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...posed in the litterature in the late nineties mainly by Bertsimas and Lo [8] and Almgren and Chriss [2]. If the original approach has been recently generalized in several directions (see for instance =-=[3, 4, 13, 14, 15, 18, 26, 29, 34]-=-), only few attempts have been made to drill down the model at the level of the interactions with the order books. The more noticeable proposal is the one by Obizhaeva and Wang [31], followed and soph...

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...cal Problems in Engineering of market price impact to solve the equilibrium of the predatory trading game and comparative statics from the HJB equation. Besides the continuous time, many scholars discuss the discrete time and the optimal strategies from the utility function maximization. Schied et al. [19] show that the utility function maximization is equal to the meanvariancemaximization and the utility functionmaximization is simplified to maximize mean-variance. They prove that the results not only meet the continuous-time model but also satisfy the discrete-time model. Lorenz and Almgren [20] find the mean-variance optimization strategies by the means of a dynamic programming principle. At the same time, several scholars investigate the situation in which the market impact function has the decay factor. Gatheral [21] studies the properties of the market impact function with the exponential decay, power-law decay, and other situations. Except the above two decay functions, Schied [22] further discusses the capped linear decay and Gaussian decay and gives some useful properties. Schied and Zhang [23] study ...

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...apers focus on this feature. In particular, Obizhaeva and Wang (2013) propose a market microstructure framework in which price impact can be understood as a consequence of fluctuations in the 1Perold (1988), for example, shows that execution costs can often erase true returns, leading to a significant “implementation shortfall.” 2Bertsimas and Lo (1998), Almgren and Chriss (2000), and Obizhaeva and Wang (2013) demonstrate the benefit from splitting orders over time under different assumptions regarding the impact of trading on current and future prices and liquidity. 3Also see Almgren (2009), Lorenz and Almgren (2012), He and Mamaysky (2005), and Schied and Schoeneborn (2009). For empirical foundations, see Bouchaud et al. (2009) for a survey, as well as references in Alfonsi et al. (2008), Alfonsi et al. (2010) and Obizhaeva and Wang (2013). For studies on how trade size affects prices see Chan and Fong (2000), Chan and Lakonishok (1995), Chordia et al. (2002) and Dufour and Engle (2000). 1 supply and demand of liquidity.4 One advantage of this approach is that the optimal strategies obtained are robust to different order book profiles. This literature highlights the fact that supply/demand dynamics are c...

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...hastic volatility and liquidity were made – [1]. Discussions on the optimization criterions and their consequences on optimal strategies are also present in the literature (see for instance [4], [7], =-=[15]-=- and [19]). The CARA (or Mean-variance) framework is predominant in the literature and it has been studied for instance in [18], and in [9] that also considers block trade pricing. Very interesting re...

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...atility and stochastic liquidity [1]. Discussions on the optimization criterions and their consequences on optimal strategies are also very present in the literature (see for instance [4], [11], [19] =-=[23]-=- and [30]). Here, we consider a Von Neumann-Morgenstern expected utility framework and the specific case of an investor with constant absolute risk aversion.3 This CARA framework has been studied in [...