Homework #6 : final examination Due on March 22nd : individual work

Size: px
Start display at page:

Download "Homework #6 : final examination Due on March 22nd : individual work"

Transcription

1 Université de ennes Année Master 2ème Mathématiques Modèles stochastiques continus ou à sauts Homework #6 : final examination Due on March 22nd : individual work Exercise Warm-up : behaviour of characteristic exponent Let ηu be the characteristic exponent of a real-valued Lévy process : ηu = ibu [ 2 Γu2 + e iuy ] iuy { y <} νdy. ηu [. Show that lim u u 2 = Γ. One can show that lim e iuy ] u u 2 iuy { y <} νdy =. 2. Show that X has bounded variations on every time interval a.s. if and only if Γ = and y νdy <. Use Lévy-Itô decomposition or the exponential formula for PM. In that case lim u ηu u 2 convergence as in the rst point. = d, where d = b y νdy is the drift. Use the dominated 3. Show that the characteristic exponent η is bounded if and only if X is a compound Poisson process. If you assume that η is bounded, show that e ηu = cosuyνdy and that e ty2 /2 νdy = 2πt e λ2 /2t e ηλdλ supe ηλ. λ Deduce further that X has bounded variations in the sense of the preceding point. Exercise 2 First round : subordinators A non-decreasing Lévy process with values in + is called a subordinator.. Show that a real valued Lévy process X = X t : t is a subordinator if its characteristic exponent has the form ηu = ibu + e iyu νdy, where b and the Lévy measure has the support in + and satises y νdy, or equivalently y νdy <. 2 + y Moreover X t = tb + s t X s. Use the positivity and the monotonicity to prove that X contains no Brownian part. Then use the existence of all moments of the third term in the Lévy-Itô decomposition of X to obtain 2.

2 2. Show that for a subordinator where the Laplace exponent φλ = ηiλ = bλ + E e λxt = e tφλ, 3 Analyse the analytic continuation of u E e iuxt. e λy νdy A subordinator X is called one-sided stable process if for each a there corresponds a constant ca such that ax t and X t ca have the same law. a Show that c in this denition is continuous ans satises the equation caã = cacã. Then deduce that ca = a α, with some α > the index. b Deduce further that φa = caφ and hence E e λxt = e t r λα, r > the rate. 5 By using the concavity of φ deduce that α,. c Prove or assume that for α,, e λy Γ α dy = λ α, y+α α where here Γ is the Euler's gamma function. Deduce that the stable subordinators with index α and rate r have the Laplace exponent 4 with Lévy measure νdy = αr Γ α. 6 y+α d An example : the stable subordinator with index 2 and rate have the probability density t f Xt s := 2 s 3/2 e t2 /4s, s. π Indeed, set g t λ := E e λxt = e λs f Xt sds, prove that g tλ = t/2 λg t λ, that g t = and deduce that g t λ = e tλ/2. Exercise 3 ising scale : transience and recurrence Let X = X t : t be a Lévy process with the characteristic exponent η. Denote by P t the associated semigroup, given by P t fx = fx + ypx t dy, with f a non-negative measurable function. ecall that the resolvent λ is given, for measurable f by λ fx = e λt P t fxdt = E x e λt fx t dt.. Let τ an exponential random time with parameter λ. Show that E fx τ = λ λ f. 2

3 2. Denote the Fourier transform of a function g L by Fgu = e i u x gxdx attention, not the same denition as in the course. Show that for every f L L, FP t fu = exp { tη u } Ffu, and F λ fu = λ + η u Ffu, 7 where t and λ >. Moreover if A denotes the generator of P t and D its domain, show that if f D and Af L, then FAfu = η uffu. 3. Let us introduce a familiy of measures called the potential measures {Ux, : x } given, for B B, by Ux, B = P x X t Bdt = E x {Xt B}dt [, ]. If T B = inf{t : X t B} denotes the rst entrance time into B, show that Ux, B = E x {Xt B}dt = Uy, BP x X TB dy, 8 where B is the closure of B. T B 4. We say that the process X is transient if the for every compact set K, Ux, K <, x, or equivalent if U, K < since Ux, K = U, K {x}. Here we denoted by B B = {x x : x B, x B }. We say that a process is recurrent if U, B = for every open ball B centered in. We want to prove that the process is either transient or recurrent. a Suppose that ε > such that U, B <, where B = ε, ε, and let B = [ ε 3, ε 3]. Justify the following relations Ux, B sup y B Uy, B = sup y B U, B {y} U, B B U, B <. Use 8 for the rst inequality. b Deduce that for every y, Ux, {y} + B < and then Ux, K < for any compact K. 5. Test of transience : the Lévy process X is transient i for some r > small enough lim sup λ r r e B λ + ηu du <. 9 a For r > arbitrary small consider f = [ r,r] [ r,r] the convolution. Clearly it can be proved seen that f 2r [ 2r,2r] is continuous non-negative with support [ 2r, 2r] and also that { [2/u sinru] 2 if u Ffu = 4r 2 otherwise 3

