Brownian Motion René L. Schilling / Lothar Partzsch ISBN: 978-3-11-030729-0 © 2014 Walter de...

Brownian MotionRené L. Schilling / Lothar PartzschISBN: 978-3-11-030729-0

© 2014 Walter de Gruyter GmbH, Berlin/Boston

Abbildungsübersicht / List of Figures

Tabellenübersicht / List of Tables

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


Basic Markov processes (M)

3


Basic sample path properties (S)

4


5

Fig. 2.1. Renewal at time . The process 𝑎 𝑊𝑡 := 𝐵 +𝑡 𝑎 − 𝐵𝑎, 0, is again a BM𝑡 ⩾ 𝑑.


Fig. 2.2. Time inversion. The process 𝑊𝑡 := 𝐵 −𝑎 𝑡 − 𝐵𝑡, [0, ], is again a BM𝑡 ∈ 𝑎 𝑑.

6


7


Fig. 3.2. The first four interpolation steps in Lévy’s construction of Brownian motion.

8


9


Fig. 3.4. The first few approximation steps in Donsker’s construction: = 5, 10, 100 and 1000.𝑛

10


Fig. 3.5. Brownian paths meeting (and missing) the sets 𝐴1,..., 𝐴5 at times 𝑡1,..., 𝑡5

11


Table 4.1. Transformations of Brownian motion.

12


Fig. 4.1. A Brownian path meeting (and missing) the sets 𝐴1,..., 𝐴5 at times 𝑡1,..., 𝑡5

13


Fig. 6.1. 𝑊𝑡 := 𝐵 +𝑡 𝑠 − 𝐵𝑠, 0, is a Brownian motion in the new coordinate system with origin ( , 𝑡 ⩾ 𝑠 𝐵𝑠).

14


Fig. 6.2. Reflection upon reaching the level for the first time.𝑏

15


Fig. 6.3. Both 𝐵𝑡 and the reflection 𝑊𝑡 are Brownian motions.

16


Fig. 6.4. A [reflected] Brownian path visiting the [reflected] interval at time .𝑡

17


Fig. 8.1. Exit time and position of a Brownian motion from the set .𝐷

18


Fig. 8.2. The outer cone condition. A point is regular, if we 𝑥 ∈ 𝐷can touch it with a (truncated) cone which is in 𝐷𝑐.

19


Fig. 8.3. Examples of singular points.

20


Fig. 8.4. Brownian motion has to exit (0,1) before it leaves (𝔹 𝔹 𝑐𝑛𝑒1, 2).

21


Fig. 8.5. A ‘cubistic’ version of Lebesgue’s spine.

22


Fig. 10.1. Position of the points and and their dyadic approximations.𝑥 𝑦

23


Fig. 10.2. Position of the points 𝑡𝑛, 𝑡 +1𝑛 and 𝑠𝑛, 𝑠 +1𝑛 relative to and .𝑡 𝑠

24


Fig. 11.1. Exhausting the interval [ , + ] by non-overlapping dyadic intervals. 𝑡 𝑡 ℎDots “•” denote dyadic numbers. Observe that at most two dyadic intervals of any

one length can occur.

25


Fig. 11.2. A typical excursion interval: 𝜉𝑡 and 𝜂𝑡 are the last zero before > 0 and the first zero after > 0.𝑡 𝑡

26


27


Fig. 14.1. For each value 𝑆𝑛( ) of the line segment connecting 𝑡 𝐵𝑛(𝜏𝑘/ ) and 𝑛 𝐵𝑛(𝜏 +1𝑘 / ) there is some such that 𝑛 𝑢 𝑆𝑛( ) = 𝑡 𝐵𝑛( ).𝑢

28


Fig. 15.1. Approximation of a simple function by a left-continuous simple function.

29


Table 17.1. Multiplication table for stochastic differentials.

30


31


32

Fig. 18.1. An increasing right-continuous function and its generalized right-continuous inverse.


33

Fig. 21.1. Derivation of the backward and forward Kolmogorov equations.


34

Fig. 22.1. Simulation of a two-dimensional -Brownian motion.𝑄


35

Fig. A.1. Two upcrossings over the interval [ , ].𝑎 𝑏


36

Fig. A.2. The set is covered by dyadic cubes of different generations.𝐾

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