The Cameron-Martin-Girsanov (CMG) Theorem
|
|
- Gwen Mills
- 6 years ago
- Views:
Transcription
1 The Cameron-Martin-Girsanov (CMG) Theorem There are many versions of the CMG Theorem. In some sense, there are many CMG Theorems. The first version appeared in ] in 944. Here we present a standard version, which gives the spirit of these theorems. Introductory material Let (Ω, F, P ) be a probability space. We can introduce other probability measures on (Ω, F) and obtain other probability spaces with the same sample space Ω and the same σ-algebra of events F. One easy way to do that is to start with a random variable Θ of (Ω, F) such that Θ P -a.s. and E P Θ] =, (.) where E P ] denotes the expectation associated to the measure P. Then, we can obtain a probability measure Q on (Ω, F) by setting Q(A) = E P Θ A ] for all A F, (.2) where A is the indicator (function) of the event A. It is easy to see that Q is a probability measure on (Ω, F) and, furthermore, that Q(A) = whenever P (A) =. (.3) Recall that if two measures P and Q are related as in formula (.3), we say that Q is absolutely continuous with respect to P and denote this as Q P. (.4) In fact, the Radon-Nicodym Theorem states that the above construction of Q is the only way to obtain probability measures, which are absolutely continuous with respect to P, namely: If Q is a probability measure on (Ω, F) and Q P, then there is a random variable Θ on (Ω, F) satisfying (.), for which Q(A) = E P Θ A ] for all A F. The random variable Θ is called the Radon-Nicodym derivative of Q with respect to P. Symbolically, Θ = dq dp and it is not hard to see that Θ is unique P -a.s. Furthermore, if we also have that P Q, then dp dq = Θ. If E Q ] is the expectation associated to the measure Q of (.2), then E Q A ] = Q(A) = E P Θ A ] for all A F,
2 and hence E Q X] = E P ΘX] for any r.v. X of (Ω, F) (.5) (if E P ΘX] does not exist, then so does E Q X]). The following proposition tells us how we can express conditional expectations associated to the measure Q of (.2) in terms of conditional expectations associated to P. Proposition. Let B be a subalgebra of F. Then E Q X B] = E P ΘX B] E P Θ B] Q-a.s. (.6) Proof. For typographical convenience let us put Ψ := E P Θ B]. (.7) Of course, Ψ P -a.s., Ψ M(B) (i.e. Ψ is B-measurable), and E P Ψ] = E P EP Θ B] ] = E P Θ] =. (.8) Because of the above we can define another probability measure on (Ω, F) as Q(A) = E P Ψ A ]. (.9) Observe that, for A B we have Q(A) = E P Θ A ] = E P EP Θ A B] ] = E P A E P Θ B] ] = E P Ψ A ], that is (in view of (.9)) which also implies immediately that We continue by setting Q(A) = Q(A) for all A B, (.) E Q X] = E QX] for all X M(B). (.) Y := E P ΘX B] and Y 2 := ΨE Q X B]. (.2) Then, formula (.6) is equivalent to Y = Y 2 Q-a.s. (.3) Notice that Y, Y 2 M(B). Hence, in view of (.), to establish (.3) it suffices to show that E QY A ] = E QY 2 A ] for all A B. (.4) 2
3 Now E QY A ] = E P ΨY A ] = E P ΨEP ΘX B] A ] = E P EP ΨΘX A B] ] = E P ΨΘX A ] = E Q ΨX A ] and (with the help of (.)) ] E QY 2 A ] = E Q ΨEQ X B] A = E Q EQ ΨX A B] ] = E Q EQ ΨX A B] ] = E Q ΨX A ]. Therefore (.4) is true and the proof is completed. Why do we need to consider more than one probability measures? First of all for educational purposes, since it helps to clarify certain concepts and issues. But, also, because such situations arise in applications. For example, two persons want to divide a pie (or a property) into two pieces so that each of them feels (s)he got a fair share. However, each person may have a different measure of fairness. 2 An example A random variable X of (Ω, F) is a F-measurable function X : Ω R. Thus, X depends on Ω and F. However, X does not depend on the probability measure put on (Ω, F). It is the distribution of X which depends on the measure. For instance, if P and Q are two probability measures on (Ω, F), then the distribution functions of X with respect to P and Q respectively are F P (x) = P {X x} and F Q (x) = Q{X x}. Of course, in general F P (x) F Q (x). Thus, e.g., the phrase X is a normal random variable (i.e. X is a normally distributed random variable ) may be misleading (or, at least, equivocal), if we work with more than one probability measures, since the distribution of X depends