Wyniki wyszukiwania - Biblioteka Nauki

1

Introduction to Stopping Time in Stochastic Finance Theory

100%

Jaeger P.

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nr 2

101-105

EN

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).

2

Conjugate priors for exponential-type processes with random initial conditions

80%

Magiera R.

Applicationes Mathematicae

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1993-1995

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tom 22

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nr 3

321-330

EN

The family of proper conjugate priors is characterized in a general exponential model for stochastic processes which may start from a random state and/or time.

3

Bayes sequential estimation procedures for exponential-type processes

80%

Magiera R.

Applicationes Mathematicae

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1993-1995

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tom 22

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nr 3

311-320

EN

The Bayesian sequential estimation problem for an exponential family of processes is considered. Using a weighted square error loss and observing cost involving a linear function of the process, the Bayes sequential procedures are derived.

4

On minimax sequential procedures for exponential families of stochastic processes

70%

Magiera R.

Applicationes Mathematicae

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1998-1999

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tom 25

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nr 1

1-18

EN

The problem of finding minimax sequential estimation procedures for stochastic processes is considered. It is assumed that in addition to the loss associated with the error of estimation a cost of observing the process is incurred. A class of minimax sequential procedures is derived explicitly for a one-parameter exponential family of stochastic processes. The minimax sequential procedures are presented in some special models, in particular, for estimating a parameter of exponential families of diffusions, for estimating the mean or drift coefficients of the class of Ornstein-Uhlenbeck processes, for estimating the drift of a geometric Brownian motion and for estimating a parameter of a family of counting processes. A class of minimax sequential rules for a compound Poisson process with multinomial jumps is also found.

5

A random walk version of Robbins' problem : small horizon

51%

Allaart P. C. , Allen A.

Mathematica Applicanda

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2019

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tom Vol. 47, no. 2

293--312

EN

In Robbins' problem of minimizing the expected rank, a finite sequence of n independent, identically distributed random variables are observed sequentially and the objective is to stop at such a time that the expected rank of the selected variable (among the sequence of all n variables) is as small as possible. In this paper we consider an analogous problem in which the observed random variables are the steps of a symmetric random walk. Assuming continuously distributed step sizes, we describe the optimal stopping rules for the cases n = 2 and n = 3 in two versions of the problem: a „full information" version in which the actual steps of the random walk are disclosed to the decision maker; and a „partial information" version in which only the relative ranks of the positions taken by the random walk are observed. When n = 3, the optimal rule and expected rank depend on the distribution of the step sizes. We give sharp bounds for the optimal expected rank in the partial information version, and fairly sharp bounds in the full information version.

PL

W problemie Robbinsa celem jest zatrzymanie sekwencyjnych obserwacji skończonego ciągu niezależnych zmiennych losowych o tym samym rozkładzie tak, aby zminimalizować oczekiwaną rangę zatrzymanej zmiennej. Niniejsza praca poświęcona jest analogonowi problemu Robbinsa, w którym obserwowane zmienne losowe są wartościami symetrycznego błądzenia losowego. Zakładamy, że długości kroków są symetrycznymi zmiennymi losowymi o rozkładzie typu ciągłego. Opisujemy optymalne reguły zatrzymania dla przypadków n = 2 i n = 3 w dwóch wersjach problemu: wersja z pełną informacją, w której rzeczywiste długości kroków losowych są jawne i znane podejmującemu decyzje statystykowi, oraz wersja z częściową informacją, w której obserwowane są tylko względne ciągi pozycji zajmowanych przez ciągły, symetryczny, spacer losowy. Dla n = 3 optymalna strategia i oczekiwana ranga zależą od rozkładu długości kroków. Otrzymano ostre oszacowania dla wartości oczekiwanej otrzymanej rangi dla wersji problemu z częściową informacją oraz lepsze oszacowania dla problemu z pełną informacją.

6

On the exponential Orlicz norms of stopped Brownian motion

41%

Peškir G.

