The topics treated in this thesis are inherently two-fold. The first part considers the problem of a market maker optimally setting bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes, but also on unobservable factors modelled by a hidden Markov chain. This stochastic control problem under partial information is solved by means of stochastic filtering, control and PDMPs theory. The value function is characterized as the unique continuous viscosity solution of its dynamic programming equation and numerically compared with its full information counterpart. The optimal full information spreads are shown to be biased when the exact market regime is unknown, as the market maker needs to adjust for additional regime uncertainty in terms of PnL sensitivity and observable order flow volatility. The second part deals with numerically solving nonzero-sum stochastic impulse control games. These offer a realistic and far-reaching modelling framework, but the difficulty in solving such problems has hindered their proliferation. A policy-iteration-type solver is proposed to solve an underlying system of quasi-variational inequalities, and it is validated numerically with reassuring results. Eventually, the focus is put on games with a symmetric structure and an improved algorithm is put forward. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. The algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise too challenging problems, and even some for which results go beyond the scope of the currently available theory.

中文翻译：

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具有应用的脉冲控制博弈的部分信息和数值方法下的最优做市

本论文中处理的主题本质上是双重的。第一部分考虑做市商在有限的时间范围内优化设置买入/卖出报价的问题，以最大化其预期效用。她收到的订单强度不仅取决于她引用的点差，还取决于由隐马尔可夫链建模的不可观察因素。这个部分信息下的随机控制问题是通过随机滤波、控制和PDMPs理论解决的。价值函数的特点是其动态规划方程的唯一连续粘度解，并与其全信息对应物进行数值比较。当确切的市场机制未知时，最优的全信息差被证明是有偏差的，因为做市商需要在盈亏敏感性和可观察的订单流波动性方面针对额外的制度不确定性进行调整。第二部分涉及数值求解非零和随机脉冲控制博弈。这些提供了一个现实且影响深远的建模框架，但解决此类问题的困难阻碍了它们的扩散。提出了一种策略迭代型求解器来求解准变分不等式的基础系统，并在数值上进行了验证，结果令人放心。最后，重点研究了具有对称结构的博弈，并提出了改进算法。对参与者策略的自然假设进行了严格的收敛分析，这些假设允许在弱链对角主导矩阵的背景下进行图论解释。