on the measure. The same remarks apply, of course, to random vectors and stochastic processes. For example, a process which is a Brownian motion with respect to a measure P, it will probably not be a Brownian motion with respect to another measure Q. Furthermore, if X and Y are random variables which are independent with respect to P, they may not be independent with respect to another measure Q. These issues do not come up if there is only one measure, but if there are more than one measures around, then they may become very important. Let us look at a specific example (taken from 3]). Suppose we have a probability space (Ω, F, P ) and a random variable Z of this space, which 3
4 has the standard normal distribution (with respect to P ). That is P {Z z} = Φ(z) = z 2π e ξ2 /2 dξ, z R. (2.) For typographical convenience we write Z P N(, ). Of course, there are many random variables of (Ω, F, P ) having the standard normal distribution (e.g., due to the symmetry of Φ(z), if Z is such a variable, then so is Z). Recall that E P Z] =, V P Z] = (2.2) (where V P ] is the variance with respect to P ), and E P e µz ] = e µ2 /2 for any µ C. (2.3) Equation (2.3) can be written as E P e µz 2 µ2] =. (2.4) Thus, for a Z as above and a µ R we can introduce the probability measure Q defined by ] Q(A) = E P e µz 2 µ2 A, A F. (2.5) A question arising here is: What is the distribution of Z with respect to Q? We have ] Q{Z x} = E P e µz 2 µ2 {Z x} = e µz 2 µ2 {z x} e z2 /2 dz 2π = x 2π e (z µ)2 2 dz, i.e., with respect to Q, Z is a normal random variable with mean µ and variance ; symbolically, Z Q N(µ, ), which can be written equivalently as (Z µ) Q N(, ), i.e. Z µ has the standard normal distribution with respect to Q. To extend this example, let the random variables Z,..., Z n be independent and identically distributed with respect to P, all having the standard normal distribution. We now set Q(A) = E P e n j= µ jz j n ] 2 j= µ2 j A, A F, (2.6) where µ,... µ n R are given constants. It is not hard to see that Q is a probability measure on (Ω, F). Again, we would like to find the (joint) distribution of Z,..., Z n with respect to Q. 4
5 For a reasonable (say bounded and continuous) function g : R n R we have (recall (.5)) E Q g(z,..., Z n )] = E P e n j= µ jz j n ] 2 j= µ2 j g(z,..., Z n ) ( = g(z,..., z n )e n j= µ jz j z n 2 ) 2 j= µ z2 n2 j e dz (2π) n/2 dz n = g(z,..., z n ) e (2π) n/2 (z µ) 2 2 e (z n µ) 2 2 dz dz n. It follows that the joint probability density function of Z,..., Z n (with respect to Q) is f Q (z,..., z n ) = (z µ) 2 e 2 e (z n µ) 2 2, (2.7) (2π) n/2 which tells us that, with respect to Q, the variables Z,..., Z n are independent and furthermore Z j Q N(µ j, ), i.e. (Z j µ j ) Q N(, ), for j =,..., n. Thus, the random variable n Z j j= n µ j (2.8) is, with respect to Q, normally distributed with mean and variance n. Roughly speaking, the Cameron-Martin-Girsanov Theorem is a continuous version of the above simple example. In fact, having this example in mind, one can guess the statement of the CMG Theorem (see the remark after Theorem in the next section). 3 The Cameron-Martin-Girsanov Theorem 3. CMG Theorem in R Consider a probability space (Ω, F, P ) and an one-dimensional Brownian motion process {B t = B(t)} t (with respect to P ) with associated filtration {F t } t. Of course F t is a subalgebra of F for every t. Next, let {X t = X(t)} t be a (real-valued) stochastic process, adapted to the filtration {F t } t. We assume that for some constant T > we have j= Having X t we introduce T X 2 t dt < P -a.s. (3.) M t := e Y t, where Y t := 2 5 X 2 s ds + X s db s (3.2)
