标题： 连续时间Kyle-Back战略 insider 均衡问题的新方法 显示英文标题

标题： A New Approach for the Continuous Time Kyle-Back Strategic Insider Equilibrium Problem

摘要： 本文研究了一个连续时间的Kyle-Back模型，这是一个内幕交易者与市场做市商之间的博弈问题。现有文献通常通过偏微分方程（PDE）方法关注均衡的存在性，该方法需要特定的马尔可夫结构，并且均衡以桥的形式存在。我们将提供一种广泛用于随机控制和随机微分博弈的新方法。我们通过一个耦合的正向倒向随机微分方程（SDE）系统来刻画所有均衡，其中正向方程是内幕信息的条件分布，而倒向方程是内幕交易者的最优值。特别是，当时间跨度较小时，我们证明了倒向正向SDE是适定的，因此博弈具有唯一的均衡。这是文献中的第一个唯一性结果，没有限制均衡到某些特殊结构。此外，这个唯一的均衡可能不是马尔可夫性的，表明在这种情况下PDE方法无法适用。接下来，我们研究博弈的集合值，粗略地说，它是所有均衡下内幕交易者的值的集合，因此本质上是唯一的。我们证明，尽管文献中的桥型均衡不满足我们均衡所需的可积性要求，但其截断形式可以作为期望的近似均衡，其值属于我们的集合值。最后，我们通过某个标准HJB方程的水平集来刻画我们的集合值。 显示英文摘要

摘要： This paper considers a continuous time Kyle-Back model which is a game problem between an insider and a market marker. The existing literature typically focuses on the existence of equilibrium by using the PDE approach, which requires certain Markovian structure and the equilibrium is in the bridge form. We shall provide a new approach which is used widely for stochastic controls and stochastic differential games. We characterize all equilibria through a coupled system of forward backward SDEs, where the forward one is the conditional law of the inside information and the backward one is the insider's optimal value. In particular, when the time duration is small, we show that the FBSDE is wellposed and thus the game has a unique equilibrium. This is the first uniqueness result in the literature, without restricting the equilibria to certain special structure. Moreover, this unique equilibrium may not be Markovian, indicating that the PDE approach cannot work in this case. We next study the set value of the game, which roughly speaking is the set of insider's values over all equilibria and thus is by nature unique. We show that, although the bridge type of equilibria in the literature does not satisfy the required integrability for our equilibria, its truncation serves as a desired approximate equilibrium and its value belongs to our set value. Finally, we characterize our set value through a level set of certain standard HJB equation.