4 is a bounded continuous non-negative function. Show that for λ >, λ f = [ 2 ] 2e 2π u sinru du. λ + η u Use Fourier inversion, 7 and the fact that λ f is a real number. b Deduce that and latter quantity is innite whenever and then X is recurrent. 2rU, [ 2r, 2r] lim λ λ f, lim sup λ c Conversely, assume that for r >, Set r r e 2r lim sup e λ 2r λ + ηu λ + ηu du = du <. { [2/x sinrx] 2 if x gx = fu = 4r 2 otherwise having its Fourier transform Fgu = 2π [ r,r] [ r,r]. Deduce un expression of λ g. One can use the same argument as in the previous point. d Prove that U, [ π 3r, π 3r ] <. For instance one can prove and use that gx r2, when x π 3r. Conclude that X is transient. Exercise 4 Final step : pathwise uniqueness Let X = X t : t be a one-dimensional symmetric stable with index α, 2 having its Lévy measure given by νdz = z α on and its generator Lfx = [fx + z fx { z } zf x] z α dz, for C 2 functions.. Set X n t = s t X s { Xs n}. Show that X n is a Lévy process and give its Lévy measure, then show that X n is a square integrable martingale. 2. Let H t be a bounded predictable process. Show that Zt n := t H sdxs n is a square integrable martingale. 3. Set U n t = X t X n t. Show that E t H s dus n CE X s { Xs >n}, for some constant C. Show that the right hand side of the latter inequality is nite and tends to, as n. 4. Deduce that the process Z t = t H sdx s is a martingale. 4 s t

5 5. Let f be a C 2 b function with bounded rst and second derivatives and set Ks, z := fz s + H s z fz s f Z s H s z. Justify each of following equalities = fz + where M t = fz t = fz + t t f Z s dz s + f Z s dz s + t t t with intensity measure dtνdz. 6. Prove that for each m, V m t = f Z s dz s + s t[fz s fz s f Z s Z s ] t Ks, znds, dz = fz + M t + t Ks, zdsνdz, Ks, zñds, dz. Here N is the PM associated to X z m Ks, zñds, dz is a martingale and that M t is the limit of martingales t f Z s dz s + Vt m. One can use the fact that for each k > m, Vt k Vt m is a martingale and show that t E Ks, z Nds, dz + dsνdz C m α, m< z for some constant C not depending on m. 7. Show that, if H s, we have t Ks, zdsνdz = H s α LfZ s. One should come back to the expression of K, perform the change of variable w = H s z and recall the expression of L. Conclude that even if H s =, fz t = fz + M t + 8. We study the uniqueness of the following SDE t where F is supposed bounded continuous such that H s α LfZ s ds. dy t = F Y t dx t, Y = y, 2 F x F y ρ x y, x, y, 3 with ρ : [, a non-decreasing continuous function, ρ =. More precisely we try to prove that if dx =, 4 ρx α then the solution of the SDE 2 is pathwise unique. + a Let Y and Y 2 be any two solutions of 2 set Z t = Yt Yt 2 and H t = F Yt F Y 2 Deduce that Z t = t H sdx s. b Dene A t = t H s α ds. Justify that A t < use the fact that F is bounded. t. 5