6 (notice that M t > P -a.s.). In differential notation we have Applying Itô Calculus to (3.2) yields dy t = 2 X2 t dt + X t db t, Y =. (3.3) dm t = e Y t dy t + 2 ey t X 2 t dt = M t X t db t, M =. (3.4) Of course, (3.4) can be written equivalently as an integral equation M t = + M s X s db s. (3.5) It follows that the Itô process M t (being a stochastic integral) is an F t - martingale (with respect to P ). In particular, E P M t ] = E P M ] = for all t, T ]. (3.6) Having M t, and in view of (3.6), we introduce the probability measures on (Ω, F) Q(A) = E P M T A ]; Q t (A) = E P M t A ], < t < T. (3.7) Observe that, for t T we have M t = E P M T F t ] (since M t is a martingale). Hence, as in (.), we have Q t (A) = Q(A) for all A F t. (3.8) Theorem (CMG in R ). Let (Ω, F, P ), B t, X t, F t, and T be as above. Set W t := B t X s ds, t, T ]. (3.9) Then, for any fixed T > the process W t, t T is an F t -Brownian motion on (Ω, F, Q) (i.e. with respect to Q). Proof. We will prove the theorem with the help of the Lévy characterization of Brownian motion in R : A continuous stochastic process W = {W t } in a probability space (Ω, F, Q) is an one-dimensional Brownian motion if and only if (i) W t is a martingale with respect to Ft W and (ii) Wt 2 t is a martingale with respect to Ft W. The proof of this characterization will not be given here (it can be found, e.g., in 3]). If we accept this characterization of Brownian motion, then, in order to prove Theorem we only need to check that W t of (3.9) satisfies (i) and (ii). 6
7 Let us check (i). Set K t := M t W t, (3.) where M t is given by (3.2). Then, Itô Calculus, (3.4), and (3.9) give dk t = W t dm t + M t dw t + dw t dm t ( ) = B t X s ds M t X t db t + M t (db t X t dt) + M t X t dt = ( B t ) ] X s ds M t X t + M t db t, (3.) which implies that K t = M t W t is an F t -martingale with respect to P. Now, for s t T, by (3.8) and Proposition we get E Q W t F s ] = E Qt W t F s ] = E P M t W t F s ] E P M t F s ] = M sw s M s = W s, Q-a.s. which says that W t is an F t -martingale with respect to Q. Checking of (ii), namely that W 2 t t is an F t -martingale with respect to Q, can be done in a similar way and is left as an exercise (EXERCISE ). Remark. The quantity M t of (3.2) can be viewed (at least in the case where X t is a deterministic function of t) as a continuous analog of the quantity e n j= µ jz j 2 n j= µ2 j, which appears in (2.6). Likewise, by writing B t = db s (assuming B = ) we can view the quantity W t of (3.9) as a continuous analog of the quantity displayed in (2.8). Theorem 2 (CMG in R d ). Let {B(t)} t, be a d-dimensional Brownian motion on (Ω, F, P ), with associated filtration {F t } t. Also, let {X(t)} t be a stochastic process with values in R d, adapted to the filtration {F t } t. We assume that for some constant T > we have Set T M t := e Yt, where Y t := 2 X t 2 dt < P -a.s. (3.2) X s 2 ds + X s db s (3.3) (notice that M t > P -a.s.). It is not hard to see that M t is an F t -martingale (with respect to P ) and E P M t ] = for all t, T ] (EXERCISE 2). If Q is the probability measure on (Ω, F) defined by Q(A) = E P M T A ], (3.4) 7
8 then, for any fixed T >, the process W t := B t is a d-dimensional F t -Brownian motion on (Ω, F, Q). X s ds, t, T ] (3.5) Proof. The proof is similar to the proof of Theorem and it is left as an exercise (EXERCISE 3). You may use (without proof) the Lévy characterization of Brownian motion in R d 3]: A continuous stochastic process W = {W (t)} in a probability space (Ω, F, Q) is an d-dimensional Brownian motion if and only if (i) W (t) is a martingale with respect to Ft W (meaning that each component W j (t), j =,..., d, is a martingale) and (ii) for any j, k {,..., d} we have that W j (t)w k (t) δ jk t is a martingale with respect to Ft W (here δ jk is the Kronecker delta). References ] R.H. Cameron and W.T. Martin, Transformation of Wiener integrals under translations, Annals of Mathematics, 45, , ] R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Inc. Belmont, CA, USA, ] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Springer, New York, 99. 4] B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Sixth Edition, Springer-Verlag, Berlin, 23. 8
The multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More informationTools of stochastic calculus
slides for the course Interest rate theory, University of Ljubljana, 212-13/I, part III József Gáll University of Debrecen Nov. 212 Jan. 213, Ljubljana Itô integral, summary of main facts Notations, basic
More informationEstimates for the density of functionals of SDE s with irregular drift
Estimates for the density of functionals of SDE s with irregular drift Arturo KOHATSU-HIGA a, Azmi MAKHLOUF a, a Ritsumeikan University and Japan Science and Technology Agency, Japan Abstract We obtain
More informationOn the quantiles of the Brownian motion and their hitting times.
