Lecture 19 L 2 -Stochastic integration
|
|
- Kerry Jordan
- 5 years ago
- Views:
Transcription
1 Lecture 19: L 2 -Stochastic integration 1 of 12 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 19 L 2 -Stochastic integration The stochastic integral for processes of finite variation For T >, let F : [, T] R be càdlàg function of finite variation, and let F : [, T] [, ) be its total variation. For a < b T we define µ((a, b]) = F(b) F(a), µ({}) = F(). A version of the Caratheodory extension theorem guarantees that the functional µ can be extended in a unique way to a signed (real) measure on B([, T]); this measure is called the Stieltjes measure induced by F, and it is typically denoted by df. It is not hard to see (we leave the details to the reader) that there exist unique càdlàg functions F +, F : [, T] R of finite variation such that 1. df + and df are finite positive measures (do nottake negative values) with df = df + df, and 2. df + and df are absolutely continuous with respect to the measure d F (the Stieltjes measure induced by the total variation F of F) and df + + df = d F. The decomposition df = df + df is called the Hahn-Jordan decomposition of the Stieltjes measure F. Definition 19.1 (Lebesgue-Stieltjes integral). A function h : [, T] R is said to be df-integrable (denoted by h L 1 (df)), if h(t) d F (t) <. For h L 1 (df), its Stieltjes integral with respect to df is defined by h(u) df(u) = h(u) df + (u) h(u) df (u). Last Updated: March 29, 215
2 Lecture 19: L 2 -Stochastic integration 2 of 12 When the function h is continuous, one can approximate the Lebesgue-Stieltjes integral by Riemann-like sums. More precisely, let = t n < tn 1 < < tn d n < t n d n +1 = T be a sequence of partitions of [, T] such that lim sup t n n k+1 tn k =, k d n and suppose that h : [, T] R is a continuous function. We leave it to the reader to show that h L 1 (df) and that h(x) df(x) = lim d n n k= If, additionally, F C 1 ([, T]), we have for all h L 1 (df). h(t) df(t) = h(t n k )(F(tn k+1 ) F(tn k )). h(t)f (t) dt, We turn to processes, now. Let {X t } t [,) be an adapted càdlàg process of finite variation, and let {H t } t [,) be a progressively measurable process such that H t (ω) d X t (ω) <, for all T, for almost all ω, where d X t (ω) is the Stieltjes measure induced by the non-decreasing function t X t (ω) (the total-variation function of X(ω)). Therefore, for almost all ω, and all t, we can define the Stieltjes integral Y t (ω) = t H t (ω) dx t (ω), where dx t (ω) is the Stieltjes measure induced by the FV function t X t (ω). We set Y t (ω) =, for all t for ω in the exceptional set. It is not hard to show that the process {Y} t [,) is an adapted and càdlàg process of finite variation, and that it is continuous when X is continuous. The process {Y t } t [,) is called the stochastic integral of {H t } t [,) with respect to {X t } t [,) and is sometimes denoted by Y = (H X). The importance of being of finite variation Unfortunately, the Lebesgue-style integration stops with functions of finite variation. Here is why. A sequence = t < t 1 < < t k < t k+1 < < of real numbers with the property that lim k t k = +, is called a partition of [, ), and the set of all partitions of [, ) is denoted by P [,). Last Updated: March 29, 215
3 Lecture 19: L 2 -Stochastic integration 3 of 12 The elements t, t 1,... of a partition are referred to as its nodes. For a partition = {t k } k N P [,), and t, we define [,t] = sup k N t k+1 t t k t and note that [,t] = max( t 1 t, t 2 t 1,..., t t k(t) ), where k(t) = sup{k N : t k < t}. A sequence { n } n N in P [,) is said to converge to identity, denoted by n Id if n [,t], for each t. Let F : [, ) R be a continuous function, and let D simp denote the set of all functions h : [, ) R for which there exists a partition = {t k } k N P [,), such that h(t) = α 1 {} (t) + For h S and t > we naturally define t h(u) df(u) = α F() + α k 1 (tk,t k+1 ](t), n N, α k R, k N. (19.1) α k (F(t t k+1 ) F(t t k )). (19.2) Note that the sum is really finite (we could have stopped at the index of the first t k > t). For a continuous function h : [, ) R and a partition P [,), we define the approximation h to h in D simp by h (t) = h()1 {} (t) + h(t k )1 (tk,t k+1 ](t), n N, (19.3) and note that h h uniformly on compact sets. We ask the following question Question What properties does F need to have for the limit t lim h (u) df(u), Id to exist for each t and each continuous h? Remark The reader should note that the question we are asking above really deals with a weak continuity property for the eventual extension of the simple integral (19.2) to a larger class of integrands h. Indeed, if a functional h h df is to be called an integral, it should be linear, and somewhat continuous. The requirement that the uniform convergence of integrands h h implies convergence of the integrals is a particularly weak form of the dominated convergence theorem. Nevertheless, as we will see, it restricts the class of functions F considerably. Last Updated: March 29, 215
