In this work, we aim to provide mathematical characterizations of effect of the existence of asset price bubbles to the financial market from various perspective, leveraging the arbitrage pricing framework and mathematical theory of financial bubbles. We analyze bubble's effect to several sets of trading strategies, including both widely employed practical strategies and those proven to replicate or super-replicate the risky asset's payoffs over the model's horizon. Additionally, we delve into the portfolio optimization problem in the presence of asset price bubbles in multiple facets. A comparative analysis of optimal portfolios is conducted in diverse settings, including complete and incomplete markets, and with or without trading constraints. Finally, we introduce a novel approach for simulating sophisticated stochastic processes that is free from space discretization. This chapter serves as a foundational framework for conducting advanced numerical and empirical research on challenges related to sophisticated stochastic dynamics. The scope of this research extends beyond, but is not confined to, the simulation of asset price bubbles.