2,194 research outputs found
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 Ï-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 â Ωfut1 â 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} â â as following: f1 : x â 100, âx â dom f, see theorem (36), f2 : x â 80, for x = 1 or x = 2 and f2 : x â 120, for x = 3 or x = 4, see theorem (37), f3 : x â 60, for x = 1, f3 : x â 80, for x = 2 and f3 : x â 100, for x = 3, f3 : x â 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 â 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 Ï-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 âCall-Optionâ 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, CzesĆaw ByliĆski, Adam Grabowski, Artur KorniĆowicz, Roman Matuszewski, Adam Naumowicz, Karol PÄ
k, 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-MĂŒnchen(25):28-34, 2012.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw ByliĆski. 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, EinfĂŒhrung 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.JĂŒrgen Kremer. EinfĂŒhrung in die diskrete Finanzmathematik. Springer-Verlag, Berlin, Heidelberg, New York, 2006.Andrzej NÄdzusiak. Ï-fields and probability. Formalized Mathematics, 1(2):401-407, 1990.Klaus Sandmann. EinfĂŒhrung in die Stochastik der FinanzmĂ€rkte. Springer-Verlag, Berlin, Heidelberg, New York, 2 edition, 2001.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990
A remarkable -finite measure associated with last passage times and penalisation problems
In this paper, we give a global view of the results we have obtained in
relation with a remarkable class of submartingales, called , and its
links with a universal sigma-finite measure and penalization problems on the
space of continuous and cadlag paths.Comment: This is a contribution for the special volume for the 60th birthday
of Eckhard Plate
Supermartingales as Radon-Nikodym densities and related measure extensions
Certain countably and finitely additive measures can be associated to a given
nonnegative supermartingale. Under weak assumptions on the underlying
probability space, existence and (non)uniqueness results for such measures are
proven.Comment: Published at http://dx.doi.org/10.1214/14-AOP956 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On the connections between PCTL and Dynamic Programming
Probabilistic Computation Tree Logic (PCTL) is a well-known modal logic which
has become a standard for expressing temporal properties of finite-state Markov
chains in the context of automated model checking. In this paper, we give a
definition of PCTL for noncountable-space Markov chains, and we show that there
is a substantial affinity between certain of its operators and problems of
Dynamic Programming. After proving some uniqueness properties of the solutions
to the latter, we conclude the paper with two examples to show that some
recovery strategies in practical applications, which are naturally stated as
reach-avoid problems, can be actually viewed as particular cases of PCTL
formulas.Comment: Submitte
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