scholarly journals Introduction to Stopping Time in Stochastic Finance Theory

2017 ◽  
Vol 25 (2) ◽  
pp. 101-105 ◽  
Author(s):  
Peter Jaeger
Keyword(s):  
Stopping Time ◽  
Share Price ◽  
Time Points ◽  
Is Element ◽  
The Moment ◽  
Definition Of ◽  
Finance Theory ◽  
Article 5 ◽  
The Given ◽  
Stochastic Finance

Summary We 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} and ST 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).

scholarly journals Modelling Real World Using Stochastic Processes and Filtration

2016 ◽  
Vol 24 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Peter Jaeger
Keyword(s):  
Stochastic Process ◽  
Real World ◽  
Random Variable ◽  
Call Option ◽  
Relevant Market ◽  
Real Random Variable ◽  
Time Points ◽  
Arbitrage Opportunity ◽  
Definition Of ◽  
The Given

Summary 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.

scholarly journals Statistical causality and local uniqueness for solutions of the martingale problem

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2851-2860
Author(s):  
Ljiljana Petrovic ◽  
Dragana Valjarevic
Keyword(s):  
Martingale Problem ◽  
Stopping Time ◽  
Stopping Times ◽  
Local Uniqueness ◽  
Causality Relationship ◽  
Definition Of ◽  
The Given

In this paper we consider the concept of statistical causality between filtrations associated with stopping times, which is based on Granger?s definition of causality. Especially, we consider a generalization of a causality relationship ?G is a cause of E within H? from fixed to stopping time. Then we apply the given causality concept to local uniqueness for the solution of the martingale problem. Also, we give some applications in finance.

Utilization of remote sensing for estimation of scale of cutting and growing of forest zones

2017 ◽  
Vol 920 (2) ◽  
pp. 57-60
Author(s):  
F.E. Guliyeva
Keyword(s):  
Remote Sensing ◽  
Fixed Time ◽  
Time Interval ◽  
Time Points ◽  
Average Value ◽  
Time Period ◽  
Remote Estimation ◽  
The Given ◽  
Forest Cutting

The study of results of relevant works on remote sensing of forests has shown that the known methods of remote estimation of forest cuts and growth don’t allow to calculate the objective average value of forests cut volume during the fixed time period. The existing mathematical estimates are not monotonous and make it possible to estimate primitively the scale of cutting by computing the ratio of data in two fixed time points. In the article the extreme properties of the considered estimates for deforestation and reforestation models are researched. The extreme features of integrated averaged values of given estimates upon limitations applied on variables, characterizing the deforestation and reforestation processes are studied. The integrated parameter, making it possible to calculate the averaged value of estimates of forest cutting, computed for all fixed time period with a fixed step is suggested. It is shown mathematically that the given estimate has a monotonous feature in regard of value of given time interval and make it possible to evaluate objectively the scales of forest cutting.

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

Alexandre Bennigsen in the Eyes of the Soviet Press

1986 ◽  
Vol 14 (1-2) ◽  
pp. 5-18
Author(s):  
Michael Rywkin
Keyword(s):  
Soviet Union ◽  
Soviet Regime ◽  
Political Opportunities ◽  
The Soviet Union ◽  
Delivery Vehicles ◽  
The Moment ◽  
General Relations ◽  
The Given

Soviet reactions to Western writings on the Soviet Union are as old as the Soviet regime itself. They are handled in an organized manner, with targets, delivery vehicles and gradation of response carefully coordinated and measured.Soviet response is, moreover, not solely connected to the perceived degree of offensiveness of the given Western work; in addition, such considerations as general relations between the USSR and the country from where the publication came, as well as political opportunities of the moment, are given even more importance than the committed “offense.”

Underground Built Heritage (UBH) as Valuable Resource in China, Japan and Italy

Heritage ◽  
2021 ◽  
Vol 4 (4) ◽  
pp. 3208-3237
Author(s):  
Roberta Varriale ◽  
Laura Genovese
Keyword(s):  
Research Council ◽  
Methodological Approach ◽  
National Research Council ◽  
Functional Classification ◽  
Static And Dynamic Analysis ◽  
Built Heritage ◽  
Living Space ◽  
Definition Of ◽  
The Given ◽  
National Systems

Recent research about the theoretical approach to elements of cultural heritage that can be included in the newly born class Underground Built Heritage (UBH), has provided several instruments for the functional classification and the static and dynamic analysis of all artefacts coherent with the given definition, while introducing several criteria for their reuse and the evaluation of connected enhancement processes as well. These guidelines can be adopted to analyze single artefacts, groups of homogenous or heterogeneous elements, and also selected territorial assets or national systems, even at a comparative level. With reference to this potential, research results from the application of this new methodological approach to the outputs of three ongoing projects by the National Research Council of Italy, all focusing on UBH, in three countries: China, Japan and Italy, are presented. With reference to the above-mentioned geographical contests, the research introduces a comparative study focusing on selected examples of artefacts that have been historically built underground to manage three functions: living space, religion and economy. This study, carried out based on data collected during onsite visits by the authors, consists in three steps: selection and analysis of case studies, definition of level of reuses on the basis of a given scale, and analysis of the different tools adopted for their conservation and enhancement. In the conclusions, possible future implementations of reuses of the analyzed elements are pointed out.