4 Lecture 19: L 2 -Stochastic integration 4 of 12 The main analytic tool we are going to use is the celebrated Banach- Steinhaus (aka uniform-boundedness theorem). Remember that for a linear operator T : X Y between two normed spaces (X, X ) and (Y, Y ), we define its operator norm A by T = sup{ T f Y : f X, f X 1}. Theorem 19.4 (Banach, Steinhaus). Let X be a Banach space, let Y be a normed space, and let (B α ) α A be a family of continuous linear operators B α : X Y. Then sup α B B α < if and only if f X, sup B α f Y <. α B Proof. (*) One direction is trivial. For the other, suppose that sup B α f Y < for each f X. α B Define the sequence {D n } n N of subsets of X by D n = { f X : B α f n}, α A which, as intersections of closed sets are themselves closed. By the assumption, n N D n = X. Since X is complete, Baire s theorem 1 implies that at least one of {D n } n N has nonempty interior. More explicitly, there exists n N, f D n and ε > such that f + εb 1 (X) D n, where B 1 (X) = { f X : f X 1} denotes the unit ball of X. Consequently, 1 Baire s (category) theorem states that the intersection of a countable collections of dense open sets in a complete metric space is dense itself. B α ( 1 ε f + f ) Y 1 ε n for all f B 1(X) and all α A. The triangle inequality now implies that B α f Y 1 ε n + B α 1 ε f n ε + 1 ε sup B α f, for all α A, f B 1 (X), α A and so sup α A B α n ε + 1 ε sup α A B α f <. Proposition Let F be a continuous function on [, 1], such that for each continuous function h : [, 1] R, the family 1 h n (u) df(u), n N, is bounded in R for each sequence { n } n N in P [,) such that n Id. Then F is a function of finite variation on [, 1]. Proof. (*) Let X = C[, 1] be the Banach space of all continuous functions on [, 1] normed with h = sup t [,1] h(t), for h C[, 1], and Last Updated: March 29, 215
5 Lecture 19: L 2 -Stochastic integration 5 of 12 let Y = R. We fix a sequence { n } n N in P [,) with n Id, and assume, without loss of generality that 1 n, for all n N. A sequence of operators can be defined by B n : X Y, B n h = 1 h n (u) df(u), n N. These operators are continuous (why?), and, by the assumption, the sequence {B n h} n N is bounded, for each h X. Therefore, by Theorem 19.4, there exists a constant K such that 1 h n (u) df(u) K h, h C[, 1], n N. (19.4) For each n N, one can construct a function h n C[, 1] such that h n () = sgn(f()) and h n (t n k ) = sgn(f(tn k+1 ) F(tn k )), for all k k n(1), where n = {t n k } k N. Moreover, such a function can be constructed with h n = 1. Using (19.4), we get K 1 h n n (u) df(u) k n (1) = F() + F(t n k+1 ) F(t n k ). We have proved that the sequence of variations of F along any sequence of partitions converging to identity remains bounded. It follows (why?) that F must be a function of finite variation. We can repeat the above argument on each [, t] instead on [, 1], and, since any convergent sequence is bounded, the answer to Question 19.2 must be It is necessary that F be of finite variation. On the other hand, when F is of finite variation, the dominated convergence theorem implies that the sequence in Question 19.2 indeed converges towards t h(u) df(u). k= Square-integrable martingales as integrators In spite of the impossibility result of the previous subsection, we will still be able to construct a satisfactory integration theory when the integrand is not of finite variation. In that case, it will be important that some randomness is present and that the class of integrands be restricted to those that do not depend on the future. This way, the terms that would otherwise accumulate and lead to the explosion in the approximating sequence will cancel each other (in the law-of-largenumbers manner). Before we move on to the general case, here is an example in a simplified (but still fully-featured) setting: Example Without knowing it, we have constructed a very simple version of a stochastic integral when we proved the martingale convergence theorem. Indeed, let {ξ n } n N be an iid sequence of coin-tosses Last Updated: March 29, 215
