p 1 ( Y p dp) 1/p ( X p dp) 1 1 p
|
|
- Sara Rice
- 5 years ago
- Views:
Transcription
1 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 <, E[sup s t X(s) p ] ( p p 1) p E[ X(t) p ] 1 2: P(λ) = P( X λ), X = sup s t X(s), Y = X(t), p E[ X p ] = = p p 1 λ p dp(λ) = p λ p 1 P(λ)dλ λ p 1 ( 1 X YdP)dλ = p Y λ p 2 dλdp λ X>λ Y X p 1 dp p p 1 ( Y p dp) 1/p ( X p dp) 1 1 p () Stochastic Calculus March 4, 29 1 / 22
2 Existence and Uniqueness Theorem σ : R d [, T ] R d d, b : R d [, T ] R d be Borel measurable, A <, σ(x, t) + b(x, t) A(1 + x ) x R d, t T and Lipschitz; σ(x, t) σ(y, t) + b(x, t) b(y, t) A x y. x R d indep of B t, E[ x 2 ] <. Then there exists a unique solution X t on [, T ] to and E[ T X t 2 dt] <. dx t = b(x t, t)dt + σ(x t, t)db t, X = x Uniqueness means that if Xt 1 and Xt 2 are two solutions then P(Xt 1 = Xt 2, t T ) = 1 () Stochastic Calculus March 4, 29 2 / 22
3 dx t = σ(t, X t )db t + b(t, X t )dt t f (t, X t ) = f (, X ) + Lf (t, x) = 1 2 t + i,j=1 i,j=1 { } s f (s, X s ) + Lf (s, X s ) ds σ ij (s, X s ) x i f (s, X s )db j s a ij (t, x) 2 f x i x j (t, x) + i=1 b i (t, x) f x i (t, x) a ij = σ ik σ jk k=1 a = σσ T () Stochastic Calculus March 4, 29 3 / 22
4 Markov property X t can be obtained by solving the stochastic differential equation up to time s < t and then solving in [s, t] with initial condition X s By uniqueness this gives the same answer Define the transition probability p(s, x, t, A) = P(X s,x t A) where X s,x t is the solution starting at x at time s From the construction we have P(X,x t A F s ) = p(s, X,x s, t, A) which is the Markov property () Stochastic Calculus March 4, 29 4 / 22
5 Diffusions A diffusion is a Markov process with transition probabilities p(s, x, t, dy) satisfying, for each δ > as h, 1 i. p(t, x, t + h, dy) continuous paths h y x δ 1 ii. (y x)p(t, x, t + h, dy) b(t, x) h y x <δ 1 iii. (y i x i )(y j x j )p(t, x,, t + h, dy) a ij (t, x) h y x <δ () Stochastic Calculus March 4, 29 5 / 22
6 t { } t f (t, X t ) f (, X ) s + L f (s, X s )ds = f (s, X s ) σdb s L = a ij (t, x) + x i x j i,j=1 M t = f (t, X t ) t { } s + L i=1 b i (t, x) x i = generator f (s, X s )ds is a martingale t { } = E[f (t, X t ) f (s, X s ) u + L f (u, X u )du F s ] s = f (t, y)p(s, x, t, y)dy f (s, x) t { u + L} f (u, y)p(s, x, u, y)dydu, s X s = x () Stochastic Calculus March 4, 29 6 / 22
7 For any f, = f (t, y)p(s, x, t, y)dy f (s, x) t { u + L} f (u, y)p(s, x, u, y)dydu s Fokker-Planck (Forward) Equation t p(s, x, t, y) = 1 2 i,j=1 i=1 = L yp(s, x, t, y) 2 y i y j (a i,j (t, y)p(s, x, t, y)) y i (b i (t, y)p(s, x, t, y)) lim p(s, x, t, y) = δ(y x). t s () Stochastic Calculus March 4, 29 7 / 22
8 Kolmogorov (Backward) Equation s p(s, x, t, y) = i,j=1 i=1 = L x p(s, x, t, y) a ij (s, x) 2 p(s, x, t, y) x i x j p(s, x, t, y) b i (s, x) x i lim p(s, x, t, y) = δ(y x). s t () Stochastic Calculus March 4, 29 8 / 22