The Home at the core of Modernity, an optimistic architecture

2021 ◽  
pp. 2-3
Author(s):  
Ana Tostões
Keyword(s):  
Daily Life ◽  
Single Family ◽  
The Core ◽  
The Moment ◽  
Definition Of ◽  
The Ideal ◽  
Modern Movement

Devoted to the theme of single-family houses, given the key role they played in the ideal definition of the Modern Movement architecture, as a symbolic and functional affirmation of the utopian turning of dreams into reality, the aim of this issue is to consider the transformation of daily life, and to address the architectural challenges that arose from the joy contained in what we might call the “architecture of happiness.” As we continue to endure a pandemic that has now lasted for more than a year, docomomo wishes to declare that “till the moment, the best vaccine to prevent contagion was invented by architects: the house”. Thus, in response to the question “How should we live?”, it is intended to debate the house and the home agenda as an important topic at the core of Modern Movement architecture. Nowadays, the growing emphasis on wellbeing goes beyond the seminal ideas that modern houses were “machines à habiter” and is closer to an idealistic vision of a stimulating shell for humans, which is shaped by imagination, experimentation, efficiency, and knowledge.

ASSESSMENT AND FORECASTING OF FREIGHT CAR TECHNICAL STATE

2021 ◽  
Vol 2021 (5) ◽  
pp. 40-47
Author(s):  
Elena Rozhkova
Keyword(s):  
Graphical Model ◽  
Technical State ◽  
Distribution Law ◽  
Repair Service ◽  
Statistic Modeling ◽  
Average Operating ◽  
Freight Car ◽  
Run Up ◽  
Definition Of ◽  
The Given

The purpose of this work is the assessment and forecasting of freight car technical state at life stages. The scientific novelty consists in the definition of car operating run-up to the first failure and between failures, and also in the definition of car life to considerable repair fulfillment from the operating run between failures and development of a graphical model of freight car reliability. As a result of the statistic modeling of gondola car operating runs up to the first failure it is defined that the given random value conforms to a normal distribution law, the first car setoff in TOR due to wear failure takes place at the operating run of 85,000 km. Besides the operating run-up to the first failure there was defined an operating run-up between failures. It is proved that the operating run-up between failures conforms to the exponential law of distribution, the mathematical expectation of which is 13,000 km. The results of investigations mentioned above formed the basis of the graphical model of car reliability. An inter-repair service life of a car can be represented as a sum of the following operating run-ups: operating run-ups to the first failure, the product of operating run-ups between failures and the number of failures and a residual operating run-up (from the utmost current repair to the nearest scheduled repair). On the basis of the mentioned it is expedient to consider a technology for the realization of an enlarged repair with the purpose of the repetition exclusion in car setoffs during the inter-repair term. On the basis of the simulator there is obtained the dependence of the frequency of car enlarged current repair fulfillment depending on average operating run-up between failures. At present an enlarged current repair must be carried out only for gondola cars in the planned order after having reached 80,000 km. The repair mentioned can be carried out both under depot conditions, and under conditions of repair workshops. The advantage of such a system of repair consists in the increase car work reliability during the inter-repair term.

Plaidoyer for the unique tourism accounts (dilemas, difficulties and questions)

1998 ◽  
Vol 4 (1) ◽  
pp. 121-134
Author(s):  
Vladimir Stipetić
Keyword(s):  
National Economy ◽  
Scientific Basis ◽  
National Accounts ◽  
System Of National Accounts ◽  
Tourism Sector ◽  
European System ◽  
Cross Country ◽  
Comparative Results ◽  
The Moment ◽  
The Given

Tourism is recognised as an important economic, social and cultural factor in modem societies. However, statisticians still face an enormous challenge when it comes to measuring the economic importance of tourism for given national economy. In his paper author examines the position of tourism in System of National Accounts (SNA-as presented in the version from 1993) and in European System of Accounts (acronym ESA, from 1995). He comes to the conclusion that is impossible to get the full answers on the importance of tourism for the given national economy within those frameworks. The main reason for such conclusion he has found in the fact that tourism is the multifacet activity, with difficult concept to define, let alone measure. He gets the proof for his conclusion comparing the different methodologies for defining who is the tourist, finding six different criterias in application at the moment. The results obtained are, of course, differing considerably, what makes them strictly incompatibles. The comparison of such data is for that reason difficult to make and the comparative results are of limited use. Author advocates a need to make on international level a Unified Tourism Economic Account (UTEA), covering the main activities of tourism sector. He regards the existing work by OECD and WTO as a good basis for further work. Only when majority countries would make UTEA, based on accepted methodology, could be the cross-country comparison made on scientific basis.

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