Basic inequalities for weighted entropies

Suhov, Yuri; Stuhl, Izabella; Sekeh, Salimeh Yasaei; Kelbert, Mark

doi:10.1007/s00010-015-0396-5

Cited by 29 publications

(45 citation statements)

References 30 publications

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“…4.4. As earlier, the staple of the proof of Theorem 4.1 is the Gibbs inequality for weighted entropies; see [14,17]. It is similar to the standard Gibbs inequality (cf.…”

Section: A General Discrete-time Settingmentioning

confidence: 88%

“…We also intend to use a recent progress in studying weighted entropies; cf. [14,15,16,17]. A strong impact on the whole direction of this research was made by [8,9] where a powerful methodology of a convex analysis has been developed (and elegantly presented) in a general form, leading -among other achievements -to existence of log-optimal portfolios.…”

Section: A Markovian Model With a Single Risky Assetmentioning

confidence: 99%

“…For the definition and basic properties of weighted entropies, see [14] and references therein. Some applications of weighted entropies are discussed in [15,16,17].…”

Section: A Markovian Model With a Single Risky Assetmentioning

confidence: 99%

“…Consequently, if α ≤ 0 for some CF q satisfying (q-p) in (14) then the risk-averse trader would restrain from investments. Define…”

Section: A Markovian Model With a Single Risky Assetmentioning

confidence: 99%

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Weighted entropy and optimal portfolios for risk-averse Kelly investments

Kelbert¹,

Stuhl

Suhov

2017

Aequat. Math.

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Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes 'weights' of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting. We focus on properties of the optimal portfolios and discuss a number of simple examples extending the well-known Kelly betting scheme.An important restriction is that the investment does not exceed the current capital value and allows the trader to cover the worst possible losses.The paper deals with a class of discrete-time models. A continuous-time extension is a topic of an ongoing study. A Markovian model with a single risky assetThis paper is an initial part of a work on log-optimal portfolios influenced by a number of earlier publications, mainly by T. Cover and co-authors. Cf. Refs [1,3] and [4], Chapter 6. Also see [10,12,18] and Ref [11], Parts II and III. We also intend to use a recent progress in studying weighted entropies; cf. [14,15,16,17]. A strong impact on the whole direction of this research was made by [8,9] where a powerful methodology of a convex analysis has been developed (and elegantly presented) in a general form, leading -among other achievements -to existence of log-optimal portfolios. See Theorem 1 from [9]. In the present article, we attempt to go beyond the issue of existence and provide a specific form of the optimal strategy.Let us discuss a finance-related context of this work. The sequential version of portfolio selection problem has received much attention in the literature not to speak about the financial practice, see [10,12,11,18] and the references therein. A simple discrete-time model of a wide use in financial engineering is where the market consists of one riskless asset and one or more risky assets. (If the riskless asset produces a zero return, we can speak of risky assets only.) Investments are made at times n − 1 = 0, 1, . . .; the returns are recorded at subsequent times n = 1, 2, . . ..We consider two investment schemes, showing that the results are valid for both schemes mutatis mutandis.Scheme I: an investor signs a deal with a broker at the time n − 1 but the actual transaction happens at the moment n when the betting results become available.Scheme II: at the moment n − 1 an investor transfers the required capital to a broker who invests this capital to buy shares or other risky assets.In fact, Scheme II can be treated as a version of Scheme I, where the number of assets increases by 1.

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“…4.4. As earlier, the staple of the proof of Theorem 4.1 is the Gibbs inequality for weighted entropies; see [14,17]. It is similar to the standard Gibbs inequality (cf.…”

Section: A General Discrete-time Settingmentioning

confidence: 88%

Section: A Markovian Model With a Single Risky Assetmentioning

confidence: 99%

“…For the definition and basic properties of weighted entropies, see [14] and references therein. Some applications of weighted entropies are discussed in [15,16,17].…”

Section: A Markovian Model With a Single Risky Assetmentioning

confidence: 99%

“…Consequently, if α ≤ 0 for some CF q satisfying (q-p) in (14) then the risk-averse trader would restrain from investments. Define…”

Section: A Markovian Model With a Single Risky Assetmentioning

confidence: 99%

See 2 more Smart Citations

Weighted entropy and optimal portfolios for risk-averse Kelly investments

Kelbert¹,

Stuhl

Suhov

2017

Aequat. Math.

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“…Also, new theoretical developments concerning weighted entropy mathematical properties are available (e.g. [41]). …”

Section: Generalized Useful Information Measures and Weighted Gini-simentioning

confidence: 99%

Combining Expected Utility and Weighted Gini-Simpson Index into a Non-Expected Utility Device

Casquilho¹

2015

TEL

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We present and discuss a conceptual decision-making procedure supported by a mathematical device combining expected utility and a generalized information measure: the weighted GiniSimpson index, linked to the scientific fields of information theory and ecological diversity analysis. After a synthetic review of the theoretical background relative to those themes, such a devicean EU-WGS framework denoting a real function defined with positive utility values and domain in the simplex of probabilities-is analytically studied, identifying its range with focus on the maximum point, using a Lagrange multiplier method associated with algorithms, exemplified numerically. Yet, this EU-WGS device is showed to be a proper analog of an expected utility and weighted entropy (EU-WE) framework recently published, both being cases of mathematical tools that can be referred to as non-expected utility methods using decision weights, framed within the field of decision theory linked to information theory. This kind of decision modeling procedure can also be interpreted to be anchored in Kurt Lewin utility's concept and may be used to generate scenarios of optimal compositional mixtures applied to generic lotteries associated with prospect theory, financial risk assessment, security quantification and natural resources management. The epistemological method followed in the reasoned choice procedure that is presented in this paper is neither normative nor descriptive in an empirical sense, but instead it is heuristic and hermeneutical in its conception.

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