9 Example. Brownian motion d = 1 2 L = 1 2 x 2 Forward p(s, x, t, y t = 1 2 p(s, x, t, y) 2 y 2, t > s p(s, x, s, y) = δ(y x) Backward p(s, x, t, y s = 1 2 p(s, x, t, y) 2 x 2, s < t, p(t, x, t, y) = δ(y x) () Stochastic Calculus March 4, 29 9 / 22
10 Example. Ornstein-Uhlenbeck Process L = σ2 2 2 x 2 αx x Forward p(s, x, t, y t = 1 2 p(s, x, t, y) 2 y 2 + (αyp(s, x, t, y), t > s, y p(s, x, s, y) = δ(y x) Backward p(s, x, t, y s = 1 2 p(s, x, t, y) 2 x 2 αx p(s, x, t, y), s < t, x p(t, x, t, y) = δ(y x) () Stochastic Calculus March 4, 29 1 / 22
11 Formal derivation of the backward equation p(s, x, t, A) = p(s, x, s + h, dy)p(s + h, y, t, A) = { } p(s, x, s + h, dy) p(s + h, y, t, A) p(s, x, t, A) = { p(s, x, t, A) p(s, x, s + h, dy) h + s p(s, x, t, A) (y i x i ) x i i=1 (y i x i )(y j x j ) 2 p(s, x, t, A) } + x i x j i,j=1 p(s, x, t, A) = s i=1 p(s, x, t, A) b i (t, x) + 1 x i 2 i,j=1 a ij (t, x) 2 p(s, x, t, A) x i x j () Stochastic Calculus March 4, / 22
12 Real derivation f (x) smooth s u = L su s < t u(t, x) = f (x) Ito s formula: u(s, X(s)) martingale up to time t u(s, x) = E s,x [u(s, X(s))] = E s,x [u(t, X(t))] = f (z)p(s, x, t, dz) Let f n (z) smooth functions tending to δ(y z) u(s, x) = p(s, x, t, y) if s u = L su s < t u(t, x) = δ(x y) () Stochastic Calculus March 4, / 22
13 Existence result from PDE Suppose that a(t, x) and b(t, x) are bounded and that there are α >, γ (, 1], C < such that for all s, t, x, y R d, i. ξ T a(t, x)ξ α ξ 2, ξ R d, ii. a(s, x) a(t, y) + b(s, x) b(t, y) C( x y γ + t s γ ). Then the backward equation has a solution and furthermore p(s, x, t, A) = p(s, x, t, y)dy with p(s, x, t, y) jointly continuous in s, x, t, y. Furthermore, p(s, x, t, y) is the unique weak solution of the forward equation,i.e. f (t, y)p(s, x, t, y)dy f (s, x) = A t s { u + L } f (u, y)p(s, x, u, y)dydu () Stochastic Calculus March 4, / 22
14 The solution X t, t of dx t = b(x t )dt + σ(x t )db t with X = x is a Markov process with infinitesimal generator Itô s formula L = a ij (x) + x i x j i,j=1 f (t, X t ) = f (, X ) + t + i,j=1 t i=1 b i (x) x i, a = σσ. { } s f (s, X s ) + Lf (s, X s ) ds σ ij (s, X s ) x i f (s, X s )db j s () Stochastic Calculus March 4, / 22
15 Under the previous conditions, the following are equivalent 1 B t, t, dx t = b(t, X t )dt + σ(t, X t )db t 2 For each λ R d, Z λ (t) = e λ{x t R t b(s,xs)ds} 1 R t 2 λt a(s,x s)λds is a martingale with respect to F t 3 For all smooth f (t, x), f (t, X t ) t is a martingale with respect to F t 4 For all smooth f (x), { s + L}f (s, X s )ds t f (X t ) Lf (X s )ds is a martingale with respect to F t () Stochastic Calculus March 4, / 22
16 Brownian motion in R d 1 B t = (Bt 1,..., Bd t ), Bi t independent Brownian motions 2 B t Markov with P(B t A B s = x) = A 1 e y x 2 (2π(t s)) d/2 2(t s) dy 3 B t has stationary independent mean zero increments with E[ B t B s 2 ] = d(t s) 4 e λ B t 1 2 λ 2t is a martingale for any λ Note that 1 does not depend on the basis: If Bt 1,..., B2 t independent and O is orthogonal, then the coordinates of OB t are independent Brownian motions in fact Theorem Suppose X 1, X 2 independent and θ Nπ/2 such that X 1 cos θ + X 2 sin θ, X 1 sin θ + X 2 cos θ independent Then X 1, X 2 are Gaussians (Maxwell) () Stochastic Calculus March 4, / 22