6 c Let a n such that a n a n+ ρx α dx = n. For each n let h n be a non-negative C 2 function with the support in [a n+, a n ], whose integral is and with h n x 2/nρx α. Why is this possible? d Denote by p t x, y the transition density for X t, that is the density of P x X t dy. Fix λ >, let r λ x = e λt p t x, dt and λ fx = fyr λ x ydy. Who is λ here? It can be proved that r λ x is bounded and is continuous in x admitted. Furthermore r λ x < r λ, if x. Finally, set f n = λ h n x. Show that if h n is C 2, then f n is C 2. One can interchanges dierentiation and integration and uses translation invariance. e Show that Lf n = L λ h n = λ λ h n h n = λf n h n. f Use Itô's product formula and to deduce E e λat f n Z t f n = E t g Conclude from the preceding two points that f n E e λat f n Z t = E e λas H s α Lf n Z s ds E t t e λas H s α h n Z s ds. e λas λ H s α f n Z s ds. Show, by using the properties of h n and the fact that H s ρz s, that the right hand side of the latter equality is less that 2t/n hence tends to, as n. h Justify that h n ydy δ weakly, as n and that f n x r λ x as n. Show that r λ E e λat r λ Z t =. i Conclude that PZ t = = for each t hence Z is identically. Exercice Bonus Last word : another approach to pathwise uniqueness Consider the same SDE 2 as in 8 of the Exercise 4 and make the same assumptions on F and ρ. ecall that N is the PM associated to X with intensity measure dtνdz.. Explain why the equation 2 can be written as Y t = y + t z t F X s zñds, dz + z > F X s znds, dz Consider again the sequence a n such that a n a n+ ρx α dx = n and also h n non-negative C 2 even functions with the support in [a n+, a n ], whose integral is and with h n x 2/nρx α. Set ux = x α and u n = u h n. Justify that u n s ux, as n. 3. In this question we prove that Lu n = cu n with c a constant independent of n. Set u ɛ x = uxe ɛ x and u ɛ n = u ɛ h n. The functions u ɛ n belongs to S. a Use Exercise 3 point 2, with same notations, and show FLu ɛ nξ = cα ξ α Fu ɛ nξ. b Show that Fu ɛ nξ = c α [ ɛ iξ α + ɛ iξ α] Fh n ξ. c For ξ compute lim ɛ [ ɛ iξ α + ɛ ξ α] and deduce Lu n = lim ɛ Lu ɛ n = c u n. 6

7 4. Denote again Y and Y 2 two solutions of 2 and set Z t = Yt Yt 2. Show that + t u n Z t u n = t F Y s F Y 2 s α Lu n Z s ds [ u n Zs + F Y s F Y 2 s z u n Z s ]Ñds, dz. 5. Show that F x F y α Lu n x, y cρ x y α h n x y c/n. 6. Set, for k, T k = inf{t > : Z t > k}. Deduce that [ ] [ t T k E u n Z t Tk u n + E Conclude that E [ Z t Tk α ] = and then then Z t = a.s. c ] n ds. 7

Definition: Lévy Process. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes. Theorem

Definition: Lévy Process. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes. Theorem Definition: Lévy Process Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes David Applebaum Probability and Statistics Department, University of Sheffield, UK July

More information

Reflected Brownian Motion

Reflected Brownian Motion Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide

More information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: 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 information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: 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 information

Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3

Brownian 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 information

Hardy-Stein identity and Square functions

Hardy-Stein identity and Square functions Hardy-Stein identity and Square functions Daesung Kim (joint work with Rodrigo Bañuelos) Department of Mathematics Purdue University March 28, 217 Daesung Kim (Purdue) Hardy-Stein identity UIUC 217 1 /

More information

Singular Integrals. 1 Calderon-Zygmund decomposition

Singular Integrals. 1 Calderon-Zygmund decomposition Singular Integrals Analysis III Calderon-Zygmund decomposition Let f be an integrable function f dx 0, f = g + b with g Cα almost everywhere, with b

More information

Stochastic Integration.

Stochastic Integration. Chapter Stochastic Integration..1 Brownian Motion as a Martingale P is the Wiener measure on (Ω, B) where Ω = C, T B is the Borel σ-field on Ω. In addition we denote by B t the σ-field generated by x(s)

More information

A slow transient diusion in a drifted stable potential

A slow transient diusion in a drifted stable potential A slow transient diusion in a drifted stable potential Arvind Singh Université Paris VI Abstract We consider a diusion process X in a random potential V of the form V x = S x δx, where δ is a positive

More information

Exercises. T 2T. e ita φ(t)dt.

Exercises. 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 information

Harmonic Functions and Brownian motion

Harmonic Functions and Brownian motion Harmonic Functions and Brownian motion Steven P. Lalley April 25, 211 1 Dynkin s Formula Denote by W t = (W 1 t, W 2 t,..., W d t ) a standard d dimensional Wiener process on (Ω, F, P ), and let F = (F

More information

Poisson random measure: motivation

Poisson 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 information

ELEMENTS OF PROBABILITY THEORY

ELEMENTS OF PROBABILITY THEORY ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable

More information

Stochastic Processes

Stochastic Processes Stochastic Processes A very simple introduction Péter Medvegyev 2009, January Medvegyev (CEU) Stochastic Processes 2009, January 1 / 54 Summary from measure theory De nition (X, A) is a measurable space

More information

Lecture 12. F o s, (1.1) F t := s>t

Lecture 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 information

n E(X t T n = lim X s Tn = X s

n 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

Potential Theory of Subordinate Brownian Motions Revisited

Potential Theory of Subordinate Brownian Motions Revisited Potential Theory of Subordinate Brownian Motions Revisited Panki Kim Renming Song and Zoran Vondraček Abstract The paper discusses and surveys some aspects of the potential theory of subordinate Brownian