6 Lecture 19: L 2 -Stochastic integration 6 of 12 (P[ξ 1 = 1] = P[ξ 1 = 1] = 1 2 ) and let {M t} t [,1] be defined as follows: M t = n S k 1 [1 2 k+1,1 2 k ) (t), where S 1 k = k ξ k, for t < 1. The martingale convergence theorem guarantees that the limit M 1 = lim t 1 M t exists a.s. The reader can check that the variation of the path M (ω) on [, 1] is given by 1 k ξ k(ω) = 1 k = +, so that no trajectory of {M t } t [,1] is of finite variation 2. On the other 2 The reader will also note that hand, let {H {M t } t [,1] is not continuous; a similar t } t [,1) be any bounded (say by K ) and {Ft M } t [,1] - example with {M t } t [,1] continuous adapted left-continuous stochastic process. On each segment [, t], can be concocted, but it would not t < 1, {M t } t [,1] is a process of finite variation and so we have the following expression t H u dm u = k(t) h k k ξ k, where h k = H 1 2 k+1 and k(t) is such that t [1 2 k+1, 1 2 k ). Moreover, h k σ(ξ 1, ξ 2,..., ξ k 1 ), so that the process N n = n h k k ξ k, is a martingale (it is, really, a martingale transform of {S n } n N ). Moreover, it is a martingale bounded in L 2 ; indeed, by the orthogonality of martingale increments, we have E[N 2 n] = n E[ h2 k ξ 2 n k 2 k ] = 1 K 2 K 2 k 2 1 k 2 <, so that sup n E[Nn] 2 <. Consequently, the martingale convergence theorem implies that the limit N = lim n N N n exists a.s., and in L 2. Equivalently, the random variable 1 t H u dm u = lim H u dm u = lim N n = N, t 1 n is well defined, even though no path of M is of finite variation. It is important to note that the sequence {N n } n N only converges a.s. There is no way to conclude that the limit exists for each ω. Indeed, take h n = 1, for all n N, and note that for those ω for which ξ k (ω) = 1, for all k N, the sequence 1 1 k ξ k (ω) = k = +, does not converge. Moreover, the series 1 k ξ k(ω) will never converge absolutely. It is the fact that the sequence {ξ k } k N fluctuates so much and the fact that h k does not see ξ k that guarantee convergence N n N, and, equivalently, allow for the integral 1 H u dm u to be well-defined. be as transparent as the one we give here. Also, if one prefers to work with [, ) instead of [, 1], one can define M t = S k1 [k 1,k) (t). Last Updated: March 29, 215
7 Lecture 19: L 2 -Stochastic integration 7 of 12 Turning to the general case, we assume that a filtered probability space (Ω, F, {F t } t [,), P), where {F t } t [,) satisfies the usual conditions, is given and fixed throughout. When the integrand is simple the definition of the stochastic integral is easy. More precisely, a stochastic process {H t } t [,) is said to be simple predictable if there exists a partition = {t n } n N P [,) and a sequence {K n } n N of random variables such that K n F tn, and H t = K n 1 (tn,t n+1 ](t). (19.5) n= The set of all simple predictable processes is denoted by H simp. The subset Hsimp of H simp consists of all processes {H t } t [,) H simp such that H H = sup simp n N K n L <. Definition 19.7 (Stochastic integration of simple processes). Suppose that {M t } t [,) is an arbitrary stochastic process, and let {H t } t [,) be a simple predictable process with representation (19.5). The (simple) stochastic integral of H with respect to M is the stochastic process {(H M) t } t [,), given by (H M) t = K n (M t tn+1 M t tn ). (19.6) n= Remark Note that simple predictable processes are predictable (measurable in the σ-algebra generated by left-continuous adapted processes). Moreover, H simp and Hsimp are vector spaces and the functional H is a norm on H simp simp (if indistinguishable processes are identified). 2. One of the best ways to visualize the stochastic integral is to remember the notion of the martingale transform from our discussion of discrete-time martingales. The gambling interpretation used there easily transfers to continuous time, at least when the integrands are simple predictable, as above. Note, though, that one has to think of gambles in an infinitesimal manner in continuous time. 3. The value at t of the stochastic integral is usually denoted by (H M) t or t H u dm u. Sometimes the terminology is abused and the random variable (H M) t (as opposed to the whole process {(H M) t } t [,) or, simply, H M) is called the stochastic integral. The stochastic integral for square-integrable martingales We turn now to a general construction. Let M 2,c denote the family of all continuous martingales M with M = such that M M 2,c := Last Updated: March 29, 215
