Introduction to Stopping Time in Stochastic Finance Theory

SummaryWe start with the definition of stopping time according to [4], p.283. We prove, that different definitions for stopping time can coincide. We give examples of stopping time using constant-functions or functions defined with the operator max or min (defined in [6], pp.37–38). Finally we give an example with some given filtration. Stopping time is very important for stochastic finance. A stopping time is the moment, where a certain event occurs ([7], p.372) and can be used together with stochastic processes ([4], p.283). Look at the following example: we install a function ST: {1,2,3,4} → {0, 1, 2} ∪ {+∞}, we define:a. ST(1)=1, ST(2)=1, ST(3)=2, ST(4)=2.b. The set {0,1,2} consists of time points: 0=now,1=tomorrow,2=the day after tomorrow. We can prove:c. {w, where w is Element of Ω: ST.w=0}=∅ & {w, where w is Element of Ω: ST.w=1}={1,2} & {w, where w is Element of Ω: ST.w=2}={3,4} andST is a stopping time. We use a function Filt as Filtration of {0,1,2}, Σ where Filt(0)=Ωnow, Filt(1)=Ωfut1 and Filt(2)=Ωfut2. From a., b. and c. we know that:d. {w, where w is Element of Ω: ST.w=0} in Ωnow and{w, where w is Element of Ω: ST.w=1} in Ωfut1 and{w, where w is Element of Ω: ST.w=2} in Ωfut2.The sets in d. are events, which occur at the time points 0(=now), 1(=tomorrow) or 2(=the day after tomorrow), see also [7], p.371. Suppose we have ST(1)=+∞, then this means that for 1 the corresponding event never occurs.As an interpretation for our installed functions consider the given adapted stochastic process in the article [5].ST(1)=1 means, that the given element 1 in {1,2,3,4} is stopped in 1 (=tomorrow). That tells us, that we have to look at the value f2(1) which is equal to 80. The same argumentation can be applied for the element 2 in {1,2,3,4}.ST(3)=2 means, that the given element 3 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(3) which is equal to 100.ST(4)=2 means, that the given element 4 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(4) which is equal to 120.In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see [7], p.372).Siegmund-Schacky-Str. 18a, 80993 Munich, GermanyGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 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.Hans Föllmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.Peter Jaeger. Modelling real world using stochastic processes and filtration. Formalized Mathematics, 24(1):1–16, 2016. doi: 10.1515/forma-2016-0001.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.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2): 329–334, 1990.25210110

An overview of probabilistic and time series models in finance

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In this paper, we study reflected backward stochastic difference equations (RBSDEs for short) with finitely many states in discrete time. The general existence and uniqueness result, as well as comparison theorems for the solutions, are established under mild assumptions. The connections between RBSDEs and optimal stopping problems are also given. Then we apply the obtained results to explore optimal stopping problems under $g$-expectation. Finally, we study the pricing of American contingent claims in our context.Comment: 29 page