More information

Filtrations, 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 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 information

Jump Processes. Richard F. Bass

Jump Processes. Richard F. Bass Jump Processes Richard F. Bass ii c Copyright 214 Richard F. Bass Contents 1 Poisson processes 1 1.1 Definitions............................. 1 1.2 Stopping times.......................... 3 1.3 Markov

More information

Introduction to Random Diffusions

Introduction to Random Diffusions Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales

More information

TD M1 EDP 2018 no 2 Elliptic equations: regularity, maximum principle

TD M1 EDP 2018 no 2 Elliptic equations: regularity, maximum principle TD M EDP 08 no Elliptic equations: regularity, maximum principle Estimates in the sup-norm I Let be an open bounded subset of R d of class C. Let A = (a ij ) be a symmetric matrix of functions of class

More information

4.6 Example of non-uniqueness.

4.6 Example of non-uniqueness. 66 CHAPTER 4. STOCHASTIC DIFFERENTIAL EQUATIONS. 4.6 Example of non-uniqueness. If we try to construct a solution to the martingale problem in dimension coresponding to a(x) = x α with

More information

Potential theory of subordinate killed Brownian motions

Potential theory of subordinate killed Brownian motions Potential theory of subordinate killed Brownian motions Renming Song University of Illinois AMS meeting, Indiana University, April 2, 2017 References This talk is based on the following paper with Panki

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 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 information

Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions

Iowa State University. Instructor: Alex Roitershtein Summer Homework #5. Solutions Math 50 Iowa State University Introduction to Real Analysis Department of Mathematics Instructor: Alex Roitershtein Summer 205 Homework #5 Solutions. Let α and c be real numbers, c > 0, and f is defined

More information

Harnack Inequalities and Applications for Stochastic Equations

Harnack 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 information

BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, March 2009

BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, March 2009 BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, 16-20 March 2009 1 1) L p viscosity solution to 2nd order semilinear parabolic PDEs

More information

A Representation of Excessive Functions as Expected Suprema

A Representation of Excessive Functions as Expected Suprema A Representation of Excessive Functions as Expected Suprema Hans Föllmer & Thomas Knispel Humboldt-Universität zu Berlin Institut für Mathematik Unter den Linden 6 10099 Berlin, Germany E-mail: foellmer@math.hu-berlin.de,

More information

Fractional Laplacian

Fractional Laplacian Fractional Laplacian Grzegorz Karch 5ème Ecole de printemps EDP Non-linéaire Mathématiques et Interactions: modèles non locaux et applications Ecole Supérieure de Technologie d Essaouira, du 27 au 30 Avril

More information

Stochastic Process (ENPC) Monday, 22nd of January 2018 (2h30)

Stochastic Process (ENPC) Monday, 22nd of January 2018 (2h30) Stochastic Process (NPC) Monday, 22nd of January 208 (2h30) Vocabulary (english/français) : distribution distribution, loi ; positive strictement positif ; 0,) 0,. We write N Z,+ and N N {0}. We use the

More information

I forgot to mention last time: in the Ito formula for two standard processes, putting

I forgot to mention last time: in the Ito formula for two standard processes, putting I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy

More information

Solution. 1 Solution of Homework 7. Sangchul Lee. March 22, Problem 1.1

Solution. 1 Solution of Homework 7. Sangchul Lee. March 22, Problem 1.1 Solution Sangchul Lee March, 018 1 Solution of Homework 7 Problem 1.1 For a given k N, Consider two sequences (a n ) and (b n,k ) in R. Suppose that a n b n,k for all n,k N Show that limsup a n B k :=

More information

lim n C1/n n := ρ. [f(y) f(x)], y x =1 [f(x) f(y)] [g(x) g(y)]. (x,y) E A E(f, f),

lim n C1/n n := ρ. [f(y) f(x)], y x =1 [f(x) f(y)] [g(x) g(y)]. (x,y) E A E(f, f), 1 Part I Exercise 1.1. Let C n denote the number of self-avoiding random walks starting at the origin in Z of length n. 1. Show that (Hint: Use C n+m C n C m.) lim n C1/n n = inf n C1/n n := ρ.. Show that

More information

Multivariate Generalized Ornstein-Uhlenbeck Processes

Multivariate Generalized Ornstein-Uhlenbeck Processes Multivariate Generalized Ornstein-Uhlenbeck Processes Anita Behme TU München Alexander Lindner TU Braunschweig 7th International Conference on Lévy Processes: Theory and Applications Wroclaw, July 15 19,

More information

Markov processes Course note 2. Martingale problems, recurrence properties of discrete time chains.

