AI 中文总结
研究具有路径依赖系数的随机微分方程的风险敏感退出时间控制问题，通过结合路径依赖偏微分方程理论等工具推导出新颖变分表示，用概率方法分析收敛性，还给出可计算示例及相关刻画。
AI 中文摘要
在这项工作中，我们研究由具有路径依赖系数的随机微分方程所控制的风险敏感退出时间控制问题的小噪声渐近性。我们的主要结果建立了对数变换后的退出时间问题到具有路径依赖系数的确定性控制问题的收敛性。为证明此结果，我们首先为具有路径依赖系数的一般对数变换随机控制问题推导了一种新颖的变分表示，结合了路径依赖偏微分方程理论和路径空间上凸期望的工具。第二步，我们使用概率方法分析所得变分公式的收敛性。为说明我们分析的范围，我们考虑一个具有记忆的随机微分方程的可计算示例，并刻画极限问题及相关控制策略。
英文摘要
In this work, we study small-noise asymptotics of risk-sensitive exit-time control problems governed by stochastic differential equations with path-dependent coefficients. Our main result establishes the convergence of the $\log$-transformed exit-time problem to a deterministic control problem with path-dependent coefficients. For its proof, we first derive a novel variational representation for general $\log$-transformed stochastic control problems with path-dependent coefficients, combining tools from the theory of path-dependent partial differential equations and convex expectations on path spaces. In a second step, we use probabilistic methods to analyze the convergence of the resulting variational formulas. To illustrate the scope of our analysis, we consider a computable example for a stochastic differential equation with memory and characterize the limiting problem and associated control strategies.
