Conditioning and initial enlargement of filtration on a Riemannian manifold

Strict Local Martingales via Filtration Enlargement

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration. We study and implement a particular type of enlargement, initial expansion of filtration, for various stochastic differential equations and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale under an equivalent probability measure. Such situations arise in the theory of Mathematical Finance.

Local solution method for the problem of enlargement of filtration

The enlargement of filtration theory is a study of semimartingales when the basic filtration changes. This theory provides particular techniques on stochastic calculus. We present here a technique, that we call the local solution method. We will show, with several examples, that the local solution method is an effective and flexible method. In particular, with the local solution method, we will give a unified proof of three of the classical formulas, namely Jacod's formula, the progressive enlargement formula and the enlargement formula with honest time. We also show that, besides its generali…

Progressive Enlargements of Filtrations with Pseudo-Honest Times and their Applications in Financial Mathematics

We deal with various alternative decompositions of F-martingales with respect to the filtration G which represents the enlargement of a filtration F by a progressive flow of observations of a random time that either belongs to the class of pseudo-honest times or satisfies the extended density hypothesis. Several related results from the existing literature are essentially extended. Results on G-semimartingale decompositions of F-local martingales are crucial for applications in financial mathematics, most notably in the context of modeling credit risk and the study of insider trading where the…

Enlargement of filtration and predictable representation property for semi-martingales

We present two examples of loss of the predictable representation property for semi-martingales by enlargement of the reference filtration. First of all we show that the predictable representation property for a square-integrable semi-martingale X does not transfer from the reference filtration F to a larger filtration G when the information starts growing up to a positive time. Then we study the case when F coincides with the natural filtration of X and G is obtained by adding the natural filtration of a second square-integrable semi-martingale, Y. We establish conditions under which the trip…

Doob's maximal identity, multiplicative decompositions and enlargements of filtrations

In the theory of progressive enlargements of filtrations, the supermartingale $Z_{t}=\mathbf{P}(g>t\mid \mathcal{F}_{t}) $ associated with an honest time g, and its additive (Doob-Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale Z_{t}, based on Doob's maximal identity. We thus give new examples of progressive enlargements. Moreover, we give, in our setting, a proof of the decomposition formula for martingales, using initial enlargement techniques, and use it to obtain some path decompo…

Enlargements of filtrations and path decompositions at non-stopping times

Azéma associated with an honest time L the supermartingale $Z_{t}^{L}=\mathbb{P}[L>t|\mathcal{F}_{t}]$ and established some of its important properties. This supermartingale plays a central role in the general theory of stochastic processes and in particular in the theory of progressive enlargements of filtrations. In this paper, we shall give an additive characterization for these supermartingales, which in turn will naturally provide many examples of enlargements of filtrations. In particular, we use this characterization to establish some path decomposition results, closely related to or re…

Integral representations of martingales for progressive enlargements of filtrations

We work in the setting of the progressive enlargement $\mathbb G$ of a reference filtration $\mathbb F$ through the observation of a random time $τ$. We study an integral representation property for some classes of $\mathbb G$-martingales stopped at $τ$. In the first part, we focus on the case where $\mathbb F$ is a Poisson filtration and we establish a predictable representation property with respect to three $\mathbb G$-martingales. In the second part, we relax the assumption that $\mathbb F$ is a Poisson filtration and we assume that $τ$ is an $\mathbb F$-pseudo-stopping time. We establish …

Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which are determined in a finite backward induction. Ou…

Progressively Enlargement of Filtrations and Control Problems for Step Processes

In the present paper we address stochastic optimal control problems for a step process $(X,\mathbb{F})$ under a progressive enlargement of the filtration. The global information is obtained adding to the reference filtration $\mathbb{F}$ the point process $H=1_{[τ,+\infty)}$. Here $τ$ is a random time that can be regarded as the occurrence time of an external shock event. We study two classes of control problems, over $[0,T]$ and over the random horizon $[0,T \wedge τ]$. We solve these control problems following a dynamical approach based on a class of BSDEs driven by the jump measure $μ^ Z$ o…

Martingale representation in progressive enlargement by the reference filtration of a semimartingale: a note on the multidimensional case

Let X and Y be an m-dimensional F-semimartingale and an n-dimensional H-semimartingale respectively on the same probability space, both enjoying the strong predictable representation property. We propose a martingale representation result under the probability measure P for the square integrable G-martingales, where G is the union of F and H. As a first application we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of the union of F1,..., Fd, where, fixed i in (1,...,d), Fi is the reference filtration of a real martingale Mi, which enjoys the Fi-predic…

Arbitrage of the first kind and filtration enlargements in semimartingale financial models

In a general semimartingale financial model, we study the stability of the No Arbitrage of the First Kind (NA1) (or, equivalently, No Unbounded Profit with Bounded Risk) condition under initial and under progressive filtration enlargements. In both cases, we provide a simple and general condition which is sufficient to ensure this stability for any fixed semimartingale model. Furthermore, we give a characterisation of the NA1 stability for all semimartingale models.

