Equilibrium asset pricing with power utility

本研究项目的目标是刻画在不同类型的均衡下产生的资产的均衡动态：分别具有内生收益、内生波动性和内生利率。第一个目标是研究完全市场和不完全市场中的无摩擦情形，并将其与现有文献进行比较。虽然指数效用的情况现在已经很好地理解了，例如参见[2]、[3]、[6]，但是幂效用的情况在很大程度上是开放的。然后，下一个目标是将结果扩展到交易成本。在这里，均值-方差效用的情况被很好地理解，见[1]和[4]，并且对于指数效用存在部分结果[5]。从数学上讲，这个问题与正倒向随机微分方程组有关。与指数效用的情况不同，在功率效用的情况下，前向方程和后向方程是耦合的。因此，一个重要的问题是证明这些FBSDE系统的适定性、存在唯一性。参考文献[1]B.Bouchard，M.Fukasawa，M.Herdegen和J.Muhle-Karbe。有交易成本的均衡收益。金融与随机，22：569-601,2018年。[2]Escauriaza，Xing和Schwarz：Radner均衡和具有不连续生成器的二次BSDE系统，2021。[3]P.Guasoni和M.H.Weber。具有不可套期保值基本面和异质代理人的不完全市场均衡。Michael J.Brennan爱尔兰金融工作论文系列，2022年.[4]M.Herdegen，J.Muhle-Karbe和D.Possamai：平衡资产定价与交易成本，金融和随机，25：231-275,2021年.[5]M.Herdegen和J.Muhle-Karbe：Radner均衡点关于小摩擦的稳定性，金融和随机，22：443-502,2018.[6]C.Kardaras，H.Xing和G.Zitkovic.指数效用接近帕累托最优的不完全随机均衡。《随机分析、过滤和随机优化》，第267-292页，2022页。

The goal of this research project is to characterise the equilibrium dynamics of assets arising in different types of equilibria: with endogenous returns, endogenous volatility, and endogenous interest rates, respectively. The first goal is to study the frictionless case in both complete and incomplete markets and compare this to the extant literature. Whereas the case of exponential utility is by now well understood, see e.g. [2], [3], [6], the case of power utility is largely open. The next goal is then to extend the results to transaction costs. Here the case of mean-variance utilities is well understood, see [1] and [4], and partial results exist for exponential utilities [5].Mathematically, the problem is linked to systems of forward-backward stochastic differential equations. Unlike in the case of exponential utilities, the forward and the backward equations are coupled in the case of power utility. So, an important problem is to prove well-posedness, existence and uniqueness of those FBSDE systems.References:[1] B. Bouchard, M. Fukasawa, M. Herdegen, and J. Muhle-Karbe. Equilibrium returns with transaction costs. Finance and Stochastics, 22:569-601, 2018.[2] Escauriaza, Xing and Schwarz: Radner equilibrium and systems of quadratic BSDEs with discontinuous generators, 2021.[3] P. Guasoni and M. H. Weber. Incomplete-market equilibrium with unhedgeable fundamentals and heterogenous agents. Michael J. Brennan Irish Finance Working Paper Series, 2022.[4] M. Herdegen, J. Muhle-Karbe and D. Possamai: Equilibrium asset pricing with transaction costs, Finance and Stochastics, 25:231-275, 2021.[5] M. Herdegen and J. Muhle-Karbe: Stability of Radner equilibria with respect to small frictions, Finance and Stochastics, 22:443-502, 2018.[6] C. Kardaras, H. Xing, and G. Zitkovic. Incomplete stochastic equilibria with exponential utilities close to pareto optimality. Stochastic Analysis, Filterting and Stochastic Optimisation, pages 267-292, 2022.