Stochastic Calculus A Practical Introduction

Notes for the following book:
Richard Durrett, Stochastic Calculus A Practical Introduction, CRC Press 2000.

1. Brownian Motion

1.1 Definition and Construction

Exercise 1.2. Show by considering increments instead of 3 that if then with probability 1, Brownian paths are not Holder continuous with exponent at any point of .
Answer: Let For let

The claim still can be proved when . Specifically, we have

Using and the scaling relation as in the book gives

To ensure the right hand side goes to zero as , we need

which implies . Obviously, the proof is completed.

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2020