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Insider’s problem in the trinomial model: a discrete jump process point of view Halconruy, Hélène E-print/Working paper (2021) In an incomplete market underpinned by the trinomial model, we consider two investors: an ordinary agent whose decisions are driven by public information and an insider who possesses from the beginning a ... [more ▼] In an incomplete market underpinned by the trinomial model, we consider two investors: an ordinary agent whose decisions are driven by public information and an insider who possesses from the beginning a surplus of information encoded through a random variable for which he or she knows the outcome. Through the definition of an auxiliary model based on a marked binomial process, we handle the trinomial model as a volatility one, and use the stochastic analysis and Malliavin calculus toolboxes available in that context. In particular, we connect the information drift, i.e. the drift to eliminate in order to preserve the martingale property within an initial enlargement of filtration in terms of Malliavin’s derivative. We solve explicitly the agent and the insider expected logarithmic utility maximization problems and provide a Ocone-Karatzas type formula for replicable claims. We identify insider’s expected additional utility with the Shannon entropy of the extra information, and examine then the existence of arbitrage opportunities for the insider. [less ▲] Detailed reference viewed: 44 (3 UL)Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation Halconruy, Hélène E-print/Working paper (2021) In this paper we develop a stochastic analysis for marked binomial processes, that can be viewed as the discrete analogues of marked Poisson processes. The starting point is the statement of a chaotic ... [more ▼] In this paper we develop a stochastic analysis for marked binomial processes, that can be viewed as the discrete analogues of marked Poisson processes. The starting point is the statement of a chaotic expansion for square-integrable (marked binomial) functionals, prior to the elaboration of a Markov-Malliavin structure within this framework. We take advantage of the new formalism to deal with two main applications. First, we revisit the Chen-Stein method for the (compound) Poisson approximation which we perform in the paradigm of the built Markov-Malliavin structure, before studying in the second one the problem of portfolio optimisation in the trinomial model. [less ▲] Detailed reference viewed: 44 (1 UL)Kernel Selection in Nonparametric Regression Halconruy, Hélène ; in Mathematical Methods of Statistics (2021) In the regression model Y=b(X)+ε, where X has a density f, this paper deals with an oracle inequality for an estimator of bf, involving a kernel in the sense of Lerasle et al. (2016), selected via the PCO ... [more ▼] In the regression model Y=b(X)+ε, where X has a density f, this paper deals with an oracle inequality for an estimator of bf, involving a kernel in the sense of Lerasle et al. (2016), selected via the PCO method. In addition to the bandwidth selection for kernel-based estimators already studied in Lacour, Massart and Rivoirard (2017) and Comte and Marie (2020), the dimension selection for anisotropic projection estimators of f and bf is covered. [less ▲] Detailed reference viewed: 37 (7 UL)Malliavin and Dirichlet structures for independent random variables Halconruy, Hélène ; in Stochastic Processes and Their Applications (2019), 129(8), 2611-2653 On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual ... [more ▼] On any denumerable product of probability spaces, we construct a Malliavin gradient and then a divergence and a number operator. This yields a Dirichlet structure which can be shown to approach the usual structures for Poisson and Brownian processes. We obtain versions of almost all the classical functional inequalities in discrete settings which show that the Efron-Stein inequality can be interpreted as a Poincaré inequality or that the Hoeffding decomposition of U-statistics can be interpreted as an avatar of the Clark representation formula. Thanks to our framework, we obtain a bound for the distance between the distribution of any functional of independent variables and the Gaussian and Gamma distributions. [less ▲] Detailed reference viewed: 38 (10 UL) |
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