Studia Mathematica

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1995-1996

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tom 117

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nr 3

253-273

EN

Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by $ψ_p(x) = exp(|x|^p)-1$ with 0 < p ≤ 2) of $max_{0≤t≤τ}|B_t|$ or $|B_τ|$ to be finite, where $B = (B_t)_{t≥0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $∥max_{0≤t≤τ}|B_t|∥_{ψ_1} < ∞$ as soon as $E(τ^{k}) = O(C^{k}k^{k})$ for some constant C > 0 as k → ∞ (or equivalently $∥τ∥_{ψ_1} < ∞$). In particular, if τ ∼ Exp(λ) or $|N(0,σ^2)|$ then the last condition is satisfied, and we obtain $∥max_{0≤t≤τ}|B_t|∥_{ψ_1} ≤ K √{E(τ)}$ with some universal constant K > 0. Moreover, this inequality remains valid for any class of stopping times τ for B satisfying $E(τ^{k}) ≤ C(Eτ)^{k}k^{k}$ for all k ≥ 1 with some fixed constant C > 0. The method of proof relies upon Taylor expansion, Burkholder-Gundy's inequality, best constants in Doob's maximal inequality, Davis' best constants in the $L^p$-inequalities for stopped Brownian motion, and estimates of the smallest and largest positive zero of Hermite polynomials. The results extend to the case of any continuous local martingale (by applying the time change method of Dubins and Schwarz).

7

An apartment problem

41%

Szajowski K.

Mathematica Applicanda

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2014

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tom Vol. 42, No. 2

193--206

EN

In the 60 - ies of the last century, several optimization problems referring to the sequential methods were investigated. These tasks may include the Robbins’ problem of optimal stopping, the secretary problem (see the discussion paper by Ferguson [18]), the parking problem or the job search problem. Subtle details of the wording in these issues cause that each of these terms include family of problems that differ significantly in detail. These issues focused attention of a large group of mathematicians. One of the related topic has been the subject of Professor Jerzy Zabczyk attention. Based on the discussions with Professor Richard Cowan1 the model of choosing the best facility available from a random number of offers was established. In contemporary classification of the best choice problems it is the noinformation, continuous time, secretary problem with the Poisson stream of options and the finite horizon.

PL

W latach 60 -tych poprzedniego wieku analizowano wielu matematyków skupiało swoja uwagę na zadaniach optymalizacyjnych nawiązujących do sekwencyjnego przeszukiwania czy obserwacji. Do tych zadań można zaliczyć problem optymalnego zatrzymania Robbinsa, problem sekretarki, (dość obszerną analizę tego zagadnienia przeprowadził Ferguson [18]), zadanie optymalnego parkowania czy też problem poszukiwania pracy. Subtelne szczegóły tych zagadnień powodują, iż każde zagadnienie z wymienionych ma liczne wersje różniące się szczegółami, które powodują, iż mamy do czynienia całą rodziną modeli. Jedno z zagadnień zainteresowało profesora Jerzy Zabczyk. W wyniku dyskusji z profesorem Richardem Cowanem (w Warszawie ) stworzyli model poszukiwania najlepszego obiektu, gdy dostępnych obiektów jest losowa liczba. Wg współczesnej klasyfikacji problemów wyboru najlepszego obiektu jest to przypadek poszukiwania najlepszego obiektu przy braku informacji, z czasem ciągłym, gdy strumień zgłoszeń jest poissonowski a horyzont jest skończony, ustalony.

8

Travel data collection using a smart phone for the estimation of multimodal travel times of intra-city public transportation

41%

Reddy K. T. V. K. , Challagulla S. P.

Archives of Civil Engineering

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2022

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tom Vol. 68, nr 3

397--409

EN

All the available modes of travel and their respective travel parameters must be known to the commuters before their trip. Otherwise they may either spend more money or more time for the trip. In addition to this, recent pandemic, rapidly spreading novel corona virus is demanding a smart solution for contactless commuting. This paper suggests a practical solution to make both the above possible and it emphasizes the applicability of two developed android applications, one for travel data collection and another to predict travel time for a multimodal trip within the study area. If the whole trip is by a single mode, the user can get the corresponding travel time estimate from “Google maps”. But, if the trip is by multiple modes, it is not possible to get the total travel time estimate for the whole trip at a time from “Google maps”. A separate travel mode for “auto” is unavailable in “Google maps” alongside drive, two-wheeler, train or bus and walk alternatives. It is also observed that the travel time estimate of “Google maps” for the city buses is inaccurate. Hence, the two modes (Buses and Autos) were chosen for the study. Unless and until the travel times and stopping times of the two modes are known, it is not possible to predict their trip times. Hence, the mobility analysis was performed for the two modes in the study area to find their respective average travel rate at peak hours, across 15 corridors and the results were presented.