Stochastic modelling for the financial markets. Part 1. Probabilistic tools

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION NATIONAL RESEARCH TOMSK STATE UNIVERSITY FACULTY OF MECHANICS AND MATHEMATICS







STOCHASTIC MODELLING FOR THE FINANCIAL MARKETS PART 1. PROBABILISTIC TOOLS




Lectures notes
for the courses "Stochastic Modelling" and "Theory of the Random Processes" taken by most Mathematics students and Economics students (Directions of training 01.03.01 - Mathematics and 38.04.01 Economics)









Tomsk 2017

   APPROVED the Department of Mathematical Analysis
   Head of the Department, Associate Professor L.S. Kopaneva



   REVIEWED and APPROVED Methodical Commission of
the Faculty of Mechanics and Mathematics
   Minutes No from ”” Febrary 2017
   Chairman of the Commission, Associate Professor O.P. Fedorova



   The main goal of these lectures notes is to give the basic notions of the stochastic calculus such that conditional expectations, predictable processes, martingales, stochastic integrals and Ito’s formula. The notes are intended for students of the Mathematics and Economical Faculties.
   This work was supported by the Ministry of Education and Science of the Russian Federation (Goszadanie No 1.172.2016 FPM).



            AUTHORS


   Professor Serguei M. Pergamenshchikov and Associate Professor
Evgeny A. Pchelintsev




© Tomsk State University, 2017

                Contents





1  Introduction                                          4
   1.1  Probability space............................... 4
   1.2  Random variables, vectors and mappings ......... 6
   1.3  Conditional expectations and conditional probabilities 7
   1.4  Stochastic basis............................... 13

2  Markovian moments                                    15

3  Stochastic processes                                 19

4  Optional and Predictable a - fields                  21

5  Martingales                                          27

6  Stochastic integral                                  32

7  Appendix                                             38
   7.1  Caratlieodory’s extension theorem ............. 38
   7.2  Radon - Nikodym theorem ....................... 40
   7.3  Kolmogorov theorem............................. 41

References                                               44


3

Introduction





            1.1  Probability space


Definition 1.1. The measurable space (Q, F, P) is called the probability space, where Q is any fixed universal Ft, F isc- field and P is a probability measure.
   It should be noted that if the set Q is finite or countable then the field (or a - field) F is defined as all subsets of the set Q, i.e. F = {A : A C Q}. Moreover, in this case the probability is defined as

P(A) = X P(M),                     (1.1)
weA
where P({w}) is defined for every ш from Q.


            Examples


   1. The Bernoulli space.
     The set Q = {0,1} and F = {Q , 0, {0} , {1}}- The probability is defined as P({0}) = p and P({1}) = 1 — p for some fixed 0 < p < 1. Note that, if p = 1/2, then we obtain the "throw a coin" model.

4

   2. The binomial space.
      The set Q = {0,1,..., n} and F = {A : A C Q}. In this case for any 0 < k < n the probability is defined as


P({k}) =

pk (1 - p)ra⁻fc

(1-2)


   3. The finite power of the Bernoulli spaces.
      The set Q = {0,1}ⁿ = {wl},. l. ₂- . where wl a re n-dimensional vectors, i.e. wl = (wz>₁,..., wl,ₙ) a nd wlj- e {0,1}. The field F = {A : A C Q} and

P(wl)= pv (1 - p)ⁿ⁻v ,              (1.3)

      where vl = Pn₌₁ ш^.
   4. The infinite power of the Bernoulli spaces.
      The set Q = {0,1}^ = {w}. In th is case w = (wl )z>₁, wl e {0,1} and the set Q is note countable, moreover, this set is isomorphes to interval [0,1] by the natural representation


x = £ wl 2⁻z e [0,1]. l>1


(1-4)

5