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# An FBSDE approach to market impact games with stochastic parameters

• Author Bio: Email: sdrapeau@saif.sjtu.edu.cn; Email: aschied@uwaterloo.ca; Email: xiongdewen@sjtu.edu.cn
• Email: peng.luo@sjtu.edu.cn (Corresponding author)
The authors thank two anonymous referees for their constructive comments and suggestions, which have significantly improved the manuscript. Samuel Drapeau gratefully acknowledges the support from the National Natural Science Foundation of China (Grant No. 11971310) and “Assessment of Risk and Uncertainty in Finance” (Grant No. AF0710020) from Shanghai Jiao Tong University. Peng Luo gratefully acknowledges the support from the National Natural Science Foundation of China (Grant No. 12101400). Peng Luo and Alexander Schied gratefully acknowledge the support from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN-2017-04054). Dewen Xiong gratefully acknowledges the support from the National Natural Science Foundation of China (Grant No. 11671257).
• In this study, we have analyzed a market impact game between n risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting in a unique Nash equilibrium.

Mathematics Subject Classification: 93E20, 60H30.

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• Figure 1.  Plot of the three agents’ inventory for different values of price impact

Figure 2.  Plot of the three agents’ inventory for different values of slippage effect

Figure 3.  Plot of the three agents’ inventory for the different start values of the first agent’s inventory with two arbitrageurs

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