8 Lecture 19: L 2 -Stochastic integration 8 of 12 sup t E[Mt 2 ] <. Here is the fundamental property of the (simple) stochastic integral with respect to M M 2,c : Proposition 19.9 (Preservation of M 2,c - simple predictable integrands). For M M 2,c and {H t } t [,) Hsimp we have H M M2,c and E[(H M) 2 t ] = E[Kn(M 2 tn+1 t M tn t) 2 ]. (19.7) n= Proof. Continuity, value at time, and the martingale property of H M are evident from the definition. To establish (19.7), we assume that t = t N for some N (otherwise just add it to the partition), and use the fact that the increments of square-integrable martingales are orthogonal, i.e., E[K n (M tn+1 M tn )K m (M tm+1 M tm )] = for n < m. Finally, (H M) M 2,c because, for t, we have E[(H M) 2 t ] = E[Kn(M 2 tn+1 M tn ) 2 ] n N H 2 H simp n = H 2 Hsimp M 2 M 2,c. E[(M tn+1 M tn ) 2 ] We turn now to the notion of quadratic variation of a continuous martingale. The main theorem listing its properties is stated without proof, but the reader is instructed to draw the parallel with the case of the Brownian motion we covered in previous lectures. Before we start, we need to introduce some notation: let A c denote the set of all continuous, adapted, nondecreasing processes {A t } t [,) with A =. We also remind the reader that, for a RCLL of RLCC process X, we define the maximal process X by Xt = sup s t X s. For a sequence {X n } n N of RCLL or RLCC processes, we say that X n converges to X, ucp uniformly on compacts in probability (ucp), and write X n X, if (X n X) t in probability, for each t. Theorem 19.1 (Quadratic variation of square-integrable martingales). For M M 2,c, there exists a unique process M Ac such that such that Mt 2 M t is a uniformly-integrable martingale. Furthermore, for each sequence { n } n N in P [,) with n Id, we have M n ucp M, where, for = {t k } k N P [,), M t := k= (M t k+1 t M tk t) 2. Last Updated: March 29, 215
9 Lecture 19: L 2 -Stochastic integration 9 of 12 The process M A c is called the quadratic variation (process) of M. It admits a natural interpretation as the increasing part in the (continuous-time) Doob-Meyer decomposition of M 2 and plays a central role in stochastic analysis. The reader should probably do Problem?? at this point. Here is how quadratic variation can be used to construct the stochastic integral: Given M M 2,c, it follows from the fact that both M and M2 M are a martingales that E[(M tk+1 M tk ) 2 F tk ] = E[M 2 t k+1 M 2 t k F tk ] = E[ M tk +1 M tk F tk ]. Therefore, the relation (19.7) can be written as [ E[(H M) 2 ] ] = E Kk 2 E[ M t k+1 M tk F tk ] n= (19.8) [ = E K 2 ( ) ] k M tk+1 M tk = E[ Ht 2 d M t ]. n= Inspired by the last expression in (19.8) above, for a progressively-measurable process H, we define H L 2 (M) E[ := Ht 2 d M t]. It is not hard to check that the family L 2 (M) of all progressively-measurable processes H for which H L 2 (M) < forms a vector space, and that L 2 (M) is a norm there. We also note that H simp L2 (M), for each M M 2,c (why?). In this, new, notation, the conclusion of (19.8) looks like this: H M M 2,c = H L 2 (M), H H simp, (19.9) and is called Itô s isometry. We use it to extend the domain of the stochastic integral from Hsimp to a much larger set. To do that, we need a simple, but very powerful, result from real analysis: Proposition Let X be a metric space and Y a complete metric space. Moreover, let f : A Y, with = A X be a uniformly-continuous function. Then, there exists a unique uniformly-continuous function f : Cl A Y such that f (x) = f (x), for all x A. Proof. For x Cl A, let {x n } n N be a sequence in A such that x n x, and let y n = f (x n ). Since x n is convergent in X and f is uniformly continuous, {y n } n N is a Cauchy sequence in Y. Since Y is complete it admits a unique limit y, and we set f (x) = y. Last Updated: March 29, 215