Markov processes Course note 2. Martingale problems, recurrence properties of discrete time chains. Institute for Applied Mathematics WS17/18 Massimiliano Gubinelli Markov processes Course note 2. Martingale problems, recurrence properties of discrete time chains. [version 1, 2017.11.1] We introduce

More information

Feller Processes and Semigroups

Feller Processes and Semigroups Stat25B: Probability Theory (Spring 23) Lecture: 27 Feller Processes and Semigroups Lecturer: Rui Dong Scribe: Rui Dong ruidong@stat.berkeley.edu For convenience, we can have a look at the list of materials

More information

1. Stochastic Processes and filtrations

1. 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 information

p 1 ( Y p dp) 1/p ( X p dp) 1 1 p

p 1 ( Y p dp) 1/p ( X p dp) 1 1 p Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p

More information

Some Properties of NSFDEs

Some Properties of NSFDEs Chenggui Yuan (Swansea University) Some Properties of NSFDEs 1 / 41 Some Properties of NSFDEs Chenggui Yuan Swansea University Chenggui Yuan (Swansea University) Some Properties of NSFDEs 2 / 41 Outline

More information

Selected Exercises on Expectations and Some Probability Inequalities

Selected Exercises on Expectations and Some Probability Inequalities Selected Exercises on Expectations and Some Probability Inequalities # If E(X 2 ) = and E X a > 0, then P( X λa) ( λ) 2 a 2 for 0 < λ

More information

Martingale Problems. Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi

Martingale Problems. Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi s Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi Lectures on Probability and Stochastic Processes III Indian Statistical Institute, Kolkata 20 24 November

More information

1 Solutions to selected problems

1 Solutions to selected problems 1 Solutions to selected problems 1. Let A B R n. Show that int A int B but in general bd A bd B. Solution. Let x int A. Then there is ɛ > 0 such that B ɛ (x) A B. This shows x int B. If A = [0, 1] and

More information

Kolmogorov s Forward, Basic Results

Kolmogorov s Forward, Basic Results 978--521-88651-2 - Partial Differential Equations for Probabilists chapter 1 Kolmogorov s Forward, Basic Results The primary purpose of this chapter is to present some basic existence and uniqueness results

More information

Sobolev Spaces. Chapter 10

Sobolev Spaces. Chapter 10 Chapter 1 Sobolev Spaces We now define spaces H 1,p (R n ), known as Sobolev spaces. For u to belong to H 1,p (R n ), we require that u L p (R n ) and that u have weak derivatives of first order in L p

More information

Brownian Motion and Stochastic Calculus

Brownian Motion and Stochastic Calculus ETHZ, Spring 17 D-MATH Prof Dr Martin Larsson Coordinator A Sepúlveda Brownian Motion and Stochastic Calculus Exercise sheet 6 Please hand in your solutions during exercise class or in your assistant s

More information

STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, = (2n 1)(2n 3) 3 1.

STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, = (2n 1)(2n 3) 3 1. STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, 26 Problem Normal Moments (A) Use the Itô formula and Brownian scaling to check that the even moments of the normal distribution

More information

MAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.

MAS3706 Topology. Revision Lectures, May I do not answer  enquiries as to what material will be in the exam. MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar

More information

Calculus of Variations. Final Examination

Calculus of Variations. Final Examination Université Paris-Saclay M AMS and Optimization January 18th, 018 Calculus of Variations Final Examination Duration : 3h ; all kind of paper documents (notes, books...) are authorized. The total score of

More information

Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process

Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process Asymptotic properties of maximum likelihood estimator for the growth rate for a jump-type CIR process Mátyás Barczy University of Debrecen, Hungary The 3 rd Workshop on Branching Processes and Related

More information

On continuous time contract theory

On 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 information

MATH 6337: Homework 8 Solutions

MATH 6337: Homework 8 Solutions 6.1. MATH 6337: Homework 8 Solutions (a) Let be a measurable subset of 2 such that for almost every x, {y : (x, y) } has -measure zero. Show that has measure zero and that for almost every y, {x : (x,

More information

Exercises: Brunn, Minkowski and convex pie

Exercises: Brunn, Minkowski and convex pie Lecture 1 Exercises: Brunn, Minkowski and convex pie Consider the following problem: 1.1 Playing a convex pie Consider the following game with two players - you and me. I am cooking a pie, which should

More information

In this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,

In this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots, Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical

More information

Math 361: Homework 1 Solutions

Math 361: Homework 1 Solutions January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x

More information

Properties of an infinite dimensional EDS system : the Muller s ratchet

Properties of an infinite dimensional EDS system : the Muller s ratchet Properties of an infinite dimensional EDS system : the Muller s ratchet LATP June 5, 2011 A ratchet source : wikipedia Plan 1 Introduction : The model of Haigh 2 3 Hypothesis (Biological) : The population

