Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
See all formats and pricing
More options …
Volume 24, Issue 1 (Mar 2016)


Modelling Real World Using Stochastic Processes and Filtration

Peter Jaeger
Published Online: 2016-08-31 | DOI: https://doi.org/10.1515/forma-2016-0001


First we give an implementation in Mizar [2] basic important definitions of stochastic finance, i.e. filtration ([9], pp. 183 and 185), adapted stochastic process ([9], p. 185) and predictable stochastic process ([6], p. 224). Second we give some concrete formalization and verification to real world examples.

In article [8] we started to define random variables for a similar presentation to the book [6]. Here we continue this study. Next we define the stochastic process. For further definitions based on stochastic process we implement the definition of filtration.

To get a better understanding we give a real world example and connect the statements to the theorems. Other similar examples are given in [10], pp. 143-159 and in [12], pp. 110-124. First we introduce sets which give informations referring to today (Ωnow, Def.6), tomorrow (Ωfut1 , Def.7) and the day after tomorrow (Ωfut2 , Def.8). We give an overview for some events in the σ-algebras Ωnow, Ωfut1 and Ωfut2, see theorems (22) and (23).

The given events are necessary for creating our next functions. The implementations take the form of: Ωnow ⊂ Ωfut1 ⊂ Ωfut2 see theorem (24). This tells us growing informations from now to the future 1=now, 2=tomorrow, 3=the day after tomorrow.

We install functions f : {1, 2, 3, 4} → ℝ as following:

f1 : x → 100, ∀x ∈ dom f, see theorem (36),

f2 : x → 80, for x = 1 or x = 2 and

f2 : x → 120, for x = 3 or x = 4, see theorem (37),

f3 : x → 60, for x = 1, f3 : x → 80, for x = 2 and

f3 : x → 100, for x = 3, f3 : x → 120, for x = 4 see theorem (38).

These functions are real random variable: f1 over Ωnow, f2 over Ωfut1, f3 over Ωfut2, see theorems (46), (43) and (40). We can prove that these functions can be used for giving an example for an adapted stochastic process. See theorem (49).

We want to give an interpretation to these functions: suppose you have an equity A which has now (= w1) the value 100. Tomorrow A changes depending which scenario occurs − e.g. another marketing strategy. In scenario 1 (= w11) it has the value 80, in scenario 2 (= w12) it has the value 120. The day after tomorrow A changes again. In scenario 1 (= w111) it has the value 60, in scenario 2 (= w112) the value 80, in scenario 3 (= w121) the value 100 and in scenario 4 (= w122) it has the value 120. For a visualization refer to the tree:

The sets w1,w11,w12,w111,w112,w121,w122 which are subsets of {1, 2, 3, 4}, see (22), tell us which market scenario occurs. The functions tell us the values to the relevant market scenario:

For a better understanding of the definition of the random variable and the relation to the functions refer to [7], p. 20. For the proof of certain sets as σ-fields refer to [7], pp. 10-11 and [9], pp. 1-2.

This article is the next step to the arbitrage opportunity. If you use for example a simple probability measure, refer, for example to literature [3], pp. 28-34, [6], p. 6 and p. 232 you can calculate whether an arbitrage exists or not. Note, that the example given in literature [3] needs 8 instead of 4 informations as in our model. If we want to code the first 3 given time points into our model we would have the following graph, see theorems (47), (44) and (41):

The function for the “Call-Option” is given in literature [3], p. 28. The function is realized in Def.5. As a background, more examples for using the definition of filtration are given in [9], pp. 185-188.

MSC: 60G05; 03B35

Keywords: stochastic process; random variable

MML: identifier: FINANCE3; version: 8.1.04 5.36.1267


  • [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Google Scholar

  • [2] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17.CrossrefGoogle Scholar

  • [3] Francesca Biagini and Daniel Rost. Money out of nothing? - Prinzipien und Grundlagen der Finanzmathematik. MATHE-LMU.DE, LMU-München(25):28-34, 2012.Google Scholar

  • [4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Google Scholar

  • [5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [6] Hans Föllmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.Google Scholar

  • [7] Hans-Otto Georgii. Stochastik, Einführung in die Wahrscheinlichkeitstheorie und Statistik. deGruyter, Berlin, 2nd edition, 2004.Google Scholar

  • [8] Peter Jaeger. Events of Borel sets, construction of Borel sets and random variables for stochastic finance. Formalized Mathematics, 22(3):199-204, 2014. doi:10.2478/forma-2014-0022.CrossrefGoogle Scholar

  • [9] Achim Klenke. Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, 2006.Google Scholar

  • [10] Jürgen Kremer. Einführung in die diskrete Finanzmathematik. Springer-Verlag, Berlin, Heidelberg, New York, 2006.Google Scholar

  • [11] Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Google Scholar

  • [12] Klaus Sandmann. Einführung in die Stochastik der Finanzmärkte. Springer-Verlag, Berlin, Heidelberg, New York, 2 edition, 2001.Google Scholar

  • [13] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990. Google Scholar

About the article

Received: 2015-12-30

Published Online: 2016-08-31

Published in Print: 2016-03-01

Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0001.

Export Citation

© by Peter Jaeger. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Peter Jaeger
Formalized Mathematics, 2017, Volume 25, Number 2

Comments (0)

Please log in or register to comment.
Log in