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Integral Representations of Martingales for Progressive Enlargements Of
INTEGRAL REPRESENTATIONS OF MARTINGALES FOR PROGRESSIVE ENLARGEMENTS OF FILTRATIONS ANNA AKSAMIT, MONIQUE JEANBLANC AND MAREK RUTKOWSKI Abstract. 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. Keywords: predictable representation property, Poisson process, random time, progressive enlargement, pseudo-stopping time Mathematics Subjects Classification (2010): 60H99 1. Introduction We are interested in the stability of the predictable representation property in a filtration en- largement setting: under the postulate that the predictable representation property holds for F and G is an enlargement of F, the goal is to find under which conditions the predictable rep- G G resentation property holds for . We focus on the progressive enlargement = (Gt)t∈R+ of a F reference filtration = (Ft)t∈R+ through the observation of the occurrence of a random time τ and we assume that the hypothesis (H′) is satisfied, i.e., any F-martingale is a G-semimartingale. It is worth noting that no general result on the existence of a G-semimartingale decomposition after τ of an F-martingale is available in the existing literature, while, for any τ, any F-martingale stopped at time τ is a G-semimartingale with an explicit decomposition depending on the Az´ema supermartingale of τ. -
Stochastic Differential Equations
Stochastic Differential Equations Stefan Geiss November 25, 2020 2 Contents 1 Introduction 5 1.0.1 Wolfgang D¨oblin . .6 1.0.2 Kiyoshi It^o . .7 2 Stochastic processes in continuous time 9 2.1 Some definitions . .9 2.2 Two basic examples of stochastic processes . 15 2.3 Gaussian processes . 17 2.4 Brownian motion . 31 2.5 Stopping and optional times . 36 2.6 A short excursion to Markov processes . 41 3 Stochastic integration 43 3.1 Definition of the stochastic integral . 44 3.2 It^o'sformula . 64 3.3 Proof of Ito^'s formula in a simple case . 76 3.4 For extended reading . 80 3.4.1 Local time . 80 3.4.2 Three-dimensional Brownian motion is transient . 83 4 Stochastic differential equations 87 4.1 What is a stochastic differential equation? . 87 4.2 Strong Uniqueness of SDE's . 90 4.3 Existence of strong solutions of SDE's . 94 4.4 Theorems of L´evyand Girsanov . 98 4.5 Solutions of SDE's by a transformation of drift . 103 4.6 Weak solutions . 105 4.7 The Cox-Ingersoll-Ross SDE . 108 3 4 CONTENTS 4.8 The martingale representation theorem . 116 5 BSDEs 121 5.1 Introduction . 121 5.2 Setting . 122 5.3 A priori estimate . 123 Chapter 1 Introduction One goal of the lecture is to study stochastic differential equations (SDE's). So let us start with a (hopefully) motivating example: Assume that Xt is the share price of a company at time t ≥ 0 where we assume without loss of generality that X0 := 1. -
PREDICTABLE REPRESENTATION PROPERTY for PROGRESSIVE ENLARGEMENTS of a POISSON FILTRATION Anna Aksamit, Monique Jeanblanc, Marek Rutkowski
PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION Anna Aksamit, Monique Jeanblanc, Marek Rutkowski To cite this version: Anna Aksamit, Monique Jeanblanc, Marek Rutkowski. PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION. 2015. hal- 01249662 HAL Id: hal-01249662 https://hal.archives-ouvertes.fr/hal-01249662 Preprint submitted on 2 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION Anna Aksamit Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom Monique Jeanblanc∗ Laboratoire de Math´ematiques et Mod´elisation d’Evry´ (LaMME), Universit´ed’Evry-Val-d’Essonne,´ UMR CNRS 8071 91025 Evry´ Cedex, France Marek Rutkowski School of Mathematics and Statistics University of Sydney Sydney, NSW 2006, Australia 10 December 2015 Abstract We study problems related to the predictable representation property for a progressive en- largement G of a reference filtration F through observation of a finite random time τ. We focus on cases where the avoidance property and/or the continuity property for F-martingales do not hold and the reference filtration is generated by a Poisson process. -
Lecture 18 : Itō Calculus
