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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

Optimal Stopping for Partial Sums   (Citations: 34)
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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. 1363-1368, 1972
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    • ...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 one-dimensional 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 Karouiet al. Max-Plus 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 Novikovet al. On a solution of the optimal stopping problem for processes with indep...

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