On the quantiles of the Brownian motion and their hitting times. Angelos Dassios London School of Economics May 23 Abstract The distribution of the α-quantile of a Brownian motion on an interval [, t]
More informationNumerical methods for solving stochastic differential equations
Mathematical Communications 4(1999), 251-256 251 Numerical methods for solving stochastic differential equations Rózsa Horváth Bokor Abstract. This paper provides an introduction to stochastic calculus
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 7 9/25/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 7 9/5/013 The Reflection Principle. The Distribution of the Maximum. Brownian motion with drift Content. 1. Quick intro to stopping times.
More informationThe Wiener Itô Chaos Expansion
1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in
More informationLecture 22 Girsanov s Theorem
Lecture 22: Girsanov s Theorem of 8 Course: Theory of Probability II Term: Spring 25 Instructor: Gordan Zitkovic Lecture 22 Girsanov s Theorem An example Consider a finite Gaussian random walk X n = n
More informationOPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS
APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze
More informationWeak solutions of mean-field stochastic differential equations
Weak solutions of mean-field stochastic differential equations Juan Li School of Mathematics and Statistics, Shandong University (Weihai), Weihai 26429, China. Email: juanli@sdu.edu.cn Based on joint works
More informationApplications of Ito s Formula
CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale
More informationMan Kyu Im*, Un Cig Ji **, and Jae Hee Kim ***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No. 4, December 26 GIRSANOV THEOREM FOR GAUSSIAN PROCESS WITH INDEPENDENT INCREMENTS Man Kyu Im*, Un Cig Ji **, and Jae Hee Kim *** Abstract.
More informationQuestion 1. The correct answers are: (a) (2) (b) (1) (c) (2) (d) (3) (e) (2) (f) (1) (g) (2) (h) (1)
Question 1 The correct answers are: a 2 b 1 c 2 d 3 e 2 f 1 g 2 h 1 Question 2 a Any probability measure Q equivalent to P on F 2 can be described by Q[{x 1, x 2 }] := q x1 q x1,x 2, 1 where q x1, q x1,x
More informationRepresenting Gaussian Processes with Martingales
Representing Gaussian Processes with Martingales with Application to MLE of Langevin Equation Tommi Sottinen University of Vaasa Based on ongoing joint work with Lauri Viitasaari, University of Saarland
More informationMFE6516 Stochastic Calculus for Finance
MFE6516 Stochastic Calculus for Finance William C. H. Leon Nanyang Business School December 11, 2017 1 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance 1 Symmetric Random Walks Scaled Symmetric
More informationContinuous Time Finance
Continuous Time Finance Lisbon 2013 Tomas Björk Stockholm School of Economics Tomas Björk, 2013 Contents Stochastic Calculus (Ch 4-5). Black-Scholes (Ch 6-7. Completeness and hedging (Ch 8-9. The martingale
More informationStochastic Calculus. Kevin Sinclair. August 2, 2016
Stochastic Calculus Kevin Sinclair August, 16 1 Background Suppose we have a Brownian motion W. This is a process, and the value of W at a particular time T (which we write W T ) is a normally distributed
More informationVerona Course April Lecture 1. Review of probability
Verona Course April 215. Lecture 1. Review of probability Viorel Barbu Al.I. Cuza University of Iaşi and the Romanian Academy A probability space is a triple (Ω, F, P) where Ω is an abstract set, F is
More informationMATH 56A SPRING 2008 STOCHASTIC PROCESSES 197
MATH 56A SPRING 8 STOCHASTIC PROCESSES 197 9.3. Itô s formula. First I stated the theorem. Then I did a simple example to make sure we understand what it says. Then I proved it. The key point is Lévy s