Lecture 18 : Itō Calculus 1. Ito's calculus In the previous lecture, we have observed that a sample Brownian path is nowhere differentiable with probability 1. In other words, the differentiation dBt dt does not exist. However, while studying Brownain motions, or when using Brownian motion as a model, the situation of estimating the difference of a function of the type f(Bt) over an infinitesimal time difference occurs quite frequently (suppose that f is a smooth function). To be more precise, we are considering a function f(t; Bt) which depends only on the second variable. Hence there exists an implicit dependence on time since the Brownian motion depends on time. dBt If the differentiation dt existed, then we can easily do this by using chain rule: dB df = t · f 0(B ) dt: dt t We already know that the formula above makes no sense. One possible way to work around this problem is to try to describe the difference df in terms of the difference dBt. In this case, the equation above becomes 0 (1.1) df = f (Bt)dBt: Our new formula at least makes sense, since there is no need to refer to the dBt differentiation dt which does not exist. The only problem is that it does not quite work. Consider the Taylor expansion of f to obtain (∆x)2 (∆x)3 f(x + ∆x) − f(x) = (∆x) · f 0(x) + f 00(x) + f 000(x) + ··· : 2 6 To deduce Equation (1.1) from this formula, we must be able to say that the significant term is the first term (∆x) · f 0(x) and all other terms are of smaller order of magnitude. -
The Ito Integral
Notes on the Itoˆ Calculus Steven P. Lalley May 15, 2012 1 Continuous-Time Processes: Progressive Measurability 1.1 Progressive Measurability Measurability issues are a bit like plumbing – a bit ugly, and most of the time you would prefer that it remains hidden from view, but sometimes it is unavoidable that you actually have to open it up and work on it. In the theory of continuous–time stochastic processes, measurability problems are usually more subtle than in discrete time, primarily because sets of measures 0 can add up to something significant when you put together uncountably many of them. In stochastic integration there is another issue: we must be sure that any stochastic processes Xt(!) that we try to integrate are jointly measurable in (t; !), so as to ensure that the integrals are themselves random variables (that is, measurable in !). Assume that (Ω; F;P ) is a fixed probability space, and that all random variables are F−measurable. A filtration is an increasing family F := fFtgt2J of sub-σ−algebras of F indexed by t 2 J, where J is an interval, usually J = [0; 1). Recall that a stochastic process fYtgt≥0 is said to be adapted to a filtration if for each t ≥ 0 the random variable Yt is Ft−measurable, and that a filtration is admissible for a Wiener process fWtgt≥0 if for each t > 0 the post-t increment process fWt+s − Wtgs≥0 is independent of Ft. Definition 1. A stochastic process fXtgt≥0 is said to be progressively measurable if for every T ≥ 0 it is, when viewed as a function X(t; !) on the product space ([0;T ])×Ω, measurable relative to the product σ−algebra B[0;T ] × FT . -
Martingale Theory
CHAPTER 1 Martingale Theory We review basic facts from martingale theory. We start with discrete- time parameter martingales and proceed to explain what modifications are needed in order to extend the results from discrete-time to continuous-time. The Doob-Meyer decomposition theorem for continuous semimartingales is stated but the proof is omitted. At the end of the chapter we discuss the quadratic variation process of a local martingale, a key concept in martin- gale theory based stochastic analysis. 1. Conditional expectation and conditional probability In this section, we review basic properties of conditional expectation. Let (W, F , P) be a probability space and G a s-algebra of measurable events contained in F . Suppose that X 2 L1(W, F , P), an integrable ran- dom variable. There exists a unique random variable Y which have the following two properties: (1) Y 2 L1(W, G , P), i.e., Y is measurable with respect to the s-algebra G and is integrable; (2) for any C 2 G , we have E fX; Cg = E fY; Cg . This random variable Y is called the conditional expectation of X with re- spect to G and is denoted by E fXjG g. The existence and uniqueness of conditional expectation is an easy con- sequence of the Radon-Nikodym theorem in real analysis. Define two mea- sures on (W, G ) by m fCg = E fX; Cg , n fCg = P fCg , C 2 G . It is clear that m is absolutely continuous with respect to n. The conditional expectation E fXjG g is precisely the Radon-Nikodym derivative dm/dn. -
Martingales by D
