We consider the so-called $\natural$-model. It is an one-default model which gives the conditional law of a random time with respect to a reference filtration. This model has been studied in the case where the parameters are continuous. In this paper we will establish the $\natural$-model in the case of jump parameters. We then prove the corresponding enlargement of filtration formula and we compute the derivative of the conditional distribution functions of the random time.
In this paper, we study backward stochastic Volterra integral equations introduced in [26, 45] and extend the existence, uniqueness or comparison results for general filtration as in [31] (not only Brownian-Poisson setting). We also consider Lp-data and explore the time regularity of the solution in the It{ô} setting, which is also new in this jump setting.
In [8] we established existence and uniqueness of solutions of backward stochastic differential equations in L^p under a monotonicity condition on the generator and in a general filtration. There was a mistake in the case 1 \textless{} p \textless{} 2. Here we give a corrected proof. Moreover the quasi-left continuity condition on the filtration is removed.
In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of $\mathbb{L}^p(p\ge2)$-solutions in the case of a fixed terminal time under suitable $p$-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension $1$.
$\mathbb{L}^p$-solutions for Reflected BSDEs with jumps in a general filtration under stochastic Lipschitz coefficients
National Institute of Statistics and Applied Economics (INSEA)（国立统计与应用经济研究所）；Cadi Ayyad University (UCA), National School of Applied Sciences of Marrakech (ENSA-M)（卡迪·阿亚德大学，马拉喀什国立应用科学学院）
A single jump filtration $({\mathscr{F}}_t)_{t\in \mathbb{R}_+}$ generated by a random variable $γ$ with values in $\overline{\mathbb{R}}_+$ on a probability space $(Ω,{\mathscr{F}},\mathsf{P})$ is defined as follows: a set $A\in {\mathscr{F}}$ belongs to ${\mathscr{F}}_t$ if $A\cap \{γ>t\}$ is either $\varnothing$ or $\{γ>t\}$. A process $M$ is proved to be a local martingale with respect to this filtration if and only if it has a representation $M_t=F(t){\mathbb{1}}_{\{t<γ\}}+L{\mathbb{1}}_{\{t\geqslant γ\}}$, where $F$ is a deterministic function and $L$ is a random variable such that $\mat…
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…
In this paper we investigate the propagation of the weak representation property (WRP) to an independently enlarged filtration. More precisely, we consider an $\mathbb{F}$-semimartingale $X$ possessing the WRP with respect to $\mathbb{F}$ and an $\mathbb{H}$-semimartingale $Y$ possessing the WRP with respect to $\mathbb{H}$. Assuming that $\mathbb{F}$ and $\mathbb{H}$ are independent, we show that the $\mathbb{G}$-semimartingale $Z=(X,Y)$ has the WRP with respect to $\mathbb{G}$, where $\mathbb{G}:=\mathbb{F}\vee\mathbb{H}$. In our setting, $X$ and $Y$ may have simultaneous jump-times. Further…
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…
We consider controller-stopper problems in which the controlled processes can have jumps. The global filtration is represented by the Brownian filtration, enlarged by the filtration generated by the jump process. We assume that there exists a conditional probability density function for the jump times and marks given the filtration of the Brownian motion and decompose the global controller-stopper problem into controller-stopper problems with respect to the Brownian filtration, which are determined by a backward induction. We apply our decomposition method to indifference pricing of American o…
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…
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 …
This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.
Let $(X_t)_{t\ge0}$ be a continuous-time, time-homogeneous strong Markov process with possible jumps and let $τ$ be its first hitting time of a Borel subset of the state space. Suppose $X$ is sampled at random times and suppose also that $X$ has not hit the Borel set by time $t$. What is the intensity process of $τ$ based on this information? This question from credit risk encompasses basic mathematical problems concerning the existence of an intensity process and filtration expansions, as well as some conceptual issues for credit risk. By revisiting and extending the famous Jeulin--Yor [Lectu…
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…
We consider filtration consistent nonlinear expectations in probability spaces satisfying only the usual conditions and separability. Under a domination assumption, we demonstrate that these nonlinear expectations can be expressed as the solutions to Backward Stochastic Differential Equations with Lipschitz continuous drivers, where both the martingale and the driver terms are permitted to jump, and the martingale representation is infinite dimensional. To establish this result, we show that this domination condition is sufficient to guarantee that the comparison theorem for BSDEs will hold, a…
We study generalized backward stochastic differential equations (BSDEs) up to a random time horizon $\vartheta$, which is not a stopping time, under minimal assumptions regarding the properties of $\vartheta$. In contrast to existing works in this area, we do not impose specific assumptions on the random time $\vartheta$ and we study the existence of solutions to BSDEs and reflected BSDEs with a random time horizon through the method of reduction. In addition, we also examine BSDEs and reflected BSDEs with a làdlàg driver where the driver is allowed to have a finite number of common jumps with…
In this paper we study progressive filtration expansions with random times. We show how semimartingale decompositions in the expanded filtration can be obtained using a natural link between progressive and initial expansions. The link is, on an intuitive level, that the two coincide after the random time. We make this idea precise and use it to establish known and new results in the case of expansion with a single random time. The methods are then extended to the multiple time case, without any restrictions on the ordering of the individual times. Finally we study the link between the expanded…
This paper answers a question of Émery [In Séminaire de Probabilités XLII (2009) 383-396 Springer] by constructing an explicit coupling of two copies of the Beneš et al. [In Applied Stochastic Analysis (1991) 121-156 Gordon & Breach] diffusion (BKR diffusion), neither of which starts at the origin, and whose natural filtrations agree. The paper commences by surveying probabilistic coupling, introducing the formal definition of an immersed coupling (the natural filtration of each component is immersed in a common underlying filtration; such couplings have been described as co-adapted or Markovi…
In this paper our first goal is to give precise definition of the Lévy bridges with random length. Our second task is to establish the Markov property of this process with respect to its completed natural filtration and thus with respect to the usual augmentation of this one. This property will be crucial for the right-continuity of completed natural filtration.
The aim objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. Among them the Markov property as well as the canonical decomposition with respect to the usual augmentation of its natural filtration, which leads us to conclude that its completed natural filtration is right continuous.
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…
In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi--left--continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges …
Motivated by the Brownian bridge on random interval considered by Bedini et al \cite{BBE}, we introduce and study Gaussian bridges with random length with special emphasis to the Markov property. We prove that if the starting process is Markov then this property was kept by the bridge with respect to the usual augmentation of its natural filtration. This leads us to conclude that the completed natural filtration of the bridge satisfies the usual conditions of right-continuity and completeness.
We study the standard property of the natural filtration associated to a 0--1 valued stationary process. In our main result we show that if the process has summable memory decay, then the associated filtration is standard. We prove it by coupling techniques. For a process whose associated filtration is standard, we construct a product type filtration extending it, based upon the usual couplings and the Vershik's criterion for standardness.
Motivated by liquidity risk in mathematical finance, D. Lacker introduced concentration inequalities for risk measures, i.e. upper bounds on the \emph{liquidity risk profile} of a financial loss. We derive these inequalities in the case of time-consistent dynamic risk measures when the filtration is assumed to carry a Brownian motion. The theory of backward stochastic differential equations (BSDEs) and their dual formulation plays a crucial role in our analysis. Natural by-products of concentration of risk measures are a description of the tail behavior of the financial loss and transport-type…
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…
