Stochastic control and first passage time probabilities in finance

许多复杂的动态系统都是用随机过程(如随机微分方程)成功地建模的，尤其是股票市场。 在金融市场中，投资者可能寻求建立一种动态策略，允许他优化预期终端财富的某个函数(例如：最小化风险函数，最大化预期效用……)。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。 这一研究项目的目的是在现实约束下开发创新的投资策略，无论是投资者施加的约束还是市场强制的约束，同时考虑具有(可能是无界的)随机参数的市场模型。 这可以通过利用倒向随机微分方程(BSDE)理论的最新结果在寻找一般随机控制问题的新的最优解方面取得突破来实现。 此外，通过努力寻找更复杂的首次通过时间问题的解决方案，这最终将导致尖端的混合策略，其中包括停止时间规则，作为谨慎投资者的财务困境的保护措施，主要关注的是他们实现目标的可能性。

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.