Martingales by D. Cox December 2, 2009 1 Stochastic Processes. Definition 1.1 Let T be an arbitrary index set. A stochastic process indexed by T is a family of random variables (Xt : t ∈ T) defined on a common probability space (Ω, F,P ). If T is clear from context, we will write (Xt). If T is one of ZZ, IN, or IN \{0}, we usually call (Xt) a discrete time process. If T is an interval in IR (usually IR or [0, ∞)), then we usually call (Xt) a continuous time process. In a sense, all of probability is about stochastic processes. For instance, if T = {1}, then we are just talking about a single random variable. If T = {1,...,n}, then we have a random vector (X1,...,Xn). We have talked about many results for i.i.d. random variables X1, X2, . Assuming an inifinite sequence of such r.v.s, T = IN \{0} for this example. Given any sequence of r.v.s X1, X2, . , we can define a partial sum process n Sn = Xi, n =1, 2,.... Xi=1 One important question that arises about stochastic processes is whether they exist or not. For example, in the above, can we really claim there exists an infinite sequence of i.i.d. random variables? The product measure theorem tells us that for any valid marginal distribution PX , we can construct any finite sequence of r.v.s with this marginal distribution. If such an infinite sequence of i.i.d. r.v.sr does not exist, we have stated a lot of meaniningless theorems. -
View on the “Impossible” Events up to Time T
Probability, Uncertainty and Quantitative Risk (2017) 2:14 Probability, Uncertainty DOI 10.1186/s41546-017-0026-3 and Quantitative Risk R E S E A RC H Open Access Financial asset price bubbles under model uncertainty Francesca Biagini · Jacopo Mancin Received: 10 January 2017 / Accepted: 3 December 2017 / © The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Abstract We study the concept of financial bubbles in a market model endowed with a set P of probability measures, typically mutually singular to each other. In this setting, we investigate a dynamic version of robust superreplication, which we use to introduce the notions of bubble and robust fundamental value in a way consistent with the existing literature in the classical case P ={P}. Finally, we provide concrete examples illustrating our results. Keywords Financial bubbles · Model uncertainty AMS classification 60G48 · 60G07 Introduction Financial asset price bubbles have been intensively studied in the mathematical litera- ture in recent years. The description of a financial bubble is usually built on two main ingredients: the market price of an asset and its fundamental value. As the first is sim- ply observed by the agent, the modeling of the intrinsic value is where the differences between various approaches arise. In the classical setup, where models with one prior are considered, the main approach is given by the martingale theory of bubbles, see Biagini et al. -
G-Expectations with Application to Risk Measures
g-Expectations with application to risk measures Sonja C. Offwood Programme in Advanced Mathematics of Finance, University of the Witwatersrand, Johannesburg. 5 August 2012 Abstract Peng introduced a typical filtration consistent nonlinear expectation, called a g-expectation in [40]. It satisfies all properties of the classical mathematical ex- pectation besides the linearity. Peng's conditional g-expectation is a solution to a backward stochastic differential equation (BSDE) within the classical framework of It^o'scalculus, with terminal condition given at some fixed time T . In addition, this g-expectation is uniquely specified by a real function g satisfying certain properties. Many properties of the g-expectation, which will be presented, follow from the spec- ification of this function. Martingales, super- and submartingales have been defined in the nonlinear setting of g-expectations. Consequently, a nonlinear Doob-Meyer decomposition theorem was proved. Applications of g-expectations in the mathematical financial world have also been of great interest. g-Expectations have been applied to the pricing of contin- gent claims in the financial market, as well as to risk measures. Risk measures were introduced to quantify the riskiness of any financial position. They also give an indi- cation as to which positions carry an acceptable amount of risk and which positions do not. Coherent risk measures and convex risk measures will be examined. These risk measures were extended into a nonlinear setting using the g-expectation. In many cases due to intermediate cashflows, we want to work with a multi-period, dy- namic risk measure. Conditional g-expectations were then used to extend dynamic risk measures into the nonlinear setting. -
Itô's Stochastic Calculus
