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

2. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Basic stochastic calculus (C) 2

3. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Basic Markov processes (M) 3

4. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Basic sample path properties (S) 4

5. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 5 Fig. 2.1. Renewal at time. The process := + −, 0, is again a BM.

6. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 2.2. Time inversion. The process := − −, ∈ [0, ], is again a BM. 6

7. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 7

8. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 3.2. The first four interpolation steps in Lévy’s construction of Brownian motion. 8

9. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 9

10. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 3.4. The first few approximation steps in Donsker’s construction: = 5, 10, 100 and 1000. 10

11. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 3.5. Brownian paths meeting (and missing) the sets 1,..., 5 at times 1,..., 5 11

12. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Table 4.1. Transformations of Brownian motion. 12

13. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 4.1. A Brownian path meeting (and missing) the sets 1,..., 5 at times 1,..., 5 13

14. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 6.1. := + −, 0, is a Brownian motion in the new coordinate system with origin (, ). 14

15. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 6.2. Reflection upon reaching the level for the first time. 15

16. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 6.3. Both and the reflection are Brownian motions. 16

17. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 6.4. A [reflected] Brownian path visiting the [reflected] interval at time. 17

18. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 8.1. Exit time and position of a Brownian motion from the set. 18

19. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 8.2. The outer cone condition. A point ∈ is regular, if we can touch it with a (truncated) cone which is in. 19

20. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 8.3. Examples of singular points. 20

21. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 8.4. Brownian motion has to exit (0,1) before it leaves ( 1, 2). 21

22. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 8.5. A ‘cubistic’ version of Lebesgue’s spine. 22

23. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 10.1. Position of the points and and their dyadic approximations. 23

24. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 10.2. Position of the points, +1 and, +1 relative to and. 24

25. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 11.1. 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

26. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 11.2. A typical excursion interval: and are the last zero before > 0 and the first zero after > 0. 26

27. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 27

28. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 14.1. For each value () of the line segment connecting (/) and ( +1 /) there is some such that () = (). 28

29. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Fig. 15.1. Approximation of a simple function by a left-continuous simple function. 29

30. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston Table 17.1. Multiplication table for stochastic differentials. 30

31. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 31

32. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 32 Fig. 18.1. An increasing right-continuous function and its generalized right-continuous inverse.

33. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 33 Fig. 21.1. Derivation of the backward and forward Kolmogorov equations.

34. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 34 Fig. 22.1. Simulation of a two-dimensional -Brownian motion.

35. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 35 Fig. A.1. Two upcrossings over the interval [, ].

36. Brownian Motion, René L. Schilling / Lothar Partzsch ISBN 978-3-11-030729-0 © 2014 Walter de Gruyter GmbH, Berlin/Boston 36 Fig. A.2. The set is covered by dyadic cubes of different generations.