Zero-sum stochastic differential games in weak formulation and related norms for semi-martingales

In this dissertation, we study three topics under a common theme: nonlinear expectation related to zero-sum stochastic differential games. To develop this nonlinear expectation, we first study the stochastic game problem where both players use feedback controls. This is in contrast with the standard literature where the setting of strategies versus controls is usually used. Such approach has the drawback of creating the asymmetry between the two players. Using feedback controls, we prove the existence of the game value where both players use controls and preserve the symmetry. Moreover, we allow for non-Markovian structure and characterize the value process as the unique viscosity solution of the path-dependent Bellman-Isaacs equation. ❧ Using the dynamic programming principle, the game value process can be viewed as a filtration consistent nonlinear expectation. Moreover, this nonlinear expectation is dominated by the G-Expectation, which is defined naturally from the game setting. It follows that the game value process is a G-submartingale. It is natural to conjecture that a G-submartingale is a semi-martingale under each probability measure that composes the G-Expectation. Therefore, we study norm estimate for semi-martingales as our second topic. We introduce two new types of norms. The first characterizes square integrable semi-martingales. The second characterizes the absolute continuity of the finite variation part with respect to the Lebesgue measure. As an application of the first norm, we obtain the Doob-Meyer decomposition for G-submartingale. ❧ Finally, we study the well-posedness problem of doubly reflected Backward Stochastic Differential Equations and establish some a priori estimates for Doubly Reflected Backward Stochastic Differential Equations without imposing the Mokobodski's condition.

本论文围绕一个核心主题展开研究，即与零和随机微分博弈相关的非线性期望（nonlinear expectation）。为构建该非线性期望，我们首先探讨双方均采用反馈控制的随机博弈问题。这与主流文献中通常采用的策略与控制的设定范式形成区别，这类方法存在缺陷：会导致两名博弈参与方间的非对称性。通过采用反馈控制，我们证明了双方均使用控制策略时博弈值的存在性，且该设定保留了博弈的对称性。此外，我们的研究允许非马尔可夫结构（non-Markovian structure），并将博弈值过程刻画为路径依赖型贝尔曼-以撒斯方程（path-dependent Bellman-Isaacs equation）的唯一粘性解（viscosity solution）。
借助动态规划原理（dynamic programming principle），博弈值过程可被视为滤波相容非线性期望。此外，该非线性期望受控于G-期望（G-Expectation），而G-期望可通过博弈设定自然定义。由此可得，博弈值过程为G-下鞅（G-submartingale）。自然可以推测：在构成G-期望的每一个概率测度下，G-下鞅均为半鞅（semi-martingale）。因此，我们将半鞅的范数估计作为第二个研究主题。我们引入了两类全新的范数：第一类范数用于刻画平方可积半鞅，第二类范数用于刻画有限变差部分关于勒贝格测度（Lebesgue measure）的绝对连续性。作为第一类范数的应用，我们推导出了G-下鞅的杜布-迈耶分解（Doob-Meyer decomposition）。
最后，我们探讨双重反射倒向随机微分方程（doubly reflected Backward Stochastic Differential Equations）的适定性问题，并在无需施加莫科博德斯基条件（Mokobodski's condition）的前提下，为该类方程建立了若干先验估计。