17 Dirichlet problem Given a bounded open subset G R d and a continuous function f : G R find a continuous function u : Ḡ R such that { u = in G u G = f u def = i=1 2 u x 2 i = 2d lim r r 2 ( 1 S(r, x) S(r,x) uds u(x) ) Lemma u harmonic in G u satisfies the mean value property: for all sufficiently small r >, 1 uds = u(x) S(r, x) S(r,x) () Stochastic Calculus March 4, / 22
18 Lemma u harmonic in G u satisfies the mean value property: for all sufficiently small r >, 1 uds = u(x) S(r, x) S(r,x) Proof. Green s identity G v udx = G u vdx + { G v u n u v n log r log x log r log δ d = 2 G = {δ < x < r}, v = x 2 d r 2 d d > 2 δ 2 d r 2 d let ρ ds () Stochastic Calculus March 4, / 22
19 B t d-dimensional Brownian motion starting at x G τ G = inf{t : B(t) G} u(x) = E x [f (B(τ G ))] Theorem If G nice then u solves the Dirichlet problem E x [f (B(τ G ))] = G f (y)π G (x, dy), π G (x, Γ) = P x (B(τ G ) Γ), Γ G Example. G = B(x, r), π G (x, Γ) = Γ, Γ S(x, r) S(x,r) Brownian motion is invariant under rotations π G (x, ) is invariant under rotations () Stochastic Calculus March 4, / 22
20 Proposition G bounded open R d, f bounded measurable on G. Then u(x) = E x [f (B(τ G ))] is harmonic in G. Proof. B = B(x, r) G τ B τ G Strong Markov property: u(b(τ S )) = E x [f (B(τ G )) F τs ] u(x) = E x [f (B(τ G ))] = E x [E x [f (B(τ G )) F τs ]] So u satisfies the mean value property in G. = E x [u(b(τ S ))] = u(y)π S (x, dy) = S 1 S u(y)ds () Stochastic Calculus March 4, 29 2 / 22 S
21 a G To complete the proof that u solves the Dirichlet problem we need Proposition lim E x[f (B(τ G ))] = f (a) x a, x G It is not always true! If lim x a P x [τ G > ɛ] =, ɛ > then for any bdd mble function x G f : G R which is continuous at a, lim x a E x [f (B(τ G ))] = f (a) x G Proof. Need: lim x a, x G P x ( B(τ G ) x < δ) = 1 P x ( B(τ G ) x < δ) P x ( sup B(t) x < δ, τ G ɛ) t ɛ P x ( sup B(t) x < δ) P x (τ G ɛ) t ɛ 1 as x a, x G then ɛ () Stochastic Calculus March 4, / 22
22 Proposition a G is regular if P a (σ G = ) = 1 σ G = inf{t > : B(t) G} a regular lim x a, x G E x [f (B τg )] = f (a) f bdd mble, cont at a Proof of Enough to prove P x (σ G < ɛ) lower semi-continuous in x Then lim sup x a P x (σ G < ɛ) P a (σ G < ɛ) = 1 and σ G τ G x G But p(, x, δ, y)p y ( s (, ɛ δ), B(s) G) continuous and P x (σ G < ɛ) as δ Examples 1 If G is a smooth manifold near a then a is regular by LIL 2 If cone C of height h > and vertex at a such that C {a} ḠC then a is a regular (exterior cone condition) 3 d 2 always,d 3 counterexamples () Stochastic Calculus March 4, / 22
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 informationMathematical 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 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 informationStochastic Differential Equations.
Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)
More 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 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 informationPoisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation
Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Jingyi Zhu Department of Mathematics University of Utah zhu@math.utah.edu Collaborator: Marco Avellaneda (Courant
More informationExercises. T 2T. e ita φ(t)dt.
Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.
More informationA Short Introduction to Diffusion Processes and Ito Calculus
A Short Introduction to Diffusion Processes and Ito Calculus Cédric Archambeau University College, London Center for Computational Statistics and Machine Learning c.archambeau@cs.ucl.ac.uk January 24,
More informationKolmogorov Equations and Markov Processes
Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define
More informationI 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 informationMA8109 Stochastic Processes in Systems Theory Autumn 2013
Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form
More informationBernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010
1 Stochastic Calculus Notes Abril 13 th, 1 As we have seen in previous lessons, the stochastic integral with respect to the Brownian motion shows a behavior different from the classical Riemann-Stieltjes
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 informationLecture 4: Introduction to stochastic processes and stochastic calculus
Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London
More information{σ x >t}p x. (σ x >t)=e at.
3.11. EXERCISES 121 3.11 Exercises Exercise 3.1 Consider the Ornstein Uhlenbeck process in example 3.1.7(B). Show that the defined process is a Markov process which converges in distribution to an N(0,σ
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationLAN property for sde s with additive fractional noise and continuous time observation
LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday,
More informationReflected 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 informationVerona Course April Lecture 1. Review of probability
Verona Course April 215. Lecture 1. Review of probability Viorel Barbu Al.I. Cuza University of Iaşi and the Romanian Academy A probability space is a triple (Ω, F, P) where Ω is an abstract set, F is
More informationHomogenization with stochastic differential equations
Homogenization with stochastic differential equations Scott Hottovy shottovy@math.arizona.edu University of Arizona Program in Applied Mathematics October 12, 2011 Modeling with SDE Use SDE to model system
More informationProperties 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 information2012 NCTS Workshop on Dynamical Systems
Barbara Gentz gentz@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ gentz 2012 NCTS Workshop on Dynamical Systems National Center for Theoretical Sciences, National Tsing-Hua University Hsinchu,
More informationIntroduction 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 informationStochastic 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 informationConvoluted Brownian motions: a class of remarkable Gaussian processes
Convoluted Brownian motions: a class of remarkable Gaussian processes Sylvie Roelly Random models with applications in the natural sciences Bogotá, December 11-15, 217 S. Roelly (Universität Potsdam) 1
More informationApplications of Ito s Formula
CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale
More information6. Brownian Motion. Q(A) = P [ ω : x(, ω) A )
6. Brownian Motion. stochastic process can be thought of in one of many equivalent ways. We can begin with an underlying probability space (Ω, Σ, P) and a real valued stochastic process can be defined
More informationAn adaptive numerical scheme for fractional differential equations with explosions
An adaptive numerical scheme for fractional differential equations with explosions Johanna Garzón Departamento de Matemáticas, Universidad Nacional de Colombia Seminario de procesos estocásticos Jointly
More informationTMS165/MSA350 Stochastic Calculus, Lecture on Applications
TMS165/MSA35 Stochastic Calculus, Lecture on Applications In this lecture we demonstrate how statistical methods such as the maximum likelihood method likelihood ratio estimation can be applied to the
More informationLocal vs. Nonlocal Diffusions A Tale of Two Laplacians
Local vs. Nonlocal Diffusions A Tale of Two Laplacians Jinqiao Duan Dept of Applied Mathematics Illinois Institute of Technology Chicago duan@iit.edu Outline 1 Einstein & Wiener: The Local diffusion 2
More informationSolution 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 informationBernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012
1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.
