Predictable Representation Property Research Articles

A Note on the Strong Predictable Representation Property and Enlargement of Filtration

The strong predictable representation property of semi-martingales and the notion of enlargement of filtration meet naturally in modeling financial markets, and theoretical problems arise. Here, first, we illustrate some of them through classical examples. Then, we review recent results obtained by studying predictable martingale representations for filtrations enlarged by means of a full process, possibly with accessible components in its jump times. The emphasis is on the non-uniqueness of the martingale enjoying the strong predictable representation property with respect to the same enlarged filtration.

Shadow martingales – a stochastic mass transport approach to the peacock problem

Given a family of real probability measures $(\mu_t)_{t\geq 0}$ increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in \cite{BeJu16,NuStTa17}. As input data we take an increasing family of measures $(\nu^\alpha)_{\alpha \in [0,1]}$ with $\nu^\alpha(\mathbb{R})=\alpha$ that are submeasures of $\mu _0$, called a parametrization of $\mu_0$. Then, for any $\alpha$ we define an evolution $(\eta^\alpha_t)_{t\geq 0}$ of the measure $\nu^\alpha=\eta^\alpha_0$ across our peacock by setting $\eta^\alpha_t$ equal to the obstructed shadow of $\nu^\alpha$ in $(\mu _s)_{s \in [0,t]}$. We identify conditions on the parametrization $(\nu^\alpha)_{\alpha\in [0,1]}$ such that this construction leads to a unique martingale measure $\pi$, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization $(\nu_{\text{lc}}^\alpha)_{\alpha \in [0,1]}$ we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem. Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.

Statistical causality and purely discontinuous local martingales

ABSTRACT The statistical concept of causality in continuous time between filtered probability spaces considered in this paper is based on Granger's definition of causality. The given concept of causality can be connected to the purely discontinuous property for martingale and filtration. If M is a purely discontinuous local martingale with respect to purely discontinuous filtration , we prove that M will remain purely discontinuous local martingale with respect to an extension of the filtration if and only if is its own cause within . Moreover, we give conditions for a natural filtration of the purely discontinuous martingale to be a purely discontinuous filtration, using the property of causality. We also, consider the connection between the concept of statistical causality and the weak predictable representation property for purely discontinuous local martingales.

Stability results for martingale representations: The general case

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 to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature and has applications to stability results for backward stochastic differential equations with jumps, and to discretization schemes for stochastic systems.

On the support of extremal martingale measures with given marginals: the countable case

AbstractWe investigate the supports of extremal martingale measures with prespecified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure Q and the denseness in $L^1(Q)$ of a suitable linear subspace, which can be seen in a financial context as the set of all semistatic trading strategies. Moreover, when the supports of both marginals are countable, we focus on the slightly stronger notion of weak exact predictable representation property (WEP) and provide two combinatorial sufficient conditions, called the ‘2-link property’ and ‘full erasability’, on how the points in the supports are linked to each other for granting extremality. When the support of the first marginal is a finite set, we give a necessary and sufficient condition for the WEP to hold in terms of the new concepts of 2-net and deadlock. Finally, we study the relation between cycles and extremality.

Predictable representation for time inhomogeneous Lévy processes and BSDEs

The aim of this paper is to give the predictable representation property associated with Levy process in the non-homogeneous case. In this latter, we establish the existence and uniqueness of solution for the Backward Stochastic Differential Equations and its relation with Partial integro-differential equations.

Martingale representations in progressive enlargement by the reference filtration of a semi-martingale: a note on the multidimensional case

ABSTRACTLet and be an m-dimensional -semi-martingale and an n-dimensional -semi-martingale, respectively, on the same probability space , both enjoying the predictable representation property. We propose two representation results for the square-integrable-martingales, where . As a first application, we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of , where, fixed , is the reference filtration of a martingale , which enjoys the -predictable representation property. This result helps us to identify a basis of martingales for the Poisson filtration enlarged by a general random time. A second application falls into the framework of credit risk modelling and in particular into the study of progressive enlargement of the market filtration by a default time. We present a new proof of the analogous of classical Kusuoka's theorem, when the risky asset price is a multidimensional semi-martingale enjoying the predictable representation property and the default time satisfies the density hypothesis.

Integral representations of martingales for progressive enlargements of filtrations

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

The strong predictable representation property in initially enlarged filtrations under the density hypothesis

We study the strong predictable representation property in filtrations initially enlarged with a random variable L. We prove that the strong predictable representation property can always be transferred to the enlarged filtration as long as the classical density hypothesis of Jacod (1985) holds. This generalizes the existing martingale representation results and does not rely on the equivalence between the conditional and the unconditional laws of L. Depending on the behavior of the density process at zero, different forms of martingale representation are established. The results are illustrated in the context of hedging contingent claims under insider information.

The Predictable Representation Property of Compensated-Covariation Stable Families of Martingales

We study the predictable representation property for certain families of square integrable martingales, which we call compensated-covariation stable families. The definition of the predictable representation property is given by means of stable subspaces. The main result is that compensated-covariation stable families of martingales which satisfy some further conditions possess the predictable representation property. As first examples, we consider continuous Gaussian families of martingales and independent families of compensated Poisson processes. Then we apply the result to the case of Lévy processes. We shall construct families of martingales relative to a Lévy filtration which possess the predictable representation property. We give several examples including Teugels martingales.

