Why does the lecturer seek solutions of the form $S=f(W,t)$?

Conditions

• The process $S$ is driven by the Brownian motion $W$.
• Itô's lemma is applicable to functions of $W$ and $t$.

Reasoning, step by step

1. Identify the goal of solving the SDE for $S$.
2. Propose the ansatz $S = f(W,t)$.
3. Recognize that this restricts the solution class to functions of $W$ and $t$.
4. Prepare to apply Itô's lemma to $f(W,t)$.

Example

The board shows 'Seek solutions $S = f(W,t)$', and the lecturer explains they will look only at functions of Brownian motion and time as the potential solution class of this SDE.

Common misconceptions

• Assuming this ansatz covers all possible solutions to the SDE, rather than a specific class of solutions.
• Forgetting that $W$ itself is a function of time, so $f(W,t)$ captures both stochastic and deterministic dependencies.

Watch the explanation

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