More informationIntertwinings 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 information13 The martingale problem
19-3-2012 Notations Ω complete metric space of all continuous functions from [0, + ) to R d endowed with the distance d(ω 1, ω 2 ) = k=1 ω 1 ω 2 C([0,k];H) 2 k (1 + ω 1 ω 2 C([0,k];H) ), ω 1, ω 2 Ω. F
More informationBirth and Death Processes. Birth and Death Processes. Linear Growth with Immigration. Limiting Behaviour for Birth and Death Processes
DTU Informatics 247 Stochastic Processes 6, October 27 Today: Limiting behaviour of birth and death processes Birth and death processes with absorbing states Finite state continuous time Markov chains
More informationExercises in stochastic analysis
Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with
More informationBrownian Motion and the Dirichlet Problem
Brownian Motion and the Dirichlet Problem Mario Teixeira Parente August 29, 2016 1/22 Topics for the talk 1. Solving the Dirichlet problem on bounded domains 2. Application: Recurrence/Transience of Brownian
More informationLecture 12: Detailed balance and Eigenfunction methods
Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),
More informationLecture 3. This operator commutes with translations and it is not hard to evaluate. Ae iξx = ψ(ξ)e iξx. T t I. A = lim
Lecture 3. If we specify D t,ω as a progressively mesurable map of (Ω [, T], F t ) into the space of infinitely divisible distributions, as well as an initial distribution for the starting point x() =
More informationPath integrals for classical Markov processes
Path integrals for classical Markov processes Hugo Touchette National Institute for Theoretical Physics (NITheP) Stellenbosch, South Africa Chris Engelbrecht Summer School on Non-Linear Phenomena in Field
More informationAn Introduction to Malliavin calculus and its applications
An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart
More informationPartial 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 informationA 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 informationBROWNIAN MOTION AND LIOUVILLE S THEOREM
BROWNIAN MOTION AND LIOUVILLE S THEOREM CHEN HUI GEORGE TEO Abstract. Probability theory has many deep and surprising connections with the theory of partial differential equations. We explore one such
More informationLecture 12: Detailed balance and Eigenfunction methods
Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility), 4.2-4.4 (explicit examples of eigenfunction methods) Gardiner
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationThe Smoluchowski-Kramers Approximation: What model describes a Brownian particle?
The Smoluchowski-Kramers Approximation: What model describes a Brownian particle? Scott Hottovy shottovy@math.arizona.edu University of Arizona Applied Mathematics October 7, 2011 Brown observes a particle
More informationBrownian Motion on Manifold
Brownian Motion on Manifold QI FENG Purdue University feng71@purdue.edu August 31, 2014 QI FENG (Purdue University) Brownian Motion on Manifold August 31, 2014 1 / 26 Overview 1 Extrinsic construction
More informationJump 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 informationGaussian processes for inference in stochastic differential equations
Gaussian processes for inference in stochastic differential equations Manfred Opper, AI group, TU Berlin November 6, 2017 Manfred Opper, AI group, TU Berlin (TU Berlin) inference in SDE November 6, 2017
More informationWeak convergence and large deviation theory
First Prev Next Go To Go Back Full Screen Close Quit 1 Weak convergence and large deviation theory Large deviation principle Convergence in distribution The Bryc-Varadhan theorem Tightness and Prohorov
More information1. Stochastic Process
HETERGENEITY IN QUANTITATIVE MACROECONOMICS @ TSE OCTOBER 17, 216 STOCHASTIC CALCULUS BASICS SANG YOON (TIM) LEE Very simple notes (need to add references). It is NOT meant to be a substitute for a real
More informationThe multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
More informationLAN property for ergodic jump-diffusion processes with discrete observations
LAN property for ergodic jump-diffusion processes with discrete observations Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Arturo Kohatsu-Higa (Ritsumeikan University, Japan) &
More informationBranching Processes II: Convergence of critical branching to Feller s CSB