More information

MATH34032 Mid-term Test 10.00am 10.50am, 26th March 2010 Answer all six question [20% of the total mark for this course]

MATH34032 Mid-term Test 10.00am 10.50am, 26th March 2010 Answer all six question [20% of the total mark for this course] MATH3432: Green s Functions, Integral Equations and the Calculus of Variations 1 MATH3432 Mid-term Test 1.am 1.5am, 26th March 21 Answer all six question [2% of the total mark for this course] Qu.1 (a)

More information

Stability of Stochastic Differential Equations

Stability of Stochastic Differential Equations Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

More information

Math The Laplacian. 1 Green s Identities, Fundamental Solution

Math The Laplacian. 1 Green s Identities, Fundamental Solution Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external

More information

Simulation methods for stochastic models in chemistry

Simulation methods for stochastic models in chemistry Simulation methods for stochastic models in chemistry David F. Anderson anderson@math.wisc.edu Department of Mathematics University of Wisconsin - Madison SIAM: Barcelona June 4th, 21 Overview 1. Notation

More information

HARMONIC ANALYSIS. Date:

HARMONIC ANALYSIS. Date: HARMONIC ANALYSIS Contents. Introduction 2. Hardy-Littlewood maximal function 3. Approximation by convolution 4. Muckenhaupt weights 4.. Calderón-Zygmund decomposition 5. Fourier transform 6. BMO (bounded

More information

PUTNAM PROBLEMS DIFFERENTIAL EQUATIONS. First Order Equations. p(x)dx)) = q(x) exp(

PUTNAM PROBLEMS DIFFERENTIAL EQUATIONS. First Order Equations. p(x)dx)) = q(x) exp( PUTNAM PROBLEMS DIFFERENTIAL EQUATIONS First Order Equations 1. Linear y + p(x)y = q(x) Muliply through by the integrating factor exp( p(x)) to obtain (y exp( p(x))) = q(x) exp( p(x)). 2. Separation of

More information

Theoretical Tutorial Session 2

Theoretical Tutorial Session 2 1 / 36 Theoretical Tutorial Session 2 Xiaoming Song Department of Mathematics Drexel University July 27, 216 Outline 2 / 36 Itô s formula Martingale representation theorem Stochastic differential equations

More information

Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula

Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula Group 4: Bertan Yilmaz, Richard Oti-Aboagye and Di Liu May, 15 Chapter 1 Proving Dynkin s formula

More information

Solution for Problem 7.1. We argue by contradiction. If the limit were not infinite, then since τ M (ω) is nondecreasing we would have

Solution for Problem 7.1. We argue by contradiction. If the limit were not infinite, then since τ M (ω) is nondecreasing we would have 362 Problem Hints and Solutions sup g n (ω, t) g(ω, t) sup g(ω, s) g(ω, t) µ n (ω). t T s,t: s t 1/n By the uniform continuity of t g(ω, t) on [, T], one has for each ω that µ n (ω) as n. Two applications

More information

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( ) Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical

More information

Mandelbrot s cascade in a Random Environment

Mandelbrot s cascade in a Random Environment Mandelbrot s cascade in a Random Environment A joint work with Chunmao Huang (Ecole Polytechnique) and Xingang Liang (Beijing Business and Technology Univ) Université de Bretagne-Sud (Univ South Brittany)

More information

1 Independent increments

1 Independent increments Tel Aviv University, 2008 Brownian motion 1 1 Independent increments 1a Three convolution semigroups........... 1 1b Independent increments.............. 2 1c Continuous time................... 3 1d Bad

More information

Some remarks on special subordinators

Some remarks on special subordinators Some remarks on special subordinators Renming Song Department of Mathematics University of Illinois, Urbana, IL 6181 Email: rsong@math.uiuc.edu and Zoran Vondraček Department of Mathematics University

More information

Uniformly Uniformly-ergodic Markov chains and BSDEs

Uniformly Uniformly-ergodic Markov chains and BSDEs Uniformly Uniformly-ergodic Markov chains and BSDEs Samuel N. Cohen Mathematical Institute, University of Oxford (Based on joint work with Ying Hu, Robert Elliott, Lukas Szpruch) Centre Henri Lebesgue,

More information

Additive Lévy Processes

Additive Lévy Processes Additive Lévy Processes Introduction Let X X N denote N independent Lévy processes on. We can construct an N-parameter stochastic process, indexed by N +, as follows: := X + + X N N for every := ( N )

