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Keywords
(7)
Expected Returns
Optimal Stopping
Optimal Stopping Time
Partial Sums
Random Variable
Stopping Rule
Independent Identically Distributed
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Optimal Stopping for Partial Sums
Optimal Stopping for Partial Sums,10.1214/aoms/1177692491,The Annals of Mathematical Statistics,D. A. Darling,T. Liggett,H. M. Taylor
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Optimal Stopping for Partial Sums
(
Citations: 34
)
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D. A. Darling
,
T. Liggett
,
H. M. Taylor
We determine $\sup E\lbrack r(S_T)\rbrack$, where $S_n$ is a sequence of
partial sums
of
independent identically distributed
random variables, for two reward functions: $r(x) = x^+$ and $r(x) = (e^x  1)^+$. The supremum is taken over all stop rules $T$. We give conditions under which the optimal expected return is finite. Under these conditions,
optimal stopping
times exist, and we determine them. The problem has an interpretation in an action timing problem in finance.
Journal:
The Annals of Mathematical Statistics
, vol. 43, no. 1972, pp. 13631368, 1972
DOI:
10.1214/aoms/1177692491
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Citation Context
(18)
...[
4
] and Mordecki [
7
]...
Paavo Salminen
.
Optimal stopping, Appell polynomials, and Wiener–Hopf factorization
...random variables (eg, Dubins and Teicher,
^{[}
^{9}
^{]}
Darling et al,
^{[}
^{7}
^{]}
Ferguson and McQueen,
^{[}
^{11}
^{]}
and Kramkov and Shiryaev
^{[}
^{14}
^{]}
); geometric Brownian motion (Shepp and Shiryaev
^{[}
^{17}
^{]}
); Lévy processes (Mordecki
^{[}
^{15}
^{]}
); and general onedimensional diffusions (eg, Dayanik and Karatzas
^{[}
^{8}
^{]}
)...
Pieter C. Allaart
.
Optimal Stopping Rules for American and Russian Options in a Correlate...
...This particularly generalizes the results of Darling, Liggett and Taylor [
10
] on American options, only valid in the discrete case, where the underlying defines a partial sum of independent and identically distributed random variables with negative drift...
...In this section we generalize the ideas of Darling, Liggett and Taylor, who considered in their paper [
10
] American Call options written on partial sums Sn of independent and identically distributed random variables with negative drift, and characterized optimal stopping times in terms of the running supremum of the underlying Sn...
...Similar ideas are used by Darling, Liggett and Taylor [
10
] to solve the optimal stopping problem in the discrete case...
Nicole El Karoui
,
et al.
MaxPlus decomposition of supermartingales and convex order. Applicati...
...techniques in order to gain more detailed information on the problem. In [
10
], Darling et al. solve...
...Alili and Kyprianou [1], Boyarchenko and Levendorski•i [7], Darling et al. [
10
], Mordecki [15])...
...(C) E £ eM ⁄ < 1. Proof. See [
10
], pp. 1367...
...problem (1.3). This theorem is essentially due to Darling et al. [
10
], where they consider the case...
...For simplicity, assume that c = 1. Then it is known from Darling et al. [
10
], pp. 1368 that the threshold s⁄ can be expressed as...
Jukka Lempa
.
On infinite horizon optimal stopping of general random walk
...The explicit solution of the problem under consideration for discrete time setting and the case = 1 was found in [7] and [
6
]...
...Here we present some details of the proof for (38) only for the case t 2 Z+ and q > 0 . The idea of our proof is similar to that one used in [
6
] and [11] and it is based on Lemma 5 and the following fact known as Lindley recursion:...
Alexander Novikov
,
et al.
On a solution of the optimal stopping problem for processes with indep...
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Citations
(34)
Optimal stopping, Appell polynomials, and Wiener–Hopf factorization
Paavo Salminen
Journal:
Stochastics An International Journal of Probability and Stochastic Processes
, vol. aheadofp, no. aheadofp, 2011
Optimal Stopping Rules for American and Russian Options in a Correlated Random Walk Model
Pieter C. Allaart
Journal:
Stochastic Models  STOCH MODELS
, vol. 26, no. 4, pp. 594616, 2010
Optimal stopping, Appell polynomials and WienerHopf factorization representations of excessive functions of L\'evy processes
Paavo Salminen
Published in 2010.
Optimal stopping problem for processes with independent increments
Georgiy M. Shevchenko
,
Anna G. Moroz
Published in 2009.
MaxPlus decomposition of supermartingales and convex order. Application to American options and portfolio insurance
(
Citations: 2
)
Nicole El Karoui
,
Asma Meziou
Journal:
Annals of Probability  ANN PROBAB
, vol. 36, no. 2, pp. 647697, 2008