论文标题
有限或无限地平线的有条件扩散过程具有吸收边界条件
Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon
论文作者
论文摘要
当无条件的过程是在半行$ x \ in] - \ infty上的扩散时，a [$在存在下吸收边界条件的位置$ x = a $时，我们构建了对应于有限或无限地平线的各种条件过程。当时间透射是有限的$ t <+\ infty $时，条件包括施加概率$ p^*（y，t）$在时间$ t $和位置$ y \ in] - \ infty，a [$，概率$γ^*（t_a）$上的位置$ y \ y y \ y y \ y y \ y y y time $ time $ t _a a [0 the] 0.当时间范围是无限的$ t =+\ infty $时，条件包括施加概率$γ^*（t_a）$在[0，+\ infty [$ insurative $ t_a \ in [$ t_a \ in [$ hypty [$ normorative $ normolorization $ [1- s^*（\ infty）] $确定条件的概率$ $ s^in [0]（in [0）时，永远存在。因此，这种无限范围$ t =+\ infty $的案例可以被重新重新重新构建为扩散过程相对于其在位置$ a $的首次计时属性的条件。该通用框架应用于明确的情况，在该情况下，无条件的过程是带有均匀漂移$μ$的布朗运动，以生成满足各种类型的条件约束的随机轨迹。最后，我们描述了与2.5级的动态大偏差和随机控制理论的联系。
When the unconditioned process is a diffusion living on the half-line $x \in ]-\infty,a[$ in the presence of an absorbing boundary condition at position $x=a$, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite $T<+\infty$, the conditioning consists in imposing the probability $P^*(y,T ) $ to be surviving at time $T$ and at the position $y \in ]-\infty,a[$, as well as the probability $γ^*(T_a ) $ to have been absorbed at the previous time $T_a \in [0,T]$. When the time horizon is infinite $T=+\infty$, the conditioning consists in imposing the probability $γ^*(T_a ) $ to have been absorbed at the time $T_a \in [0,+\infty[$, whose normalization $[1- S^*(\infty )]$ determines the conditioned probability $S^*(\infty ) \in [0,1]$ of forever-survival. This case of infinite horizon $T=+\infty$ can be thus reformulated as the conditioning of diffusion processes with respect to their first-passage-time properties at position $a$. This general framework is applied to the explicit case where the unconditioned process is the Brownian motion with uniform drift $μ$ in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, we describe the links with the dynamical large deviations at Level 2.5 and the stochastic control theory.