More information

CIMPA SCHOOL, 2007 Jump Processes and Applications to Finance Monique Jeanblanc

CIMPA SCHOOL, 2007 Jump Processes and Applications to Finance Monique Jeanblanc CIMPA SCHOOL, 27 Jump Processes and Applications to Finance Monique Jeanblanc 1 Jump Processes I. Poisson Processes II. Lévy Processes III. Jump-Diffusion Processes IV. Point Processes 2 I. Poisson Processes

More information

Exponential functionals of Lévy processes

Exponential functionals of Lévy processes Exponential functionals of Lévy processes Víctor Rivero Centro de Investigación en Matemáticas, México. 1/ 28 Outline of the talk Introduction Exponential functionals of spectrally positive Lévy processes

More information

ERRATA: Probabilistic Techniques in Analysis

ERRATA: Probabilistic Techniques in Analysis ERRATA: Probabilistic Techniques in Analysis ERRATA 1 Updated April 25, 26 Page 3, line 13. A 1,..., A n are independent if P(A i1 A ij ) = P(A 1 ) P(A ij ) for every subset {i 1,..., i j } of {1,...,

More information

u xx + u yy = 0. (5.1)

u xx + u yy = 0. (5.1) Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function

More information

Nonlinear representation, backward SDEs, and application to the Principal-Agent problem

Nonlinear representation, backward SDEs, and application to the Principal-Agent problem Nonlinear representation, backward SDEs, and application to the Principal-Agent problem Ecole Polytechnique, France April 4, 218 Outline The Principal-Agent problem Formulation 1 The Principal-Agent problem

More information

Existence and uniqueness of solutions for nonlinear ODEs

Existence and uniqueness of solutions for nonlinear ODEs Chapter 4 Existence and uniqueness of solutions for nonlinear ODEs In this chapter we consider the existence and uniqueness of solutions for the initial value problem for general nonlinear ODEs. Recall

More information

Elliptic Operators with Unbounded Coefficients

Elliptic Operators with Unbounded Coefficients Elliptic Operators with Unbounded Coefficients Federica Gregorio Universitá degli Studi di Salerno 8th June 2018 joint work with S.E. Boutiah, A. Rhandi, C. Tacelli Motivation Consider the Stochastic Differential

More information

Richard F. Bass Krzysztof Burdzy University of Washington

Richard F. Bass Krzysztof Burdzy University of Washington ON DOMAIN MONOTONICITY OF THE NEUMANN HEAT KERNEL Richard F. Bass Krzysztof Burdzy University of Washington Abstract. Some examples are given of convex domains for which domain monotonicity of the Neumann

More information

Math 172 Problem Set 5 Solutions

Math 172 Problem Set 5 Solutions Math 172 Problem Set 5 Solutions 2.4 Let E = {(t, x : < x b, x t b}. To prove integrability of g, first observe that b b b f(t b b g(x dx = dt t dx f(t t dtdx. x Next note that f(t/t χ E is a measurable

More information

for all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true

for all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true 3 ohn Nirenberg inequality, Part I A function ϕ L () belongs to the space BMO() if sup ϕ(s) ϕ I I I < for all subintervals I If the same is true for the dyadic subintervals I D only, we will write ϕ BMO

More information

LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION

LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION We will define local time for one-dimensional Brownian motion, and deduce some of its properties. We will then use the generalized Ray-Knight theorem proved in

More information

8 Singular Integral Operators and L p -Regularity Theory

8 Singular Integral Operators and L p -Regularity Theory 8 Singular Integral Operators and L p -Regularity Theory 8. Motivation See hand-written notes! 8.2 Mikhlin Multiplier Theorem Recall that the Fourier transformation F and the inverse Fourier transformation

More information

OBSTACLE PROBLEMS FOR NONLOCAL OPERATORS: A BRIEF OVERVIEW

OBSTACLE PROBLEMS FOR NONLOCAL OPERATORS: A BRIEF OVERVIEW OBSTACLE PROBLEMS FOR NONLOCAL OPERATORS: A BRIEF OVERVIEW DONATELLA DANIELLI, ARSHAK PETROSYAN, AND CAMELIA A. POP Abstract. In this note, we give a brief overview of obstacle problems for nonlocal operators,

More information

Martin boundary for some symmetric Lévy processes

Martin boundary for some symmetric Lévy processes Martin boundary for some symmetric Lévy processes Panki Kim Renming Song and Zoran Vondraček Abstract In this paper we study the Martin boundary of open sets with respect to a large class of purely discontinuous

More information

On the submartingale / supermartingale property of diffusions in natural scale

On the submartingale / supermartingale property of diffusions in natural scale On the submartingale / supermartingale property of diffusions in natural scale Alexander Gushchin Mikhail Urusov Mihail Zervos November 13, 214 Abstract Kotani 5 has characterised the martingale property

