MAIN REFERENCE: Zdzisław Brzeźniak, Tomasz Zastawniak, Basic Stochastic Processes: A Course Through Exercises, Springer 1999. 

Other references:

1. A First Course in Stochastic Calculus by Louis-Pierre Arguin, AMS 2022

2. Stochastic Differential Equations: An Introduction with Applications by B.Oksendal, Springer 1998

3. A good reference in Greek by Cheliotis http://users.uoa.gr/~dcheliotis/SimeioseisStocCalc.pdf

Syllabus: Axiomatic probability theory. Conditional Expectation. Martingales. Brownian Motion. Ito integral & Ito calculus. Basic Stochastic Differential Equations 

Interesting and possibly helpful senior thesis in Greek https://ir.lib.uth.gr/xmlui/bitstream/handle/11615/52080/20209.pdf?sequence=1

FINAL EXAM:  A sheet (A4) of handwritten notes is allowed. Basic definitions, basic theorems, solutions of exercises similar to those in the list below.

Material for the exam (BZ):  

Chapter 1.  All sections. All exercises.

Chapter 2.  All sections. All exercises except 2.17.

Chapter3: All sections. All exercises. 

Chapter 6: Only sections 6.1, 6.3, 6.4. Exercises 6.19, 6.20, 6.21, 6.25, 6.26, 6.27, 6.28, 6.29, 6.30, 6.35.

Chapter 7: All sections. All exercises except 7.10. 

COVERED MATERIAL

σ-fields, Probability function, Borel sets, Lebesgue measure, Borel-Cantelli lemma, Random variables (section 1.1  from BZ)

Probability distribution functions, Density functions (section 1.2  from BZ)

Expected values, Conditional Probability, Independence  (sections 1.2, 1.3  from BZ). Conditional Expectation. Conditioning on an event (section 2.1 from BZ)

Conditioning on a discrete random variable  (section 2.2 from BZ) 

Conditioning on an arbitrary random variable: examples and exercises  (section 2.3 from BZ) 

Conditioning on a σ-field. General Properties (sections 2.4-2.5 from BZ). 

Exercises (section 2.6 from BZ)

Martingales (sections 3.1-3.3 from BZ)

Games of Chance. Stopping Times. Optional Stopping Theorem (sections 3.4-3.6 from BZ)

Continuous time (section 6.1 from BZ)

Brownian motion (section 6.3 from BZ)

Increments of Brownian Motion  (section 6.3 from BZ)

Sample paths (section 6.3 from BZ)

Ito integral: Random step processes  (section 7.1 from BZ)

Ito integral: general case (section 7.1 from BZ)

Ito integral: martingale (section 7.1 from BZ)

Ito integral: continuous paths (section 7.1 from BZ).  Examples (section 7.2 from BZ)

Ito integral: basic properties (section 7.3 from BZ). 

Stochastic Differential and Ito formula (section 7.4 from BZ). 

Stochastic Differential Equations  (section 7.5 from BZ)