Every financial model that involves randomness specifies an SDE. But writing is only meaningful if a process satisfying this equation actually exists. The question "does there exist a solution?" is non-trivial: some SDEs have no solution, some have infinitely many, and some have a unique solution that exists only up to a random time.
The answer for the models used in finance is reassuring: under a Lipschitz condition on the drift and diffusion coefficients plus a linear growth condition, the SDE has a unique strong solution for all time. The Picard iteration that proves this in the deterministic setting extends verbatim to the stochastic case, with norm replacing the supremum norm.
This note also contrasts strong solutions (built on a given Brownian motion) with weak solutions (where the probability space and BM are part of the solution), and explains why the distinction matters for simulation: when a strong solution exists, you can simulate by simulating and applying a deterministic functional.
Learning objectives. By the end of this note you should be able to:
- State the definition of a strong solution to an SDE and explain the "causality" requirement.
- State and apply the existence-uniqueness theorem (Itô's theorem) under Lipschitz and linear growth.
- Verify that GBM and the OU/Vasicek process satisfy the Lipschitz condition.
- Distinguish strong solutions from weak solutions and give a canonical example of each.
- Apply Picard iteration to estimate the error of successive approximations.
In this lesson
- Setting and Definitions
- The Existence-Uniqueness Theorem
- Key Examples
- Weak Solutions
- Simulation Consequence: Strong Solutions
Before you start
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