, Brownian Motion (3rd edition)
An Guide to Random Processes and Stochastic Calculus
de Gruyter Graduate, Berlin 2021
ISBN: 978-3-11-074125-4
Solution Manual
René L. Schilling
Dresden, August 2021. Last update February 4, 2022
,R.L. Schilling: Brownian Motion (3rd edn)
Acknowledgement. I am grateful to Björn Böttcher, David Berger, Katharina Fischer, Julian
Hollender, Franziska Kühn, Cailing Li, Felix Lindner, Lothar Partzsch, Vera Schade, Michael
Schwarzenberger and Paolo diTella who supported me in the preparation of this solution manual.
Daniel Tillich and Robert Baumgarth pointed out quite a few misprints and minor errors, as
well as alternative solutions.
Dresden, August 2021 René Schilling
2
, Contents
1 Robert Brown’s new thing 5
2 Brownian motion as a Gaussian process 15
3 Constructions of Brownian motion 29
4 The canonical model 39
5 Brownian motion as a martingale 49
6 Brownian motion as a Markov process 63
7 Brownian motion and transition semigroups 77
8 The PDE connection 99
9 The variation of Brownian paths 111
10 Regularity of Brownian paths 119
11 Brownian motion as a random fractal 125
12 The growth of Brownian paths 131
13 Strassen’s functional law of the iterated logarithm 137
14 Skorokhod representation 145
15 Stochastic integrals: L2 –theory 147
16 Stochastic integrals: Localization 161
17 Stochastic integrals: Martingale drivers 165
18 Itô’s formula 169
19 Applications of Itô’s formula 183
20 Wiener Chaos and iterated Wiener–Itô integrals 195
3
An Guide to Random Processes and Stochastic Calculus
de Gruyter Graduate, Berlin 2021
ISBN: 978-3-11-074125-4
Solution Manual
René L. Schilling
Dresden, August 2021. Last update February 4, 2022
,R.L. Schilling: Brownian Motion (3rd edn)
Acknowledgement. I am grateful to Björn Böttcher, David Berger, Katharina Fischer, Julian
Hollender, Franziska Kühn, Cailing Li, Felix Lindner, Lothar Partzsch, Vera Schade, Michael
Schwarzenberger and Paolo diTella who supported me in the preparation of this solution manual.
Daniel Tillich and Robert Baumgarth pointed out quite a few misprints and minor errors, as
well as alternative solutions.
Dresden, August 2021 René Schilling
2
, Contents
1 Robert Brown’s new thing 5
2 Brownian motion as a Gaussian process 15
3 Constructions of Brownian motion 29
4 The canonical model 39
5 Brownian motion as a martingale 49
6 Brownian motion as a Markov process 63
7 Brownian motion and transition semigroups 77
8 The PDE connection 99
9 The variation of Brownian paths 111
10 Regularity of Brownian paths 119
11 Brownian motion as a random fractal 125
12 The growth of Brownian paths 131
13 Strassen’s functional law of the iterated logarithm 137
14 Skorokhod representation 145
15 Stochastic integrals: L2 –theory 147
16 Stochastic integrals: Localization 161
17 Stochastic integrals: Martingale drivers 165
18 Itô’s formula 169
19 Applications of Itô’s formula 183
20 Wiener Chaos and iterated Wiener–Itô integrals 195
3
