Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

Area of Science:

• Stochastic Processes
• Mathematical Physics
• Probability Theory

Background:

• Diffusion processes are fundamental in modeling random phenomena.
• Absorbing boundary conditions introduce complexities in process behavior.
• Conditioning processes alters their statistical properties based on specific outcomes.

Purpose of the Study:

• To construct conditioned diffusion processes on a half-line with absorbing boundaries.
• To analyze processes under finite and infinite time horizons.
• To investigate the impact of absorption time and survival probabilities on process conditioning.

Main Methods:

• Development of mathematical framework for conditioned diffusion processes.
• Imposition of probability distributions for survival and absorption times.
• Application to Brownian motion with uniform drift for trajectory generation.

Main Results:

• Construction of conditioned processes for finite (T<+∞) and infinite (T=+∞) horizons.
• Reformulation of infinite horizon conditioning in terms of first-passage-time properties.
• Generation of stochastic trajectories satisfying specific conditioning constraints.

Conclusions:

• The study provides a general framework for conditioning diffusion processes.
• Conditioning can be achieved by specifying absorption time and survival distributions.
• Connections are established with large deviation theory and stochastic control.