布朗滤波初始放大下的倒向随机微分方程与反射倒向随机微分方程

Backward and reflected backward stochastic differential equations under initial enlargement of Brownian filtrations

该研究探讨布朗滤波初始放大对BSDEs和RBSDEs的影响，建立相关表示性质与恒等式，通过反例说明奇异方程解的唯一性问题，发展正则BSDE的条件参数化并讨论扩展方向。

Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization

The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are als…

Filtrations

In this article, we define the notion of a filtration and then give the basic theorems on initial and progressive enlargements of filtrations.

Progressive enlargement of filtrations and Backward SDEs with jumps

This work deals with backward stochastic differential equation (BSDE) with random marked jumps, and their applications to default risk. We show that these BSDEs are linked with Brownian BSDEs through the decomposition of processes with respect to the progressive enlargement of filtrations. We show that the equations have solutions if the associated Brownian BSDEs have solutions. We also provide a uniqueness theorem for BSDEs with jumps by giving a comparison theorem based on the comparison for Brownian BSDEs. We give in particular some results for quadratic BDSEs. As applications, we study the…

Enlargement of Filtrations: An Exposition of Core Ideas with Financial Examples

In this paper we provide an exhaustive survey of the current state of the mathematics of filtration enlargement and an interpretation of the key results of the literature from the viewpoint of mathematical finance. The emphasis is on providing a well-structured compendium of known mathematical results that can be used by researchers in mathematical finance. We mainly state the results and discuss their role and significance, with references provided for the omitted proofs. The discussion of mathematical results is accompanied by numerous examples from mathematical finance.

Initial Enlargement in a Markov chain market model

Enlargement of filtrations is a classical topic in the general theory of stochastic processes. This theory has been applied to stochastic finance in order to analyze models with insider information. In this paper we study initial enlargement in a Markov chain market model, introduced by R. Norberg. In the enlargened filtration several things can happen: some of the jumps times can be accessible or predictable, but in the orginal filtration all the jumps times are totally inaccessible. But even if the jumps times change to accessible or predictable, the insider does not necessarily have arbitra…

渐进扩大滤流上鞅表示的薄-厚方法

Thin-thick approach to martingale representations on progressively enlarged filtrations

该研究针对随机时间渐进扩大参考滤流的可料表示性问题，通过将随机时间分解为与F停时重叠的薄部分和避开F停时的厚部分，证明了F^τ上的鞅表示定理，拓展了F-避开随机时间的相关结果，并给出了Lévy过程滤流扩大的应用示例。

Martingale representation property in progressively enlarged filtrations

Consider $\mathbb{G}$ the progressive enlargement of a filtration $\mathbb{F}$ with a random time $τ$. Assuming that, in $\mathbb{F}$, the martingale representation property holds, we examine conditions under which the martingale representation property holds also in $\mathbb{G}$. A general methodology is developed in this paper, with results covering every known (classical or recent) examples.

Risk measures and progressive enlargement of filtration: a BSDE approach

We consider dynamic risk measures induced by Backward Stochastic Differential Equations (BSDEs) in enlargement of filtration setting. On a fixed probability space, we are given a standard Brownian motion and a pair of random variables $(τ, ζ) \in (0,+\infty) \times E$, with $E \subset \mathbb{R}^m$, that enlarge the reference filtration, i.e., the one generated by the Brownian motion. These random variables can be interpreted financially as a default time and an associated mark. After introducing a BSDE driven by the Brownian motion and the random measure associated to $(τ, ζ)$, we define the …

Harnesses, Levy bridges and Monsieur Jourdain

Relations between so-called harness processes and initial enlargements of the filtration of a Levy process with its positions at fixed times are investigated.