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 semi-martingale X does not transfer from the reference filtration to a larger filtration if the information starts growing up to a positive time. Then we study the case when there exists a second special semi-martingale Y enjoying the predictable representation property with respect to . We establish conditions under which the triplet (X, Y, [X, Y]) enjoys the predictable representation property with respect to .

The predictable representation property of compensated-covariation stable families of martingales

Изучается свойство предсказуемого представления некоторых семейств квадратично интегрируемых мартингалов, которые мы называем устойчивыми семействами с компенсированными ковариациями. Определение свойства предсказуемого представления дается с помощью устойчивых подпространств. Основной результат состоит в том, что устойчивые семейства с компенсированными ковариациями, удовлетворяющие некоторым дополнительным условиям, обладают свойством предсказуемого представления. В качестве первых примеров мы рассматриваем непрерывные гауссовские семейства мартингалов и независимые семейства компенсированных пуассоновских процессов. Затем мы применяем наш результат к процессам Леви. Мы строим семейства мартингалов относительно фильтрации Леви, обладающие свойством предсказуемого представления. Мы рассматриваем несколько примеров, в том числе мартингалы Тойгельса.

On the predictable representation property of martingales associated with Lévy processes

We investigate the predictable representation property (PRP) in the frame of Lévy processes. To give a general definition of the PRP, we make use of the theory of stable subspaces. Let L be a Lévy process with Lévy measure . The main result is that any total system in leads to a family of martingales with the PRP.

Chaos representations for marked point processes

We show that for a large class of marked point processes there exists a random measure m with the predictable representation property such that iterated integrals with respect to m span the space of square integrable random variables.

A Note On Emissions Trading: The Pricing Of Carbon Derivatives

Due to extreme volatility in the prices of CO2 permits observed in the European markets, effective hedging techniques such as futures contracts and option instruments rapidly developed, together with approximations to the dynamics of the underlying. To bridge the gap between theory and practice, we are going to devise a market model consistent with the features exhibited by equilibrium models. By definition, the prices of CO2 permits are bounded and so should be their dynamics, so that the spot prices do not have the predictable representation property. Rather than working directly with a bounded process, we model the carbon permits prices as a function of a positive unbounded process, and show that there is no equivalent probability measure such that the discounted spot price is a martingale. Consequently, investors in the emission allowances markets can generate arbitrage opportunities, which explains all the contradictory results recently obtained by various authors to justify the observed discrepancies between the futures and spot prices. Relying on the actuarial pricing approach of marked-to-model, we select freely two particular growth rates for the CO2 spot price process, compute their corresponding dynamics, and rewrite the contingent claims on the carbon spot prices in terms of the unbounded process. As a result, one can define the model of his choice for the unbounded process and compute in a simple way the claims on emission allowances.

On filtration enlargements and purely discontinuous martingales

Let M be a purely discontinuous martingale relative to a filtration ( F t ) . Given an arbitrary extension ( G t ) of the filtration ( F t ) , we will provide sufficient conditions for M to be a semimartingale relative to ( G t ) . Moreover we describe methods of how to find the Doob–Meyer decomposition with respect to the enlarged filtration. To this end we prove a new and more explicit version of the predictable representation property of Poisson random measures. Finally some concrete examples will show how the method developed may be applied.

From Dynamic Linear Evaluation Rule to Dynamic Capital Asset Pricing Model

Abstract In this article, we first present a dynamic linear evaluation mechanism of contingent claims. Then, we deduce the dynamic capital asset pricing model from this dynamic linear evaluation rule in the market driven by Levy processes. We mainly use the predictable representation property in weak form and Girsanov theorem of Levy processes to obtain the results.

Quantum stopping times and quasi-left continuity

We study the general properties of quantum stopping times on Hilbert spaces equipped with a filtration. We define and investigate notions such as the spaces of anterior events, the spaces of strictly anterior events and above all we define the property S< T for two stopping times together with the notion of predictable quantum stopping times. It is well-known that the natural filtration of any normal martingale with the predictable representation property is quasi-left continuous; with the help of our new notions we prove that this property is actually an intrinsic property of the symmetric Fock space Φ over L 2( R +) . We also apply these definitions to the case of a non commutative stochastic base. We show, in this context, that the fermionic Fock space over L 2( R +) , the quasi-free boson and fermion spaces are also quasi-left continuous.

On the robustness of backward stochastic differential equations

In this paper, we study the robustness of backward stochastic differential equations (BSDEs for short) w.r.t. the Brownian motion; more precisely, we will show that if W n is a martingale approximation of a Brownian motion W then the solution to the BSDE driven by the martingale W n converges to the solution of the classical BSDE, namely the BSDE driven by W. The particular case of the scaled random walks has been studied in Briand et al. (Electron. Comm. Probab. 6 (2001) 1). Here, we deal with a more general situation and we will not assume that the W n has the predictable representation property: this yields an orthogonal martingale in the BSDE driven by W n . As a byproduct of our result, we obtain the convergence of the “Euler scheme” for BSDEs corresponding to the case where W n is a time discretization of W.

The Relations between Minimal Martingale Measure and Minimal Entropy Martingale Measure

We consider incomplete markets, where each risky asset fluctuation is a continuous semimartingale, and study a subset of Equivalent Local Martingale Measures in which Minimal Martingale Measure minimizes relative entropy.We also discuss, as special cases, some models with the risky assetfluctuation represented as a solution of some stochastic differential equations.Finally, we mention that the predictable representation property is essentialin order that Minimal Martingale Measure coincides with Minimal Entropy Martingale Measure.