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationMultiple points of the Brownian sheet in critical dimensions
Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang Ecole Polytechnique Fédérale de Lausanne Based on joint work with: Carl Mueller Multiple points of the Brownian sheet in critical
More informationMalliavin Calculus in Finance
Malliavin Calculus in Finance Peter K. Friz 1 Greeks and the logarithmic derivative trick Model an underlying assent by a Markov process with values in R m with dynamics described by the SDE dx t = b(x
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
More informationCOVARIANCE IDENTITIES AND MIXING OF RANDOM TRANSFORMATIONS ON THE WIENER SPACE
Communications on Stochastic Analysis Vol. 4, No. 3 (21) 299-39 Serials Publications www.serialspublications.com COVARIANCE IDENTITIES AND MIXING OF RANDOM TRANSFORMATIONS ON THE WIENER SPACE NICOLAS PRIVAULT
More informationON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER
ON THE FIRST TIME THAT AN ITO PROCESS HITS A BARRIER GERARDO HERNANDEZ-DEL-VALLE arxiv:1209.2411v1 [math.pr] 10 Sep 2012 Abstract. This work deals with first hitting time densities of Ito processes whose
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationBernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012
1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.
More informationExercises in stochastic analysis
Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with
More informationCHANGE OF MEASURE. D.Majumdar
CHANGE OF MEASURE D.Majumdar We had touched upon this concept when we looked at Finite Probability spaces and had defined a R.V. Z to change probability measure on a space Ω. We need to do the same thing
More informationSEPARABLE TERM STRUCTURES AND THE MAXIMAL DEGREE PROBLEM. 1. Introduction This paper discusses arbitrage-free separable term structure (STS) models
SEPARABLE TERM STRUCTURES AND THE MAXIMAL DEGREE PROBLEM DAMIR FILIPOVIĆ Abstract. This paper discusses separable term structure diffusion models in an arbitrage-free environment. Using general consistency
More informationOn Stochastic Adaptive Control & its Applications. Bozenna Pasik-Duncan University of Kansas, USA
On Stochastic Adaptive Control & its Applications Bozenna Pasik-Duncan University of Kansas, USA ASEAS Workshop, AFOSR, 23-24 March, 2009 1. Motivation: Work in the 1970's 2. Adaptive Control of Continuous
More informationItô s formula. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Itô s formula Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Itô s formula Probability Theory
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationWiener Measure and Brownian Motion
Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u
More informationJae Gil Choi and Young Seo Park
Kangweon-Kyungki Math. Jour. 11 (23), No. 1, pp. 17 3 TRANSLATION THEOREM ON FUNCTION SPACE Jae Gil Choi and Young Seo Park Abstract. In this paper, we use a generalized Brownian motion process to define
More informationHarnack Inequalities and Applications for Stochastic Equations
p. 1/32 Harnack Inequalities and Applications for Stochastic Equations PhD Thesis Defense Shun-Xiang Ouyang Under the Supervision of Prof. Michael Röckner & Prof. Feng-Yu Wang March 6, 29 p. 2/32 Outline
More informationMA50251: Applied SDEs
MA5251: Applied SDEs Alexander Cox February 12, 218 Preliminaries These notes constitute the core theoretical content of the unit MA5251, which is a course on Applied SDEs. Roughly speaking, the aim of
More informationIntroduction to Diffusion Processes.
