Stochastic control and first passage time probabilities in finance

许多复杂的动态系统都可以通过随机过程成功地建模（例如：随机微分方程），尤其是股票市场。在金融市场中，投资者可能会寻求建立一个动态策略，使他能够优化预期最终财富的某个函数（例如：最小化风险函数，最大化期望效用......）。考虑到这一点，他必须根据不断更新的信息流做出决定，而不能准确预测未来的市场波动。该研究计划旨在制定创新的投资策略下的现实约束，无论是由投资者或市场强制执行，同时考虑市场模型（可能无限）随机参数。这可以实现突破，在寻找新的最优解一般随机控制问题，使用最近的结果在BERRY（向后随机微分方程）理论。此外，通过努力解决更复杂的首次通过时间问题，这将最终导致尖端的混合策略，其中停止时间规则被纳入作为谨慎投资者的财务困境的保障措施，其主要重点是他们的目标实现概率。

Many complex dynamic systems are modeled successfully by stochastic processes (ex.: stochastic differential equations) and this is particularly the case for stock markets. In a financial market, an investor might seek to establish a dynamic strategy allowing him to optimize a certain function of the expected terminal wealth (ex.: miminizing risk functions, maximizing expected utility ...). With this in mind, he must base his decisions on a continuously updated flow of information without being able to exactly predict future market fluctuations. This research program aims at developing innovative investment strategies under realistic constraints, either imposed by the investor or enforced by the market, while considering market models with (possibly unbounded) stochastic parameters. This can be achieved with breakthroughs in finding new optimal solutions to general stochastic control problems using more recent results in BSDE (backward stochastic differential equations) theory. Furthermore, by striving for solutions to more intricate first-passage time problems, this will ultimately lead to cutting-edge hybrid strategies where stopping time rules are incorporated as a safeguard against financial distress for cautious investors which main focus is their goal-achieving probability.