Optimal Portfolios for Different Anticipating Integrals under Insider Information

Escudero, Carlos; Ranilla-Cortina, Sandra

doi:10.3390/math9010075

Cited by 11 publications

(8 citation statements)

References 30 publications

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“…is the solution to equation (7). Moreover the solution is unique by the Wanatabe-Yamada theorem [29] as was claimed in [6].…”

Section: Energy Power and Forward Stochastic Differential Equationsmentioning

confidence: 87%

“…However, at least mathematically speaking, such an equation is well posed, even if its solution is not adapted, if interpreted according to some anticipating stochastic integral. The problem of the interpretation of noise in the anticipating setting is summarized in [2,5,7]. We leave as an open question the application of such techniques in systems in which the fluctuation-dissipation relation is of interest.…”

Section: Discussionmentioning

confidence: 99%

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Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

Escudero¹

2021

Preprint

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We derive the fluctuation-dissipation relation and explore its connection with the equipartition theorem and Maxwell-Boltzmann statistics through the use of different stochastic analytical techniques. Our first approach is the theory of backward stochastic differential equations, which arises naturally in this context, and facilitates the understanding of the interplay between these classical results of statistical mechanics. The second approach consists in deriving forward stochastic differential equations for the energy of an electric system according to both Itô and Stratonovich stochastic calculus rules. While the Itô equation possesses a unique solution, which is the physically relevant one, the Stratonovich equation admits this solution along with infinitely many more, none of which has a physical nature. Despite of this fact, some, but not all of them, obey the fluctuation-dissipation relation.

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“…is the solution to equation (7). Moreover the solution is unique by the Wanatabe-Yamada theorem [29] as was claimed in [6].…”

Section: Energy Power and Forward Stochastic Differential Equationsmentioning

confidence: 87%

Section: Discussionmentioning

confidence: 99%

Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

Escudero¹

2021

Preprint

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“…Moreover, it contrasts with previous results that studied a simplified version of this problem. In [7,8], in which only buy-andhold strategies with no shorting allowed were permitted, but risk aversion was replaced by risk neutrality (for both insider and honest traders), it was shown that the insider gets more wealth (and therefore more utility) almost surely; something more intuitive under the current assumptions. The positive, albeit possibly small, probability of the honest trader victory over the insider seems to be related to the logarithmic utility.…”

Section: Discussionmentioning

confidence: 97%

“…For this reason it is not surprising that it has been frequently used to treat insider information ( [2], [3], [4], [6], [5], [10], [12]). As noted in the previous section, the forward integral is genuinely different from the Skorokhod one, and we recommend the former and not the latter that has to be used in the mathematical modeling of insider trading in view of previous results [7,1,8].…”

Section: Merton Portfolio Problem With Insider Informationmentioning

confidence: 96%

Chances for the honest in honest versus insider trading

Elizalde¹,

Escudero²

2021

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We study a Black-Scholes market with a finite time horizon and two investors: an honest and an insider trader. We analyze it with anticipating stochastic calculus in two steps. First, we recover the classical result on portfolio optimization that shows that the expected logarithmic utility of the insider is strictly greater than that of the honest trader. Then, we prove that, whenever the market is viable, the honest trader can get a higher logarithmic utility, and therefore more wealth, than the insider with a strictly positive probability. Our proof relies on the analysis of a sort of forward integral variant of the Doléans-Dade exponential process. The main financial conclusion is that the logarithmic utility is perhaps too conservative for some insiders.

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“…Thus the dynamic of the risky-asset is also an anticipating SDE. Many other extensions can be found in, for example, [12,9,13,14,15], which are all based on the forward integral. Although many insider-trading problems have been studied, the basic theory of forward integrals is not completed since it is difficult to obtain the properties of forward integrals without other theories.…”

Section: Introductionmentioning

confidence: 99%

Malliavin calculus and its application to robust optimal portfolio for an insider

Yu¹,

Cheng²

2022

Preprint

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Insider information and model uncertainty are two unavoidable problems for the portfolio selection theory in reality. This paper studies the robust optimal portfolio strategy for an investor who owns general insider information under model uncertainty. On the aspect of the mathematical theory, we improve some properties of the forward integral and use Malliavin calculus to derive the anticipating Itô formula . Then we use forward integrals to formulate the insider-trading problem with model uncertainty. We give the half characterization of the robust optimal portfolio and obtain the semimartingale decomposition of the driving noise W with respect to the insider information filtration, which turns the problem turns to the nonanticipative stochastic differential game problem. We give the total characterization by the stochastic maximum principle. When considering two typical situations where the insider is 'small' and 'large', we give the corresponding BSDEs to characterize the robust optimal portfolio strategy, and derive the closed form of the portfolio and the value function in the case of the small insider by the Donsker δ functional. We present the simulation result and give the economic analysis of optimal strategies under different situations.

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Optimal Portfolios for Different Anticipating Integrals under Insider Information

Cited by 11 publications

References 30 publications

Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

Chances for the honest in honest versus insider trading

Malliavin calculus and its application to robust optimal portfolio for an insider

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