10 Lecture 19: L 2 -Stochastic integration 1 of 12 To show that f is well-defined, we need to argue that y does not depend on the choice of the sequence {x n } n N. Indeed, let {x n} n N be another sequence in A with x n x. Then d(x n, x n), and, so, by uniform continuty, d( f (x n ), f (x n)) =, for each y. It follows that y = lim n f (x n ) = lim n f (x n). Next, we argue that that f is uniformly continuous. Given ε >, let δ > be such that d(x, ˆx) < δ implies d( f (x), f ( ˆx)) < ε, for all x, ˆx A. For any x, x Cl A with d(x, x ) < δ, we can choose two sequences {x n } n N and {x n} n N in A such that x n x and x n x and d(x n, x n) < δ, for all n. It follows that d( f (x n ), f (x n)) < ε, for all n, and, so, d( f (x), f (x )) ε, estalishing the uniform continuity of f on Cl A. Finally, given x A, we may take x n = x, for all n, to conclude that f (x) = f (x). The completeness of the space Y is crucial for the Proposition to work. To check that it is applicable in our case, we need the following result: Proposition (M 2,c, M 2,c ) is a Banach space. Proof. Simple properties of square-integrable martingales suffice to show that M 2,c is a linear space, and that M 2,c satisfies all the axioms of a norm. The more delicate matter is completeness. We start with a Cauchy sequence {M n } n N and conclude immediately that their last elements {M} n n N form a Cauchy sequence in L 2 (F). Thus, M n M in L 2, for some M L 2 (F). We define {M} t [,) as the RCLL version of the Lévy martingale M t = E[M F t ] and use Doob s inequality to get that sup Mt n M t 2 M L n M L 2, 2 t Hence, possibly through a subsequence, we have sup Mt n M t, a.s. t It follows that almost all trajectories of M are uniform limits of the corresponding trajectories of M n. By assumption, all M n are continuous martingales, and uniform limits of continuous functions are continuous, so M is a continuous martingale and M = lim n M n in M 2,c. If we combine the Itô s isometry (19.9) (which yields the uniform continuity of the map H H M on Hsimp ) with Proposition (which asserts that the target space is complete) and use Proposition we immediately conclude that the map H H M M 2,c Last Updated: March 29, 215
11 Lecture 19: L 2 -Stochastic integration 11 of 12 admits a unique linear and continuous extension of the simple predictable integral from H simp to its closure in L2 (M), with values in M 2,c. The only question still remaining is to give a nice description of the closure of Hsimp in L2 (M): Proposition For any M M 2,c, H simp is dense in L2 (M). Proof. Without loss of generality, we assume that E[ M ] = 1. Let { n } n N be a sequence of nested partitions in P [,) such that n Id; for concreteness, we take n = {k2 n : k n2 n }). For each n, let P n be the σ-algebra on the product [, ) Ω generated by all simple predictable processes H of the form H t = n2 n K k 1 (k2 n,(k+1)2 n ](t), K k F k2 n, for all k. (19.1) k= In fact, it is not hard to see that a process H is measurable with respect to P n, precisely when it is of the form (19.1). Also, since all left-continuous processes can be obtained as pointwise limits of the processes of the form (19.1), we have P = σ( n P n ), where P denotes the predictable σ-algebra. The product space Ω = [, ) Ω, together with the σ-algebra P, the filtration {P n } n N, and the probability measure P, given by P [A] = E[ 1 A (t, ω) d M t (ω)], for A P, forms a filtered probability space. Moreover, random variables on this space correspond precisely to the predictable processes on the original space, and their expectations to the expected d M -integrals, i.e., if E denotes the expectation under P, we have In particular, we have Given H L 2 (P ), define E [H] = E[ H t d M t ] where. H n H in L 2 (P ) iff H n H in L 2 (M). H n = E [ H P n ]. This is a square-integrable martingale and H P, so H n H in L 2 (P ). Therefore, by the above characterization of convergence in L 2 (M), we have H n H in L 2 (M), as required. Last Updated: March 29, 215
12 Lecture 19: L 2 -Stochastic integration 12 of 12 We can now summarize all our findings in a compact statement: Theorem Given M M 2,c, there exist a unique linear isometry L 2 (M) H H M M 2,c which extends the simple predictable integral (19.6). Last Updated: March 29, 215
Lecture 21 Representations of Martingales
Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let
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 informationThe Arzelà-Ascoli Theorem
John Nachbar Washington University March 27, 2016 The Arzelà-Ascoli Theorem The Arzelà-Ascoli Theorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationStochastic integration. P.J.C. Spreij
Stochastic integration P.J.C. Spreij this version: April 22, 29 Contents 1 Stochastic processes 1 1.1 General theory............................... 1 1.2 Stopping times...............................
