Modelling Real World Using Stochastic Processes and Filtration

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.Jaeger Peter - Siegmund-Schacky-Str. 18a 80993 Munich, GermanyGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.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.Francesca Biagini and Daniel Rost. Money out of nothing? - Prinzipien und Grundlagen der Finanzmathematik. MATHE-LMU.DE, LMU-München(25):28-34, 2012.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Hans Föllmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.Hans-Otto Georgii. Stochastik, Einführung in die Wahrscheinlichkeitstheorie und Statistik. deGruyter, Berlin, 2nd edition, 2004.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.Achim Klenke. Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, 2006.Jürgen Kremer. Einführung in die diskrete Finanzmathematik. Springer-Verlag, Berlin, Heidelberg, New York, 2006.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Klaus Sandmann. Einführung in die Stochastik der Finanzmärkte. Springer-Verlag, Berlin, Heidelberg, New York, 2 edition, 2001.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990

Wölfe

In internationalen Übereinkommen wie der Berner Konvention, der Fauna-Flora-Habitat-Richtlinie und der Biodiversitäts-Konvention von Rio verpflichten sich die Staaten zu Maßnahmen zur Regenerierung bedrohter Arten und Lebensräume. Danach hat sich auch die Bundesrepublik Deutschland verpflichtet, die Rückkehr des Wolfes zu unterstützen. Leider stehen dieser Rückkehr nicht unerhebliche Hindernisse im Wege. Hier sind vor allem die zunehmende Zerstückelung des Lebensraumes durch Straßen, Eisenbahntrassen und andere Infrastruktureinrichtungen zu nennen. Verluste durch den Verkehr, genauso wie solche aufgrund direkter (illegaler) Verfolgung oder Verwechslung mit wildernden Hunden, schränken eine mögliche Populationsentwicklung nicht unwesentlich ein. Die Menschen müssen lernen, sich auf die Wiederkehr des Wolfes einzustellen, wieder mit dem Wolf zu leben. Im Bereich der Viehhaltung sind vor allem Konflikte mit Schafhaltern zu erwarten. Insbesondere Schafe, die nicht ausreichend gesichert sind, können gefährdet sein. Hier müssen Konzepte gefunden werden, in denen bei der Viehhaltung - in Abhängigkeit von der jeweils spezifischen Situation - neuere Schutzstrategien mit alten, traditionellen Schutzvorkehrungen (die zum Teil schon in Vergessenheit geraten sind) kombiniert werden, um das örtlich Geeignete zu realisieren. So können, wie sich in anderen Ländern gezeigt hat, durch Maßnahmen wie die Errichtung von Elektro- und/oder Lappenzäunen, die Unterbringung der Tiere in Nachteinständen, die Beaufsichtigung durch Hirten, ebenso wie durch den Einsatz von Herdenschutzhunden oder verschiedene Formen der Vergrämung, Verluste weitgehend vermieden werden. Derzeit werden für etwaige Viehverluste durch Wölfe in diversen europäischen Staaten (z.Z. noch nicht in der BRD) staatliche Entschädigungen geleistet, die eventuell durch private Mittel von NGO’s wie z.B. die GzSdW oder anderen Naturschutzorganisationen ergänzt werden könnten. Eine Wiederbesiedelung von Wölfen hat auch Einfluss auf das Jagdwesen. Deshalb wird es wichtig sein, gemeinsam mit der Jägerschaft, den Forstleuten und Wildbiologen das gesamte Mensch-Wald-Wildsystem als Ganzes langfristig zu beobachten, um daraus die richtigen Maßnahmen ableiten zu können

Elementary Introduction to Stochastic Finance in Discrete Time

This article gives an elementary introduction to stochastic finance (in discrete time). A formalization of random variables is given and some elements of Borel sets are considered. Furthermore, special functions (for buying a present portfolio and the value of a portfolio in the future) and some statements about the relation between these functions are introduced. For details see: [8] (p. 185), [7] (pp. 12, 20), [6] (pp. 3-6).Ludwig Maximilians University of Munich, GermanyGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Hans Föllmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.Hans-Otto Georgii. Stochastik, Einführung in die Wahrscheinlichkeitstheorie und Statistik. deGruyter, Berlin, 2 edition, 2004.Achim Klenke. Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, 2006.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Andrzej Nędzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Konrad Raczkowski and Andrzej Nędzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Introduction to Stochastic Finance: Random Variables and Arbitrage Theory

Using the Mizar system [1], [5], we start to show, that the Call-Option, the Put-Option and the Straddle (more generally defined as in the literature) are random variables ([4], p. 15), see (Def. 1) and (Def. 2). Next we construct and prove the simple random variables ([2], p. 14) in (Def. 8). In the third section, we introduce the definition of arbitrage opportunity, see (Def. 12). Next we show, that this definition can be characterized in a different way (Lemma 1.3. in [4], p. 5), see (17). In our formalization for Lemma 1.3 we make the assumption that ϕ is a sequence of real numbers (there are only finitely many valued of interest, the values of ϕ in Rd). For the definition of almost sure with probability 1 see p. 6 in [2]. Last we introduce the risk-neutral probability (Definition 1.4, p. 6 in [4]), here see (Def. 16). We give an example in real world: Suppose you have some assets like bonds (riskless assets). Then we can fix our price for these bonds with x for today and x · (1 + r) for tomorrow, r is the interest rate. So we simply assume, that in every possible market evolution of tomorrow we have a determinated value. Then every probability measure of Ωfut1 is a risk-neutral measure, see (21). This example shows the existence of some risk-neutral measure. If you find more than one of them, you can determine – with an additional conidition to the probability measures – whether a market model is arbitrage free or not (see Theorem 1.6. in [4], p. 6.) A short graph for (21): Suppose we have a portfolio with many (in this example infinitely many) assets. For asset d we have the price π(d) for today, and the price π(d) (1 + r) for tomorrow with some interest rate r > 0. Let G be a sequence of random variables on Ωfut1, Borel sets. So you have many functions fk : {1, 2, 3, 4}→ R with G(k) = fk and fk is a random variable of Ωfut1, Borel sets. For every fk we have fk(w) = π(k)·(1+r) for w {1, 2, 3, 4}. Today Tomorrow only one scenario {w21={1,2} w22={3,4} for all d∈N holds π(d) {fd(w)=G(d)(w)=π(d)⋅(1+r), w∈w21 or w∈w22, r>0 is the interest rate. Here, every probability measure of Ωfut1 is a risk-neutral measure.Siegmund-Schacky-Str. 18a, 80993 Munich, GermanyGrzegorz 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.Heinz Bauer. Wahrscheinlichkeitstheorie. de Gruyter-Verlag, Berlin, New York, 2002.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. The measurability of extended real valued functions. Formalized Mathematics, 9(3):525–529, 2001.Hans Föllmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Peter Jaeger. Elementary introduction to stochastic finance in discrete time. Formalized Mathematics, 20(1):1–5, 2012. doi:10.2478/v10037-012-0001-5.Peter Jaeger. Modelling real world using stochastic processes and filtration. Formalized Mathematics, 24(1):1–16, 2016. doi:10.1515/forma-2016-0001.2611