We study risk‐sharing equilibria with general convex costs on the agents' trading rates. For an infinite‐horizon model with linear state dynamics and exogenous volatilities, we prove that the equilibrium returns mean‐revert around their frictionless counterparts—the deviation has Ornstein‐Uhlenbeck dynamics for quadratic costs whereas it follows a doubly‐reflected Brownian motion if costs are proportional. More general models with arbitrary state dynamics and endogenous volatilities lead to multidimensional systems of nonlinear, fully‐coupled forward‐backward SDEs. These fall outside the scope of known well‐posedness results, but can be solved numerically using the simulation‐based deep‐learning approach of Han, Jentzen, and E (2018). In a calibration to time series of prices and trading volume, realistic liquidity premia are accompanied by a moderate increase in volatility. The effects of different cost specifications are rather similar, justifying the use of quadratic costs as a proxy for other less tractable specifications.

中文翻译：

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带有一般交易成本的资产定价：理论和数值

我们研究了风险共享的均衡性，即代理人交易费率具有一般性的凸面成本。对于具有线性状态动力学和外生波动率的无限地平线模型，我们证明了均衡收益在无摩擦对应物附近均值回归-二次成本的偏差具有Ornstein-Uhlenbeck动力学，而如果成本为，则遵循双反射布朗运动。成比例的。具有任意状态动力学和内生波动性的更通用的模型导致了非线性，完全耦合的前向后向SDE的多维系统。这些不在已知的适定性结果的范围之内，但可以使用Han，Jentzen和E（2018）的基于仿真的深度学习方法进行数值求解。在校准价格和交易量的时间序列时，现实的流动性溢价伴随着波动性的适度增加。不同成本规格的影响相当相似，因此有理由使用二次成本替代其他较难处理的规格。