Introduction to Diffusion Processes. Arka P. Ghosh Department of Statistics Iowa State University Ames, IA 511-121 apghosh@iastate.edu (515) 294-7851. February 1, 21 Abstract In this section we describe
More informationStochastic Integration and Stochastic Differential Equations: a gentle introduction
Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More information(B(t i+1 ) B(t i )) 2
ltcc5.tex Week 5 29 October 213 Ch. V. ITÔ (STOCHASTIC) CALCULUS. WEAK CONVERGENCE. 1. Quadratic Variation. A partition π n of [, t] is a finite set of points t ni such that = t n < t n1
More informationA NOTE ON STATE ESTIMATION FROM DOUBLY STOCHASTIC POINT PROCESS OBSERVATION. 0. Introduction
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVI, Number 1, March 1 A NOTE ON STATE ESTIMATION FROM DOUBLY STOCHASTIC POINT PROCESS OBSERVATION. Introduction In this note we study a state estimation
More informationLecture 17 Brownian motion as a Markov process
Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is
More informationAn Introduction to Malliavin calculus and its applications
An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart
More informationAn Analytic Method for Solving Uncertain Differential Equations
Journal of Uncertain Systems Vol.6, No.4, pp.244-249, 212 Online at: www.jus.org.uk An Analytic Method for Solving Uncertain Differential Equations Yuhan Liu Department of Industrial Engineering, Tsinghua
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationLAN property for sde s with additive fractional noise and continuous time observation
LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday,
More informationLecture 9. d N(0, 1). Now we fix n and think of a SRW on [0,1]. We take the k th step at time k n. and our increments are ± 1
Random Walks and Brownian Motion Tel Aviv University Spring 011 Lecture date: May 0, 011 Lecture 9 Instructor: Ron Peled Scribe: Jonathan Hermon In today s lecture we present the Brownian motion (BM).
More informationPotentials of stable processes
Potentials of stable processes A. E. Kyprianou A. R. Watson 5th December 23 arxiv:32.222v [math.pr] 4 Dec 23 Abstract. For a stable process, we give an explicit formula for the potential measure of the
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationThe Pedestrian s Guide to Local Time
The Pedestrian s Guide to Local Time Tomas Björk, Department of Finance, Stockholm School of Economics, Box 651, SE-113 83 Stockholm, SWEDEN tomas.bjork@hhs.se November 19, 213 Preliminary version Comments
More informationIntroduction to Stochastic SIR Model
Introduction to Stochastic R Model Chiu- Yu Yang (Alex), Yi Yang R model is used to model the infection of diseases. It is short for Susceptible- Infected- Recovered. It is important to address that R
More informationStochastic integral. Introduction. Ito integral. References. Appendices Stochastic Calculus I. Geneviève Gauthier.
Ito 8-646-8 Calculus I Geneviève Gauthier HEC Montréal Riemann Ito The Ito The theories of stochastic and stochastic di erential equations have initially been developed by Kiyosi Ito around 194 (one of
More informationMAXIMAL COUPLING OF EUCLIDEAN BROWNIAN MOTIONS
MAXIMAL COUPLING OF EUCLIDEAN BOWNIAN MOTIONS ELTON P. HSU AND KAL-THEODO STUM ABSTACT. We prove that the mirror coupling is the unique maximal Markovian coupling of two Euclidean Brownian motions starting
More informationPenalization of the Wiener Measure and Principal Values
Penalization of the Wiener Measure and Principal Values by Catherine DONATI-MARTIN and Yueyun HU Summary. Let Q µ be absolutely continuous with respect to the Wiener measure with density D t exp hb sdb
More informationON THE STRUCTURE OF GAUSSIAN RANDOM VARIABLES
ON THE STRUCTURE OF GAUSSIAN RANDOM VARIABLES CIPRIAN A. TUDOR We study when a given Gaussian random variable on a given probability space Ω, F,P) is equal almost surely to β 1 where β is a Brownian motion
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationDensity models for credit risk