Chapter 4 Branching Processes II: Convergence of critical branching to Feller s CSB Figure 4.1: Feller 4.1 Birth and Death Processes 4.1.1 Linear birth and death processes Branching processes can be studied
More information4.6 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 informationStarting from Heat Equation
Department of Applied Mathematics National Chiao Tung University Hsin-Chu 30010, TAIWAN 20th August 2009 Analytical Theory of Heat The differential equations of the propagation of heat express the most
More informationSolutions to the Exercises in Stochastic Analysis
Solutions to the Exercises in Stochastic Analysis Lecturer: Xue-Mei Li 1 Problem Sheet 1 In these solution I avoid using conditional expectations. But do try to give alternative proofs once we learnt conditional
More informationKolmogorov equations in Hilbert spaces IV
March 26, 2010 Other types of equations Let us consider the Burgers equation in = L 2 (0, 1) dx(t) = (AX(t) + b(x(t))dt + dw (t) X(0) = x, (19) where A = ξ 2, D(A) = 2 (0, 1) 0 1 (0, 1), b(x) = ξ 2 (x
More informationTheoretical 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 informationStability 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 informationSimulation of conditional diffusions via forward-reverse stochastic representations
Weierstrass Institute for Applied Analysis and Stochastics Simulation of conditional diffusions via forward-reverse stochastic representations Christian Bayer and John Schoenmakers Numerical methods for
More informationPotential 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 informationThe Wiener Itô Chaos Expansion
1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in
More informationSimulation 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 informationWeek 9 Generators, duality, change of measure
Week 9 Generators, duality, change of measure Jonathan Goodman November 18, 013 1 Generators This section describes a common abstract way to describe many of the differential equations related to Markov
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationProblem 1. Construct a filtered probability space on which a Brownian motion W and an adapted process X are defined and such that
Stochatic Calculu Example heet 4 - Lent 5 Michael Tehranchi Problem. Contruct a filtered probability pace on which a Brownian motion W and an adapted proce X are defined and uch that dx t = X t t dt +
More 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 informationLower Tail Probabilities and Normal Comparison Inequalities. In Memory of Wenbo V. Li s Contributions
Lower Tail Probabilities and Normal Comparison Inequalities In Memory of Wenbo V. Li s Contributions Qi-Man Shao The Chinese University of Hong Kong Lower Tail Probabilities and Normal Comparison Inequalities
More informationConvergence of Markov Processes. Amanda Turner University of Cambridge
Convergence of Markov Processes Amanda Turner University of Cambridge 1 Contents 1 Introduction 2 2 The Space D E [, 3 2.1 The Skorohod Topology................................ 3 3 Convergence of Probability
More informationAlbert N. Shiryaev Steklov Mathematical Institute. On sharp maximal inequalities for stochastic processes
Albert N. Shiryaev Steklov Mathematical Institute On sharp maximal inequalities for stochastic processes joint work with Yaroslav Lyulko, Higher School of Economics email: albertsh@mi.ras.ru 1 TOPIC I:
More informationMSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS. Contents
MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS ANDREW TULLOCH Contents 1. Lecture 1 - Tuesday 1 March 2 2. Lecture 2 - Thursday 3 March 2 2.1. Concepts of convergence 2 3. Lecture 3 - Tuesday 8 March
More informationDiscretization of SDEs: Euler Methods and Beyond
Discretization of SDEs: Euler Methods and Beyond 09-26-2006 / PRisMa 2006 Workshop Outline Introduction 1 Introduction Motivation Stochastic Differential Equations 2 The Time Discretization of SDEs Monte-Carlo
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More informationOptimal Stopping and Maximal Inequalities for Poisson Processes
Optimal Stopping and Maximal Inequalities for Poisson Processes D.O. Kramkov 1 E. Mordecki 2 September 10, 2002 1 Steklov Mathematical Institute, Moscow, Russia 2 Universidad de la República, Montevideo,
More informationA 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 informationModel Specification Testing in Nonparametric and Semiparametric Time Series Econometrics. Jiti Gao