More informationDoléans measures. Appendix C. C.1 Introduction
Appendix C Doléans measures C.1 Introduction Once again all random processes will live on a fixed probability space (Ω, F, P equipped with a filtration {F t : 0 t 1}. We should probably assume the filtration
More information16.1. Signal Process Observation Process The Filtering Problem Change of Measure
1. Introduction The following notes aim to provide a very informal introduction to Stochastic Calculus, and especially to the Itô integral. They owe a great deal to Dan Crisan s Stochastic Calculus and
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
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 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 informationStochastic Processes II/ Wahrscheinlichkeitstheorie III. Lecture Notes
BMS Basic Course Stochastic Processes II/ Wahrscheinlichkeitstheorie III Michael Scheutzow Lecture Notes Technische Universität Berlin Sommersemester 218 preliminary version October 12th 218 Contents
More informationTools from Lebesgue integration
Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given
More informationEconomics 204 Fall 2011 Problem Set 2 Suggested Solutions
Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit
More informationn E(X t T n = lim X s Tn = X s
Stochastic Calculus Example sheet - Lent 15 Michael Tehranchi Problem 1. Let X be a local martingale. Prove that X is a uniformly integrable martingale if and only X is of class D. Solution 1. If If direction:
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 informationOn pathwise stochastic integration
On pathwise stochastic integration Rafa l Marcin Lochowski Afican Institute for Mathematical Sciences, Warsaw School of Economics UWC seminar Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic
More informationElementary properties of functions with one-sided limits
Elementary properties of functions with one-sided limits Jason Swanson July, 11 1 Definition and basic properties Let (X, d) be a metric space. A function f : R X is said to have one-sided limits if, for
More informationPart III Stochastic Calculus and Applications
Part III Stochastic Calculus and Applications Based on lectures by R. Bauerschmidt Notes taken by Dexter Chua Lent 218 These notes are not endorsed by the lecturers, and I have modified them often significantly
More informationStochastic Calculus (Lecture #3)
Stochastic Calculus (Lecture #3) Siegfried Hörmann Université libre de Bruxelles (ULB) Spring 2014 Outline of the course 1. Stochastic processes in continuous time. 2. Brownian motion. 3. Itô integral:
More informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationStochastic Processes. Winter Term Paolo Di Tella Technische Universität Dresden Institut für Stochastik
Stochastic Processes Winter Term 2016-2017 Paolo Di Tella Technische Universität Dresden Institut für Stochastik Contents 1 Preliminaries 5 1.1 Uniform integrability.............................. 5 1.2
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationMA651 Topology. Lecture 10. Metric Spaces.
MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky
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 informationANALYSIS WORKSHEET II: METRIC SPACES
ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair
More informationd(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N
Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x
More informationMATH 202B - Problem Set 5
MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there
More informationCharacterisation of Accumulation Points. Convergence in Metric Spaces. Characterisation of Closed Sets. Characterisation of Closed Sets
Convergence in Metric Spaces Functional Analysis Lecture 3: Convergence and Continuity in Metric Spaces Bengt Ove Turesson September 4, 2016 Suppose that (X, d) is a metric space. A sequence (x n ) X is
More informationPart V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory
Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite
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 informationA NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES
A NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES STEFAN TAPPE Abstract. In a work of van Gaans (25a) stochastic integrals are regarded as L 2 -curves. In Filipović and Tappe (28) we have shown the connection
More informationStochastic Calculus. Alan Bain
Stochastic Calculus Alan Bain 1. Introduction The following notes aim to provide a very informal introduction to Stochastic Calculus, and especially to the Itô integral and some of its applications. They
More informationExercises to Applied Functional Analysis
Exercises to Applied Functional Analysis Exercises to Lecture 1 Here are some exercises about metric spaces. Some of the solutions can be found in my own additional lecture notes on Blackboard, as the
More informationFrom now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1.
