Discrete Time Stochastic Processes

Discrete time stochastic processes are mathematical models that describe the evolution of a random variable over specific, distinct intervals of time. In finance, these processes are used to simulate how asset prices move in steps, providing the foundation for tree-based pricing models.

By breaking continuous time into small, manageable increments, these models make it possible to calculate probabilities and expected values that would be computationally difficult to solve in continuous time. Each step represents a potential change in the state of the market, governed by defined probability distributions.

These models are essential for constructing the transition probabilities in trinomial trees, ensuring that the simulated price movements are consistent with the asset's observed volatility and drift. They provide a structured framework for analyzing risk, managing portfolios, and pricing path-dependent derivatives where the sequence of events matters as much as the final outcome.