Stochastic Integrals and Two Filtrations

In the definition of the stochastic integral, apart from the integrand and the integrator, there is an underlying filtration that plays a role. Thus, it is natural to ask: {\it Does the stochastic integral depend upon the filtration?} In other words, if we have two filtrations, $({\mathcal F}_\centerdot)$ and $({\mathcal G}_\centerdot)$, a process $X$ that is semimartingale under both the filtrations and a process $f$ that is predictable for both the filtrations, then are the two stochastic integrals - $Y=\int f\,dX$, with filtration $({\mathcal F}_\centerdot)$ and $Z=\int f\,dX$, with filtrat…

随机Lipschitz系数下带违约时间的双重反射倒向随机微分方程：滤链与广义Dynkin博弈

Doubly reflected BSDEs with default time under stochastic Lipschitz coefficients: Filtration links and generalized Dynkin games

该研究在渐进扩大滤子下，针对带违约时间的随机Lipschitz系数DRBSDEs，建立解的存在唯一性，探讨其与参考滤子中方程的关系，并刻画了相关非线性Dynkin博弈的价值过程与鞍点。

熵与时间递增披露的离散信息的附加效用

Entropy and additional utility of a discrete information disclosed progressively in time

本文研究了时间递增披露信息的熵及其效用，分析了其在渐进设定下的martingale空间稳定性，并计算了离散信息的附加对数效用。

内幕交易下财富最大化的 HJB 方程

HJB equation for maximization of wealth under insider trading

本文结合信息流扩张与随机控制，推广验证定理并利用前向积分刻画 G-适应最优控制与 Brown 运动半鞅性的等价关系，用于分析内幕交易市场的财富最大化问题。

Optimal investment under multiple defaults risk: A BSDE-decomposition approach

We study an optimal investment problem under contagion risk in a financial model subject to multiple jumps and defaults. The global market information is formulated as a progressive enlargement of a default-free Brownian filtration, and the dependence of default times is modeled by a conditional density hypothesis. In this Ito-jump process model, we give a decomposition of the corresponding stochastic control problem into stochastic control problems in the default-free filtration, which are determined in a backward induction. The dynamic programming method leads to a backward recursive system …

粗糙随机微分方程系统 I：弱存在性与 Yamada--Watanabe 定理

Systems of rough stochastic differential equations I: weak existence and Yamada--Watanabe

本文证明粗糙随机微分方程系统在无界系数下的弱存在性，并建立 Yamada--Watanabe 定理：弱存在性与路径唯一性蕴含强解及联合分布唯一性。

Reflected backward stochastic differential equations under stopping with an arbitrary random time

This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of \begin{eqnarray*} \begin{cases} dY_t=f(t,Y_t, Z_t)d(t\wedgeτ)+Z_tdW_t^τ+dM_t-dK_t,\quad Y_τ=ξ, Y\geq S\quad\mbox{on}\quad \Lbrack0,τ\Lbrack,\quad \displaystyle\int_0^τ(Y_{s-}-S_{s-})dK_s=0\quad P\mbox{-a.s..}\end{cases} \end{eqnarray*} Here $τ$ is an arbitrary random time that might not be a stopping time for the filtration $\mathbb F$ generated by the Brownian motion $W$. We consider the filtration $\mathbb G$ resulting from the progressive enlargement of $\mathbb F$ with $τ$ wher…

On the spatial dynamics of the solution to the stochastic heat equation

We consider the solution of $\partial_t u=\partial_x^2 u+\partial_x\partial_t B,\,(x,t)\in R\times(0,\infty)$, subject to $u(x,0)=0,\,x\in R$, where $B$ is a Brownian sheet. We show that $u$ also satisfies $\partial_x^2 u +[\,(-\partial_t^2)^{1/2}+\sqrt{2}\partial_x(-\partial_t^2)^{1/4}\,]\,u^a= \partial_x\partial_t{\tilde B}$ in $R\times(0,\infty)$ where $u^a$ stands for the extension of $u(x,t)$ to $(x,t)\in R^2$ which is antisymmetric in $t$ and $\tilde{B}$ is another Brownian sheet. The new SPDE allows us to prove the strong Markov property of the pair $(u,\partial_x u)$ when seen as a pro…

关于随机偏微分方程及其通过条件Feynman-Kac公式在衍生品定价中的应用

On Stochastic Partial Differential Equations and their applications to Derivative Pricing through a conditional Feynman-Kac formula

本文通过条件Feynman-Kac公式将金融衍生品价格表示为迭代条件期望，证明内层条件期望满足一个随机偏微分方程，并引入新的数值方法。

条件性随机稳定性与均匀超几何集上的平衡态

Conditioned stochastic stability of equilibrium states on uniformly hyperbolic sets

该研究通过条件性随机稳定性分析，揭示了均匀超几何集上平衡态在噪声趋于零时的收敛性质，并扩展至Axiom A微分同胚的瞬态混沌场景。