Chapter 1 Metric spaces 1.1 Metric and convergence We will begin with some basic concepts. Definition 1.1. (Metric space) Metric space is a set X, with a metric satisfying: 1. d(x, y) 0, d(x, y) = 0 x
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 2 November 6 November Deadline to hand in the homeworks: your exercise class on week 9 November 13 November Exercises (1) Let X be the following space of piecewise
More information7 Complete metric spaces and function spaces
7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m
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 informationContinuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
More informationLecture 4 Lebesgue spaces and inequalities
Lecture 4: Lebesgue spaces and inequalities 1 of 10 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 4 Lebesgue spaces and inequalities Lebesgue spaces We have seen how
More information01. Review of metric spaces and point-set topology. 1. Euclidean spaces
(October 3, 017) 01. Review of metric spaces and point-set topology Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 017-18/01
More informationAnalysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t
Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using
More informationI. ANALYSIS; PROBABILITY
ma414l1.tex Lecture 1. 12.1.2012 I. NLYSIS; PROBBILITY 1. Lebesgue Measure and Integral We recall Lebesgue measure (M411 Probability and Measure) λ: defined on intervals (a, b] by λ((a, b]) := b a (so
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More information16 1 Basic Facts from Functional Analysis and Banach Lattices
16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach Steinhaus theorem, or the Uniform Boundedness
More information56 4 Integration against rough paths
56 4 Integration against rough paths comes to the definition of a rough integral we typically take W = LV, W ; although other choices can be useful see e.g. remark 4.11. In the context of rough differential
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationBasics of Stochastic Analysis
Basics of Stochastic Analysis c Timo Seppäläinen This version November 16, 214 Department of Mathematics, University of Wisconsin Madison, Madison, Wisconsin 5376 E-mail address: seppalai@math.wisc.edu
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationChapter 3: Baire category and open mapping theorems
MA3421 2016 17 Chapter 3: Baire category and open mapping theorems A number of the major results rely on completeness via the Baire category theorem. 3.1 The Baire category theorem 3.1.1 Definition. A
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 informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More information1.2 Fundamental Theorems of Functional Analysis
1.2 Fundamental Theorems of Functional Analysis 15 Indeed, h = ω ψ ωdx is continuous compactly supported with R hdx = 0 R and thus it has a unique compactly supported primitive. Hence fφ dx = f(ω ψ ωdy)dx
More informationAn essay on the general theory of stochastic processes
Probability Surveys Vol. 3 (26) 345 412 ISSN: 1549-5787 DOI: 1.1214/1549578614 An essay on the general theory of stochastic processes Ashkan Nikeghbali ETHZ Departement Mathematik, Rämistrasse 11, HG G16
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationLecture Notes. Functional Analysis in Applied Mathematics and Engineering. by Klaus Engel. University of L Aquila Faculty of Engineering
Lecture Notes Functional Analysis in Applied Mathematics and Engineering by Klaus Engel University of L Aquila Faculty of Engineering 2012-2013 http://univaq.it/~engel ( = %7E) (Preliminary Version of
More informationSummary of Real Analysis by Royden
Summary of Real Analysis by Royden Dan Hathaway May 2010 This document is a summary of the theorems and definitions and theorems from Part 1 of the book Real Analysis by Royden. In some areas, such as
More informationMeasures. Chapter Some prerequisites. 1.2 Introduction
Lecture notes Course Analysis for PhD students Uppsala University, Spring 2018 Rostyslav Kozhan Chapter 1 Measures 1.1 Some prerequisites I will follow closely the textbook Real analysis: Modern Techniques
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationFiltrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition
Filtrations, Markov Processes and Martingales Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition David pplebaum Probability and Statistics Department,
More informationContinuity of convex functions in normed spaces
Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
More informationNormed and Banach Spaces
(August 30, 2005) Normed and Banach Spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ We have seen that many interesting spaces of functions have natural structures of Banach spaces:
More informationB. Appendix B. Topological vector spaces
B.1 B. Appendix B. Topological vector spaces B.1. Fréchet spaces. In this appendix we go through the definition of Fréchet spaces and their inductive limits, such as they are used for definitions of function
More information(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε
1. Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
More informations P = f(ξ n )(x i x i 1 ). i=1
Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological
More informationHILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define
HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,
More informationContents. Index... 15
Contents Filter Bases and Nets................................................................................ 5 Filter Bases and Ultrafilters: A Brief Overview.........................................................
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 informationRisk-Minimality and Orthogonality of Martingales
Risk-Minimality and Orthogonality of Martingales Martin Schweizer Universität Bonn Institut für Angewandte Mathematik Wegelerstraße 6 D 53 Bonn 1 (Stochastics and Stochastics Reports 3 (199, 123 131 2
More informationMeasures. 1 Introduction. These preliminary lecture notes are partly based on textbooks by Athreya and Lahiri, Capinski and Kopp, and Folland.
Measures These preliminary lecture notes are partly based on textbooks by Athreya and Lahiri, Capinski and Kopp, and Folland. 1 Introduction Our motivation for studying measure theory is to lay a foundation
More informationClass Notes for MATH 255.
