我们提出了暗示条件完全支持（CFS）属性的条件，由Guasoni，Rásonyi和Schachermayer [Ann. Appl. Probab.，18（2008），pp. 491- 520]，对于过程Z：= H + K \cdot W，其中W是布朗运动，H是连续过程，过程H和K是渐进的或与W无关.此外，在后一种情况下的一个额外的假设，K是有限的变化，我们提出的条件下，Z CFS也当W被替换为一个一般的连续过程CFS。作为这些结果的应用，我们证明了几个随机波动率模型和某些随机微分方程的解具有CFS。
英文摘要：
We present conditions that imply the conditional full support (CFS) property, introduced by Guasoni, Rásonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008), pp. 491--520], for processes Z := H + K \cdot W, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.
Conditional Expectation Backward Stochastic Differential Equations and Related Backward Stochastic Differential Equations with Conditional Reflection
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究院）；Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学非线性预期前沿科学中心（教育部））；Shandong Province Key Laboratory of Financial Risk, Shandong University（山东大学山东省金融风险重点实验室）
In this small note we explicit the relation between Ito and Stratonovich integrals when conditional measure flow components are present in the integrands. The `correction' term involves Lions-type measure derivatives and clarifies which cross-correlations need to be taken into account. We cast the framework in relation to SDEs of mean-field type depending on conditional flows of measure. The result being trivial under full flows of measure.
The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this extends the usual representation results for separable spaces.
Conditional Reflected Backward Stochastic Differential Equations with Two Barriers
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University（山东大学数学交叉科学研究院）；Frontiers Science Center for Nonlinear Expectations (Ministry of Education), Shandong University（山东大学非线性预期前沿科学中心（教育部））
In this paper we consider a class of {\it conditional McKean-Vlasov SDEs} (CMVSDE for short). Such an SDE can be considered as an extended version of McKean-Vlasov SDEs with common noises, as well as the general version of the so-called {\it conditional mean-field SDEs} (CMFSDE) studied previously by the authors [1, 14], but with some fundamental differences. In particular, due to the lack of compactness of the iterated conditional laws, the existing arguments of Schauder's fixed point theorem do not seem to apply in this situation, and the heavy nonlinearity on the conditional laws caused by …
We study a stochastic spatial epidemic model where the $N$ individuals carry two features: a position and an infection state, interact and move in $\R^d$. In this Markovian model, the evolution of the infection states are described with the help of the Poisson Point Processes , whereas the displacement of the individuals are driven by mean field advection, a (state dependence) diffusion and also a common noise, so that the spatial dynamic is a random process. We prove that when the number $N$ of individual goes to infinity, the conditional propagation of chaos holds : conditionnally to the com…
The article is devoted to the integration order replacement technique for iterated Ito stochastic integrals and iterated stochastic integrals with respect to martingales. We consider the class of iterated Ito stochastic integrals, for which with probability 1 the formulas of integration order replacement corresponding to the rules of classical integral calculus are reasonable. The theorems on integration order replacement for the class of iterated Ito stochastic integrals is proven. Many examples of this theorems usage have been considered. These results are generalized for the class of iterat…
The purpose of this paper is to study optimal control of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker-Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate an Hamilton-Jacobi-Bellman (HJB) equation for the optimal control of conditional McKean-Vlasov jump diffusions. Th…
A Conditional Probability Hierarchy for Stochastic Choice
Lingnan College, Laboratory of Mezzoeconomics and Regional Industrial Coordinated Development, Shenzhen Institute of Economics of Lingnan College, Sun Yat-sen University（岭南学院，中观经济与区域产业协调发展实验室，岭南学院深圳经济研究院，中山大学）；Stern School of Business, Tandon School of Engineering, NYU Shanghai, New York University（斯特恩商学院，坦顿工程学院，纽约大学上海分校，纽约大学）
We derive quantitative criteria for the existence of density for stochastic line integrals and iterated line integrals along solutions of hypoelliptic differential equations driven by fractional Brownian motion. As an application, we also study the signature uniqueness problem for these rough differential equations.
In this paper we obtain an Itô differential representation for a class of singular stochastic Volterra integral equations. As an application, we investigate the rate of convergence in the small time central limit theorem for the solution.
This note aims to give an explicit solution for backward stochastic Volterra integral equations with linear time delayed generators. The process $Y$ is expressed by an integral whose kernel is explicitly given. The processes $Z$ is expressed by Hida-Malliavin derivatives involving $Y$.
This paper aims to study a new class of integral equations called backward doubly stochastic Volterra integral equations (BDSVIEs, for short). The notion of symmetrical martingale solutions (SM-solutions, for short) is introduced for BDSVIEs. And the existence and uniqueness theorem for BDSVIEs in the sense of SM-solutions is established.
This article presents a coupling approach for the approximation of iterated stochastic integrals of length three. The generation of such integrals is the central problem of higher-order pathwise approximations for SDEs, which still lacks a satisfactory answer due to the restriction of dimensionality and computational load. Here we start from the Fourier representation of the triple stochastic integral and investigate the global behaviour of the joint density of the representation. Finally in the main result we give a coupling in the quadratic Vaserstein distance.
We present alternative definitions of the stochastic integral introduced by Ayew and Kuo and of the Hitsuda-Skorokhod integral extended to domains in $L^p$-spaces, $p \geq 1$. Our approach is motivated by the S-transform characterization of the Hitsuda-Skorokhod integral and based on simple processes of stochastic exponential type. We prove that the new stochastic integral extends the mentioned stochastic integrals above and we outline their connection.
The generalized mean-square fractional integrals $\mathcal{J}_{ρ,λ,u+;ω}^σ$ and $\mathcal{J}_{ρ,λ,v-;ω}^σ$ of the stochastic process $X$ are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite--Hadamard inequality is establish via generalized stochastic fractional integrals.
We prove an existence and uniqueness result for the obstacle problem for quasilinear stochastic integral-partial differential equations. Our method is based on the probabilistic interpretation of the solution using backward doubly SDEs with jumps.
In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual Itô's framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted $L^2$-processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as …
