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## Weak convergence of filtrations and of -fields

Hoover [68], following remarks by M. Barlow and S. Jacka, introduced the weak convergence of -fields and of filtrations in 1991. The next big step was in 2000 with the seminal paper of Antonelli and Kohatsu-Higa [7]. This was quickly followed by the work of Coquet, Mémin and Mackevicius [29] and by Coquet, Mémin and Slominsky [28]. We will recall fundamental results on the topic but we refer the interested reader to [28] and [29] for details. In these papers, all filtrations are indexed by a compact time interval [0; T] and we work also within a finite time horizon framework and assume that a probability space ( ;H; P) and a positive integer T are given. All filtrations considered are assumed to be completed by the P-null sets of H. By the natural filtration of a process X, we mean the right-continuous filtration associated with the natural filtration of X. The concepts of weak convergence of -fields and of filtrations rely on the topology imposed on the space of càdlàg processes and we use the Skorohod J1 topology as it is done in [28]. An outline of this section is the following. In subsection 1.2.4.1, we recall basic facts on the weak convergence of -fields and establish fundamental lemmas for subsequent use. A crucial lemma is given in subsection 1.2.4.2, and allows to approximate a given G stopping time with a sequence of Gn stopping times n given that the -fields Gn t converge weakly to Gt for each t. The last subsection provides a sufficient condition for the semimartingale property to hold for a given càdlàg adapted process based on the weak convergence of -fields. The sufficient condition we provide at this point is unlikely to hold in a filtration expansion context, however the proof of this result underlines what can go wrong under the more natural assumptions considered in the sections 1.4 and 1.5

### Progressive filtration expansion under initial-type assumptions

In this section 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 we exhibit and use is, on an intuitive level, that these two filtrations 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. We assume that one knows what the decomposition is in some larger filtration that coincide with the one we are interested in after the random time. Usually, one can take for this larger filtration the initially expanded one. That is the reason we mention in the title under initial-type assumptions. 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 filtrations from the point of view of filtration shrinkage. As the main analysis progresses, we indicate how the techniques can be generalized to other types of expansions, mainly to what we called in the previous section the (;X)-progressive filtration expansion and to progressive filtration expansion with counting processes. These results will be essential for the filtration expansion with a process we focus on in the next section.

#### Connection to filtration shrinkage

It has been observed that the optional projection of a local martingaleM onto a filtration to which it is not adapted may lose the local martingale property, see Föllmer and Protter [50]. In that paper, the following general condition was given that guarantees that the local martingale property is preserved (see [50], Theorem 3.7): Lemma 29 Consider two filtrations F G and a G local martingale N. If there exists a sequence of F stopping times that reduce N, then its optional projection onto F is again a local martingale.

Using this result we establish that certain local martingales that arise in the context of filtration expansion in fact retain their local martingale property when projected to various subfiltrations. In particular, if M is an F local martingale and a G-semimartingale, then oMG, the optional projection onto F of the local martingale part in the G semimartingale decomposition of M, always remains a local martingale.

Theorem 21 Consider three filtrations E F G, and let M be an E local martingale. Suppose M is also a G semimartingale with canonical decomposition M = N + A. Then the optional projection of N onto F, when it exists, is again a local martingale. Remark. Note that M remains a special semimartingale in G, given that it remains a semimartingale.

**Table of contents :**

**1 Filtration expansions and semimartingales **

1.1 Introduction

1.2 Mathematical preliminaries

1.2.1 General theory of stochastic processes

1.2.2 Initial filtration expansion with a random variable

1.2.3 Progressive filtration expansions

1.2.4 Weak convergence of filtrations and of -fields

1.3 Progressive filtration expansion under initial-type assumptions

1.3.1 Linking progressive and initial filtration expansion with one random time

1.3.2 The case of multiple non ranked random times

1.3.3 Connection to filtration shrinkage

1.4 Filtration expansion with processes

1.4.1 A semimartingale convergence result and applications to filtration expansion

1.4.2 Results based on Jacod’s type criterion for the increments of the process

1.4.3 Examples: Time reversed diffusions and Kohatsu-Higa’s example

1.5 Toward dynamic models for insider trading

1.5.1 Filtration expansion results based on a Jacod’s criterion for hitting times

1.5.2 Filtration expansion results based on a honest times assumption

1.5.3 Insider models with arbitrage: Jeulin’s example and extensions

**2 Compensators of random times and credit contagion **

2.1 Introduction

2.2 Information induced credit contagion in structural models

2.2.1 The base model for a single time

2.2.2 Multiple firms : A conditional independence model

2.2.3 A first structural model with credit contagion effect

2.2.4 A structural credit contagion model in finite time horizon

2.2.5 Structural models with random default barriers

2.3 Credit contagion under the conditional density assumption

2.3.1 The conditional density assumption : the case of two non ranked random times

2.3.2 Extension to multiple non ranked random times

2.3.3 Modeling of conditional densities

2.3.4 An application to risk management

2.3.5 The conditional density assumption and credit contagion

2.3.6 A toy structural model and credit contagion

**3 Bubbles: martingale theory and real time detection **

3.1 Introduction

3.2 The martingale theory of asset bubbles

3.3 Mathematical preliminaries

3.3.1 Strict local martingales

3.3.2 Estimation of the volatility function in diffusion models

3.4 How to detect an asset bubble in real time

3.4.1 The methodology

3.4.2 Method 1: Parametric Estimation

3.4.3 Method 2: RKHS theory

3.4.4 The dotcom bubble

3.5 Is there a bubble in LinkedIn’s stock price? A real case study

3.5.1 Real time detection : LinkedIn’s case

3.5.2 « Is there a bubble in LinkedIn’s stock price? » in the news

**4 Discretely and continuously sampled variance swaps **

4.1 Intoduction

4.2 Framework and mathematical preliminaries

4.2.1 Variance swaps

4.2.2 Mathematical preliminaries

4.3 Approximation using the quadratic variation

4.3.1 Finiteness of expectations

4.3.2 Bounds on the approximation error

4.4 Examples

4.4.1 Strict local martingales

4.4.2 Stochastic volatility of volatility

4.4.3 Time changed geometric Brownian motion

4.4.4 The 3=2-stochastic volatility model

**Bibliography**