Brownian Motion René L. Schilling / Lothar Partzsch ISBN: 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Abbildungsübersicht / List of.

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Presentation transcript:

Brownian Motion René L. Schilling / Lothar Partzsch ISBN: © 2014 Walter de Gruyter GmbH, Berlin/Boston Abbildungsübersicht / List of Figures Tabellenübersicht / List of Tables

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Basic stochastic calculus (C) 2

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Basic Markov processes (M) 3

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Basic sample path properties (S) 4

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 5 Fig Renewal at time. The process := + −, 0, is again a BM.

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Time inversion. The process := − −, ∈ [0, ], is again a BM. 6

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 7

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig The first four interpolation steps in Lévy’s construction of Brownian motion. 8

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 9

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig The first few approximation steps in Donsker’s construction: = 5, 10, 100 and

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Brownian paths meeting (and missing) the sets 1,..., 5 at times 1,..., 5 11

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Table 4.1. Transformations of Brownian motion. 12

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig A Brownian path meeting (and missing) the sets 1,..., 5 at times 1,..., 5 13

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig := + −, 0, is a Brownian motion in the new coordinate system with origin (, ). 14

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Reflection upon reaching the level for the first time. 15

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Both and the reflection are Brownian motions. 16

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig A [reflected] Brownian path visiting the [reflected] interval at time. 17

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Exit time and position of a Brownian motion from the set. 18

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig The outer cone condition. A point ∈ is regular, if we can touch it with a (truncated) cone which is in. 19

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Examples of singular points. 20

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Brownian motion has to exit (0,1) before it leaves ( 1, 2). 21

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig A ‘cubistic’ version of Lebesgue’s spine. 22

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Position of the points and and their dyadic approximations. 23

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Position of the points, +1 and, +1 relative to and. 24

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Exhausting the interval [, + ℎ ] by non-overlapping dyadic intervals. Dots “” denote dyadic numbers. Observe that at most two dyadic intervals of any one length can occur. 25

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig A typical excursion interval: and are the last zero before > 0 and the first zero after > 0. 26

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 27

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig For each value () of the line segment connecting (/) and ( +1 /) there is some such that () = (). 28

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig Approximation of a simple function by a left-continuous simple function. 29

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston Table Multiplication table for stochastic differentials. 30

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 31

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 32 Fig An increasing right-continuous function and its generalized right-continuous inverse.

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 33 Fig Derivation of the backward and forward Kolmogorov equations.

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 34 Fig Simulation of a two-dimensional -Brownian motion.

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 35 Fig. A.1. Two upcrossings over the interval [, ].

Brownian Motion, René L. Schilling / Lothar Partzsch ISBN © 2014 Walter de Gruyter GmbH, Berlin/Boston 36 Fig. A.2. The set is covered by dyadic cubes of different generations.