Density models for credit risk Nicole El Karoui, Ecole Polytechnique, France Monique Jeanblanc, Université d Évry; Institut Europlace de Finance Ying Jiao, Université Paris VII Workshop on stochastic calculus
More informationStochastic Numerical Analysis
Stochastic Numerical Analysis Prof. Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Stoch. NA, Lecture 3 p. 1 Multi-dimensional SDEs So far we have considered scalar SDEs
More informationExercise 4. An optional time which is not a stopping time
M5MF6, EXERCICE SET 1 We shall here consider a gien filtered probability space Ω, F, P, spporting a standard rownian motion W t t, with natral filtration F t t. Exercise 1 Proe Proposition 1.1.3, Theorem
More informationLecture 12. F o s, (1.1) F t := s>t
Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let
More informationProblem 1. Construct a filtered probability space on which a Brownian motion W and an adapted process X are defined and such that
Stochatic Calculu Example heet 4 - Lent 5 Michael Tehranchi Problem. Contruct a filtered probability pace on which a Brownian motion W and an adapted proce X are defined and uch that dx t = X t t dt +
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 22 12/09/2013. Skorokhod Mapping Theorem. Reflected Brownian Motion
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.7J Fall 213 Lecture 22 12/9/213 Skorokhod Mapping Theorem. Reflected Brownian Motion Content. 1. G/G/1 queueing system 2. One dimensional reflection mapping
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationSupplement A: Construction and properties of a reected diusion
Supplement A: Construction and properties of a reected diusion Jakub Chorowski 1 Construction Assumption 1. For given constants < d < D let the pair (σ, b) Θ, where Θ : Θ(d, D) {(σ, b) C 1 (, 1]) C 1 (,
More informationStochastic Differential Equations
CHAPTER 1 Stochastic Differential Equations Consider a stochastic process X t satisfying dx t = bt, X t,w t dt + σt, X t,w t dw t. 1.1 Question. 1 Can we obtain the existence and uniqueness theorem for
More informationCitation Osaka Journal of Mathematics. 41(4)
TitleA non quasi-invariance of the Brown Authors Sadasue, Gaku Citation Osaka Journal of Mathematics. 414 Issue 4-1 Date Text Version publisher URL http://hdl.handle.net/1194/1174 DOI Rights Osaka University
More informationB8.3 Mathematical Models for Financial Derivatives. Hilary Term Solution Sheet 2
B8.3 Mathematical Models for Financial Derivatives Hilary Term 18 Solution Sheet In the following W t ) t denotes a standard Brownian motion and t > denotes time. A partition π of the interval, t is a
More informationPoisson random measure: motivation
: motivation The Lévy measure provides the expected number of jumps by time unit, i.e. in a time interval of the form: [t, t + 1], and of a certain size Example: ν([1, )) is the expected number of jumps
More informationSome Terminology and Concepts that We will Use, But Not Emphasize (Section 6.2)
Some Terminology and Concepts that We will Use, But Not Emphasize (Section 6.2) Statistical analysis is based on probability theory. The fundamental object in probability theory is a probability space,
More informationMA8109 Stochastic Processes in Systems Theory Autumn 2013
Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More informationWhite noise generalization of the Clark-Ocone formula under change of measure
White noise generalization of the Clark-Ocone formula under change of measure Yeliz Yolcu Okur Supervisor: Prof. Bernt Øksendal Co-Advisor: Ass. Prof. Giulia Di Nunno Centre of Mathematics for Applications
More informationContinuous-time Gaussian Autoregression Peter Brockwell, Richard Davis and Yu Yang, Statistics Department, Colorado State University, Fort Collins, CO
Continuous-time Gaussian Autoregression Peter Brockwell, Richard Davis and Yu Yang, Statistics Department, Colorado State University, Fort Collins, CO 853-877 Abstract The problem of tting continuous-time
More informationPREDICTABLE REPRESENTATION PROPERTY OF SOME HILBERTIAN MARTINGALES. 1. Introduction.