More information

Exercises in stochastic analysis

Exercises 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 information

MATH 6605: SUMMARY LECTURE NOTES

MATH 6605: SUMMARY LECTURE NOTES MATH 6605: SUMMARY LECTURE NOTES These notes summarize the lectures on weak convergence of stochastic processes. If you see any typos, please let me know. 1. Construction of Stochastic rocesses A stochastic

More information

MATH 425, FINAL EXAM SOLUTIONS

MATH 425, FINAL EXAM SOLUTIONS MATH 425, FINAL EXAM SOLUTIONS Each exercise is worth 50 points. Exercise. a The operator L is defined on smooth functions of (x, y by: Is the operator L linear? Prove your answer. L (u := arctan(xy u

More information

Tyler Hofmeister. University of Calgary Mathematical and Computational Finance Laboratory

Tyler Hofmeister. University of Calgary Mathematical and Computational Finance Laboratory JUMP PROCESSES GENERALIZING STOCHASTIC INTEGRALS WITH JUMPS Tyler Hofmeister University of Calgary Mathematical and Computational Finance Laboratory Overview 1. General Method 2. Poisson Processes 3. Diffusion

More information

Errata for Stochastic Calculus for Finance II Continuous-Time Models September 2006

Errata for Stochastic Calculus for Finance II Continuous-Time Models September 2006 1 Errata for Stochastic Calculus for Finance II Continuous-Time Models September 6 Page 6, lines 1, 3 and 7 from bottom. eplace A n,m by S n,m. Page 1, line 1. After Borel measurable. insert the sentence

More information

A Lévy-Fokker-Planck equation: entropies and convergence to equilibrium

A Lévy-Fokker-Planck equation: entropies and convergence to equilibrium 1/ 22 A Lévy-Fokker-Planck equation: entropies and convergence to equilibrium I. Gentil CEREMADE, Université Paris-Dauphine International Conference on stochastic Analysis and Applications Hammamet, Tunisia,

More information

STOCHASTIC ANALYSIS FOR JUMP PROCESSES

STOCHASTIC ANALYSIS FOR JUMP PROCESSES STOCHASTIC ANALYSIS FOR JUMP PROCESSES ANTONIS PAPAPANTOLEON Abstract. Lecture notes from courses at TU Berlin in WS 29/1, WS 211/12 and WS 212/13. Contents 1. Introduction 2 2. Definition of Lévy processes

More information

Convergence of Feller Processes

Convergence of Feller Processes Chapter 15 Convergence of Feller Processes This chapter looks at the convergence of sequences of Feller processes to a iting process. Section 15.1 lays some ground work concerning weak convergence of processes

More information

Intertwinings for Markov processes

Intertwinings for Markov processes Intertwinings for Markov processes Aldéric Joulin - University of Toulouse Joint work with : Michel Bonnefont - Univ. Bordeaux Workshop 2 Piecewise Deterministic Markov Processes ennes - May 15-17, 2013

More information

A new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009

A new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009 A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1 Performance

More information

Green s Functions and Distributions

Green s Functions and Distributions CHAPTER 9 Green s Functions and Distributions 9.1. Boundary Value Problems We would like to study, and solve if possible, boundary value problems such as the following: (1.1) u = f in U u = g on U, where

More information

Stochastic Calculus February 11, / 33

Stochastic Calculus February 11, / 33 Martingale Transform M n martingale with respect to F n, n =, 1, 2,... σ n F n (σ M) n = n 1 i= σ i(m i+1 M i ) is a Martingale E[(σ M) n F n 1 ] n 1 = E[ σ i (M i+1 M i ) F n 1 ] i= n 2 = σ i (M i+1 M

More information

Lecture 17 Brownian motion as a Markov process

Lecture 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 information

where F denoting the force acting on V through V, ν is the unit outnormal on V. Newton s law says (assume the mass is 1) that

where F denoting the force acting on V through V, ν is the unit outnormal on V. Newton s law says (assume the mass is 1) that Chapter 5 The Wave Equation In this chapter we investigate the wave equation 5.) u tt u = and the nonhomogeneous wave equation 5.) u tt u = fx, t) subject to appropriate initial and boundary conditions.

More information

Weak convergence and Brownian Motion. (telegram style notes) P.J.C. Spreij

Weak convergence and Brownian Motion. (telegram style notes) P.J.C. Spreij Weak convergence and Brownian Motion (telegram style notes) P.J.C. Spreij this version: December 8, 2006 1 The space C[0, ) In this section we summarize some facts concerning the space C[0, ) of real

More information