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Stochastic Processes and their Applications 120 (2010) 622–652 www.elsevier.com/locate/spa Ito’sˆ stochastic calculus: Its surprising power for applications Hiroshi Kunita Kyushu University Available online 1 February 2010 Abstract We trace Ito’sˆ early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Ito’sˆ formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Ito’sˆ formula in mathematical finance in the 1970s. Throughout the paper, we treat Ito’sˆ jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions. c 2010 Elsevier B.V. All rights reserved. MSC: 60-03; 60H05; 60H30; 91B28 Keywords: Ito’sˆ formula; Stochastic differential equation; Jump–diffusion; Black–Scholes equation; Merton’s equation 0. Introduction This paper is written for Kiyosi Itoˆ of blessed memory. Ito’sˆ famous work on stochastic integrals and stochastic differential equations started in 1942, when mathematicians in Japan were completely isolated from the world because of the war. During that year, he wrote a colloquium report in Japanese, where he presented basic ideas for the study of diffusion processes and jump–diffusion processes. Most of them were completed and published in English during 1944–1951. Ito’sˆ work was observed with keen interest in the 1960s both by probabilists and applied mathematicians. Ito’sˆ stochastic integrals based on Brownian motion were extended to those E-mail address: [email protected]. -
Stochastic) Process {Xt, T ∈ T } Is a Collection of Random Variables on the Same Probability Space (Ω, F,P
Random Processes Florian Herzog 2013 Random Process Definition 1. Random process A random (stochastic) process fXt; t 2 T g is a collection of random variables on the same probability space (Ω; F;P ). The index set T is usually represen- ting time and can be either an interval [t1; t2] or a discrete set. Therefore, the stochastic process X can be written as a function: X : R × Ω ! R; (t; !) 7! X(t; !) Stochastic Systems, 2013 2 Filtration Observations: • The amount of information increases with time. • We use the concept of sigma algebras. • We assume that information is not lost with increasing time • Therefore the corresponding σ-algebras will increase over time when there is more information available. • ) This concept is called filtration. Definition 2. Filtration/adapted process A collection fFtgt≥0 of sub σ-algebras is called filtration if, for every s ≤ t, we have Fs ⊆ Ft. The random variables fXt : 0 ≤ t ≤ 1g are called adapted to the filtration Ft if, for every t, Xt is measurable with respect to Ft. Stochastic Systems, 2013 3 Filtration Example 1. Suppose we have a sample space of four elements: Ω = f!1;!2;!3;!4g. At time zero, we don't have any information about which T f g ! has been chosen. At time 2 we know wether we will have !1;!2 or f!3;!4g. r D = f!1g rB r E = f!2g r A r r F = f!3g C r G = f!4g - t T 0 2 T Abbildung 1: Example of a filtration Stochastic Systems, 2013 4 Difference between Random process and smooth functions Let x(·) be a real, continuously differentiable function defined on the interval [0;T ]. -
MARTINGALE THEORY for FINANCE 0. Motivation 1 1. Review of Basic Probability and Measure Theory 3 2. Conditional Expectations 12
MARTINGALE THEORY FOR FINANCE SEBASTIAN ANDRES CONTENTS 0. Motivation1 1. Review of basic probability and measure theory3 2. Conditional expectations 12 3. Martingales in discrete time 18 4. Arbitrage theory in discrete time 26 5. The Cox-Ross-Rubinstein binomial model 38 6. Brownian motion 43 7. Martingales in continuous time 49 8. Continuous semimartingales and quadratic variation 53 9. The Cameron-Martin theorem 57 10. The Black-Scholes model 59 References 64 0. MOTIVATION An investor needs a certain quantity of a share (or currency, good, ...), however not right now at time t = 0 but at a later time t = 1. The price of the share S(!) at t = 1 is random and uncertain, but already now at time t = 0 one has to make some calcultations with it, which leads to the presence of risk. (Example: 500 USD are roughly 375 GBP today, but in one year?) A possible solution for the investor would be to purchase a financial derivative such as • Forward contract: The owner of a forward contract has the right and the obligation to buy a share at time t = 1 for a delivery price K specified at time t = 0. Thus, the owner of the forward contract gains the difference between the actual market price S(!) and the delivery price K if S(!) is larger than K. If S(!) < K, the owner loses the amount K − S(!) to the issuer of the forward contract. Hence, a forward contract corresponds to the random payoff C(!) = S(!) − K: • Call option: The owner of a call option has the right but not the obligation to buy a share at time t = 1 for a the strike price K specified at time t = 0.