Acta Math. Univ. Comenianae Vol. LXXVII, 1(28), pp. 123 128 123 PREDICTABLE REPRESENTATION PROPERTY OF SOME HILBERTIAN MARTINGALES M. EL KADIRI Abstract. We prove as for the real case that a martingale
More informationThe Azéma-Yor Embedding in Non-Singular Diffusions
Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 9 10/2/2013. Conditional expectations, filtration and martingales
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 9 10/2/2013 Conditional expectations, filtration and martingales Content. 1. Conditional expectations 2. Martingales, sub-martingales
More informationarxiv: v2 [math.pr] 22 Aug 2009
On the structure of Gaussian random variables arxiv:97.25v2 [math.pr] 22 Aug 29 Ciprian A. Tudor SAMOS/MATISSE, Centre d Economie de La Sorbonne, Université de Panthéon-Sorbonne Paris, 9, rue de Tolbiac,
More informationOn the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family
More informationFinite element approximation of the stochastic heat equation with additive noise
p. 1/32 Finite element approximation of the stochastic heat equation with additive noise Stig Larsson p. 2/32 Outline Stochastic heat equation with additive noise du u dt = dw, x D, t > u =, x D, t > u()
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationNested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model
Nested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model Xiaowei Chen International Business School, Nankai University, Tianjin 371, China School of Finance, Nankai
More informationGeneralized Gaussian Bridges
TOMMI SOTTINEN ADIL YAZIGI Generalized Gaussian Bridges PROCEEDINGS OF THE UNIVERSITY OF VAASA WORKING PAPERS 4 MATHEMATICS 2 VAASA 212 Vaasan yliopisto University of Vaasa PL 7 P.O. Box 7 (Wolffintie
More informationSTOCHASTIC CALCULUS JASON MILLER AND VITTORIA SILVESTRI
STOCHASTIC CALCULUS JASON MILLER AND VITTORIA SILVESTRI Contents Preface 1 1. Introduction 1 2. Preliminaries 4 3. Local martingales 1 4. The stochastic integral 16 5. Stochastic calculus 36 6. Applications
More informationA MODEL FOR THE LONG-TERM OPTIMAL CAPACITY LEVEL OF AN INVESTMENT PROJECT
A MODEL FOR HE LONG-ERM OPIMAL CAPACIY LEVEL OF AN INVESMEN PROJEC ARNE LØKKA AND MIHAIL ZERVOS Abstract. We consider an investment project that produces a single commodity. he project s operation yields
More informationSmoothness of the distribution of the supremum of a multi-dimensional diffusion process
Smoothness of the distribution of the supremum of a multi-dimensional diffusion process Masafumi Hayashi Arturo Kohatsu-Higa October 17, 211 Abstract In this article we deal with a multi-dimensional diffusion
More informationFunctional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals
Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico
More informationOn semilinear elliptic equations with measure data
On semilinear elliptic equations with measure data Andrzej Rozkosz (joint work with T. Klimsiak) Nicolaus Copernicus University (Toruń, Poland) Controlled Deterministic and Stochastic Systems Iasi, July
More informationLecture 4: Introduction to stochastic processes and stochastic calculus
Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London
More informationHomework # , Spring Due 14 May Convergence of the empirical CDF, uniform samples
Homework #3 36-754, Spring 27 Due 14 May 27 1 Convergence of the empirical CDF, uniform samples In this problem and the next, X i are IID samples on the real line, with cumulative distribution function
More informationGeneralized Gaussian Bridges of Prediction-Invertible Processes
Generalized Gaussian Bridges of Prediction-Invertible Processes Tommi Sottinen 1 and Adil Yazigi University of Vaasa, Finland Modern Stochastics: Theory and Applications III September 1, 212, Kyiv, Ukraine
More information1 Solution to Problem 2.1
Solution to Problem 2. I incorrectly worked this exercise instead of 2.2, so I decided to include the solution anyway. a) We have X Y /3, which is a - function. It maps the interval, ) where X lives) onto
More informationExercises. T 2T. e ita φ(t)dt.
Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.
More informationA Short Introduction to Diffusion Processes and Ito Calculus
A Short Introduction to Diffusion Processes and Ito Calculus Cédric Archambeau University College, London Center for Computational Statistics and Machine Learning c.archambeau@cs.ucl.ac.uk January 24,
More informationKrzysztof Burdzy University of Washington. = X(Y (t)), t 0}
VARIATION OF ITERATED BROWNIAN MOTION Krzysztof Burdzy University of Washington 1. Introduction and main results. Suppose that X 1, X 2 and Y are independent standard Brownian motions starting from 0 and
More informationStochastic Differential Equations.
Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)
More informationStochastic Integration and Continuous Time Models
Chapter 3 Stochastic Integration and Continuous Time Models 3.1 Brownian Motion The single most important continuous time process in the construction of financial models is the Brownian motion process.
More informationAn Overview of the Martingale Representation Theorem
An Overview of the Martingale Representation Theorem Nuno Azevedo CEMAPRE - ISEG - UTL nazevedo@iseg.utl.pt September 3, 21 Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 1 / 25 Background
More informationGAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM
GAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM STEVEN P. LALLEY 1. GAUSSIAN PROCESSES: DEFINITIONS AND EXAMPLES Definition 1.1. A standard (one-dimensional) Wiener process (also called Brownian motion)
More information