱动的随机偏微分方程，证明温和解存在性、给出解的长期正则性与Itô公式，针对柱形噪声特性发展了新的论证方法。
Within the framework of the previous paper [8]. we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman-Kac formula and a representation formula for higher derivatives of analytic functions are also given.
The objects under investigation are the stochastic integrals with respect to free Levy processes. We define such integrals for square-integrable integrands, as well as for a certain general class of bounded integrands. Using the product form of the Ito formula, we prove the full functional Ito formula in this context.
We consider the solution $u(x,t)$ to a stochastic heat equation. For fixed $x$, the process $F(t)=u(x,t)$ has a nontrivial quartic variation. It follows that $F$ is not a semimartingale, so a stochastic integral with respect to $F$ cannot be defined in the classical Itô sense. We show that for sufficiently differentiable functions $g(x,t)$, a stochastic integral $\int g(F(t),t)\,dF(t)$ exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variabl…
Itô perspective on variance renormalisation
研究伊藤解在非线性随机热方程中的极限行为，揭示在超临界噪声下，通过方差重整化得到随机热方程的解，且指数3/4与常规标度论证不符。
Let $X$ be a (two-sided) fractional Brownian motion of Hurst parameter $H\in (0,1)$ and let $Y$ be a standard Brownian motion independent of $X$. Fractional Brownian motion in Brownian motion time (of index $H$), recently studied in \cite{13}, is by definition the process $Z=X\circ Y$. It is a continuous, non-Gaussian process with stationary increments, which is selfsimilar of index $H/2$. The main result of the present paper is an Itô's type formula for $f(Z_t)$, when $f:\R\to\R$ is smooth and $H\in [1/6,1)$. When $H>1/6$, the change-of-variable formula we obtain is similar to that of the cla…
We present several Ito-Wentzell formulae on Wiener spaces for real-valued functional random field of Ito type that depend on measure flows. We distinguish the full- and the marginal-measure flow cases in the spirit of mean-field games. Derivatives with respect to the measure components are understood in the sense of Lions.
In the present paper, a discrete version of Itô's formula for a class of multi-dimensional random walk is introduced and applied to the study of a discrete-time complete market model which we call He's framework. The formula unifies continuous-time and discrete-time settings and by regarding the latter as the finite difference scheme of the former, the order of convergence is obtained. The result shows that He's framework cannot be of order 1 scheme except for the one dimensional case.
We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G--Brownian motion and derive the related Ito's formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expe…
Jump Itô-type formula with arbitrary regularity
本文提出一种适用于具有有限p-变差且存在跳跃的路径的伊藤型公式，区分了粗糙积分与跳跃修正项，并在连续和 càdlàg 情况下分别推导了相应公式，同时给出了随机微分方程的链式法则和路径上的对数财富分解。
Itô formula for planarly branched rough paths
本文针对包含更多随机过程的平面分支粗糙路径，证明了粗糙度$\frac{1}{4}< α\leq \frac{1}{2}$条件下的Itô公式，扩展了粗糙路径理论。
We consider autonomous stochastic ordinary differential equations (SDEs) and weak approximations of their solutions for a general class of sufficiently smooth path-dependent functionals f. Based on tools from functional Itô calculus, such as the functional Itô formula and functional Kolmogorov equation, we derive a general representation formula for the weak error $E(f(X_T)-f(\tilde X_T))$, where $X_T$ and $\tilde X_T$ are the paths of the solution process and its approximation up to time T. The functional $f:C([0,T],R^d)\to R$ is assumed to be twice continuously Fréchet differentiable with de…
An Operator Ito Formula for Volterra Gaussian Processes: The Intrinsic Bracket via Causal Derivation-Divergence Factorization
本文提出了一种适用于Volterra高斯过程的Ito公式，通过因果导数-发散因子分解方法，推导了能量函数校正，并证明了内在括号的不变性和稳定性。
A pathwise Ito formula for weakly differentiable functions
该研究推广了Föllmer的逐段伊藤公式，构造了依赖路径的索伯列夫空间，证明了变量替换公式，在布朗运动情形下验证了函数的空间归属，并得到多维Föllmer–Protter公式的逐段版本。
For stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Ito flow map is given. A similar formula is also obtained for solutions of linear matrix-valued SDEs driven by arbitrary semimartingales. The computation relies on the lift to quasi-shuffle algebras of formulas involving products of Ito integrals of semimartingales. Whereas the Chen-Strichartz formula computing the logarithm of the Stratonovich flow map is classically expanded as a formal sum indexed by permutations, the analogous formula in Ito calculus is naturally indexed by surjections.…
Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.
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