Class Notes for MATH 255. by S. W. Drury Copyright c 2006, by S. W. Drury. Contents 0 LimSup and LimInf Metric Spaces and Analysis in Several Variables 6. Metric Spaces........................... 6.2 Normed
More informationEconomics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011
Economics 204 Summer/Fall 2011 Lecture 5 Friday July 29, 2011 Section 2.6 (cont.) Properties of Real Functions Here we first study properties of functions from R to R, making use of the additional structure
More informationREAL AND COMPLEX ANALYSIS
REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any
More information1.1. MEASURES AND INTEGRALS
CHAPTER 1: MEASURE THEORY In this chapter we define the notion of measure µ on a space, construct integrals on this space, and establish their basic properties under limits. The measure µ(e) will be defined
More informationSome SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen
Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text
More informationFOUNDATIONS OF MARTINGALE THEORY AND STOCHASTIC CALCULUS FROM A FINANCE PERSPECTIVE
FOUNDATIONS OF MARTINGALE THEORY AND STOCHASTIC CALCULUS FROM A FINANCE PERSPECTIVE JOSEF TEICHMANN 1. Introduction The language of mathematical Finance allows to express many results of martingale theory
More informationMeasurable functions are approximately nice, even if look terrible.
Tel Aviv University, 2015 Functions of real variables 74 7 Approximation 7a A terrible integrable function........... 74 7b Approximation of sets................ 76 7c Approximation of functions............
More informationSpaces of continuous functions
Chapter 2 Spaces of continuous functions 2.8 Baire s Category Theorem Recall that a subset A of a metric space (X, d) is dense if for all x X there is a sequence from A converging to x. An equivalent definition
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 informationCLASS NOTES FOR APRIL 14, 2000
CLASS NOTES FOR APRIL 14, 2000 Announcement: Section 1.2, Questions 3,5 have been deferred from Assignment 1 to Assignment 2. Section 1.4, Question 5 has been dropped entirely. 1. Review of Wednesday class
More informationn [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1)
1.4. CONSTRUCTION OF LEBESGUE-STIELTJES MEASURES In this section we shall put to use the Carathéodory-Hahn theory, in order to construct measures with certain desirable properties first on the real line
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More informationPathwise Construction of Stochastic Integrals
Pathwise Construction of Stochastic Integrals Marcel Nutz First version: August 14, 211. This version: June 12, 212. Abstract We propose a method to construct the stochastic integral simultaneously under
More informationNotes 1 : Measure-theoretic foundations I
Notes 1 : Measure-theoretic foundations I Math 733-734: Theory of Probability Lecturer: Sebastien Roch References: [Wil91, Section 1.0-1.8, 2.1-2.3, 3.1-3.11], [Fel68, Sections 7.2, 8.1, 9.6], [Dur10,
More informationAn introduction to some aspects of functional analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms
More informationProbability Theory II. Spring 2016 Peter Orbanz
Probability Theory II Spring 2016 Peter Orbanz Contents Chapter 1. Martingales 1 1.1. Martingales indexed by partially ordered sets 1 1.2. Martingales from adapted processes 4 1.3. Stopping times and
More informationFUNCTIONAL ANALYSIS CHRISTIAN REMLING
FUNCTIONAL ANALYSIS CHRISTIAN REMLING Contents 1. Metric and topological spaces 2 2. Banach spaces 12 3. Consequences of Baire s Theorem 30 4. Dual spaces and weak topologies 34 5. Hilbert spaces 50 6.
More informationNotes for Functional Analysis
Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) September 29, 2015 1 Lecture 09 1.1 Equicontinuity First let s recall the conception of equicontinuity for family of functions that we learned
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationReal Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
More informationLecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differenti. equations.
Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differential equations. 1 Metric spaces 2 Completeness and completion. 3 The contraction
More informationMH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then
MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever
More informationINTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRATION
1 INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRATION Eduard EMELYANOV Ankara TURKEY 2007 2 FOREWORD This book grew out of a one-semester course for graduate students that the author have taught at
More informationMAA6617 COURSE NOTES SPRING 2014
MAA6617 COURSE NOTES SPRING 2014 19. Normed vector spaces Let X be a vector space over a field K (in this course we always have either K = R or K = C). Definition 19.1. A norm on X is a function : X K
More informationIntroduction to Functional Analysis
Introduction to Functional Analysis Carnegie Mellon University, 21-640, Spring 2014 Acknowledgements These notes are based on the lecture course given by Irene Fonseca but may differ from the exact lecture
More information