Model Specification Testing in Nonparametric and Semiparametric Time Series Econometrics Jiti Gao Department of Statistics School of Mathematics and Statistics The University of Western Australia Crawley
More informationMean-field SDE driven by a fractional BM. A related stochastic control problem
Mean-field SDE driven by a fractional BM. A related stochastic control problem Rainer Buckdahn, Université de Bretagne Occidentale, Brest Durham Symposium on Stochastic Analysis, July 1th to July 2th,
More informationDensities for the Navier Stokes equations with noise
Densities for the Navier Stokes equations with noise Marco Romito Università di Pisa Universitat de Barcelona March 25, 2015 Summary 1 Introduction & motivations 2 Malliavin calculus 3 Besov bounds 4 Other
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 informationLecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University
Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral
More informationLinear Ordinary Differential Equations
MTH.B402; Sect. 1 20180703) 2 Linear Ordinary Differential Equations Preliminaries: Matrix Norms. Denote by M n R) the set of n n matrix with real components, which can be identified the vector space R
More informationNonlinear 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 informationSome Tools From Stochastic Analysis
W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click
More informationInfinite-dimensional methods for path-dependent equations
Infinite-dimensional methods for path-dependent equations (Università di Pisa) 7th General AMaMeF and Swissquote Conference EPFL, Lausanne, 8 September 215 Based on Flandoli F., Zanco G. - An infinite-dimensional
More informationFeller 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 informationSMSTC (2007/08) Probability.
SMSTC (27/8) Probability www.smstc.ac.uk Contents 12 Markov chains in continuous time 12 1 12.1 Markov property and the Kolmogorov equations.................... 12 2 12.1.1 Finite state space.................................
More informationSloppy derivations of Ito s formula and the Fokker-Planck equations
Sloppy derivations of Ito s formula and the Fokker-Planck equations P. G. Harrison Department of Computing, Imperial College London South Kensington Campus, London SW7 AZ, UK email: pgh@doc.ic.ac.uk April
More informationSmoluchowski Diffusion Equation
Chapter 4 Smoluchowski Diffusion Equation Contents 4. Derivation of the Smoluchoswki Diffusion Equation for Potential Fields 64 4.2 One-DimensionalDiffusoninaLinearPotential... 67 4.2. Diffusion in an
More informationGeneralized Gaussian Bridges of Prediction-Invertible Processes
Generalized Gaussian Bridges of Prediction-Invertible Processes Tommi Sottinen 1 and Adil Yazigi University of Vaasa, Finland Modern Stochastics: Theory and Applications III September 1, 212, Kyiv, Ukraine
More informationWEYL S LEMMA, ONE OF MANY. Daniel W. Stroock
WEYL S LEMMA, ONE OF MANY Daniel W Stroock Abstract This note is a brief, and somewhat biased, account of the evolution of what people working in PDE s call Weyl s Lemma about the regularity of solutions
More informationSDE Coefficients. March 4, 2008
SDE Coefficients March 4, 2008 The following is a summary of GARD sections 3.3 and 6., mainly as an overview of the two main approaches to creating a SDE model. Stochastic Differential Equations (SDE)
More informationApproximation of BSDEs using least-squares regression and Malliavin weights
Approximation of BSDEs using least-squares regression and Malliavin weights Plamen Turkedjiev (turkedji@math.hu-berlin.de) 3rd July, 2012 Joint work with Prof. Emmanuel Gobet (E cole Polytechnique) Plamen
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More informationSelected 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 informationMarkov processes and queueing networks
Inria September 22, 2015 Outline Poisson processes Markov jump processes Some queueing networks The Poisson distribution (Siméon-Denis Poisson, 1781-1840) { } e λ λ n n! As prevalent as Gaussian distribution
More informationThe Chaotic Character of the Stochastic Heat Equation
The Chaotic Character of the Stochastic Heat Equation March 11, 2011 Intermittency The Stochastic Heat Equation Blowup of the solution Intermittency-Example ξ j, j = 1, 2,, 10 i.i.d. random variables Taking
More informationlim 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