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The lecturer seeks solutions of the form $S=f(W,t)$ to restrict the search space to functions of Brownian motion and time, allowing the application of Itô's lemma to convert the stochastic differential equation into a system of partial differential equations.

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

The lecturer seeks solutions of the form $S=f(W,t)$ to restrict the search space to functions of Brownian motion and time, allowing the application of Itô's lemma to convert the stochastic differential equation into a system of partial differential equations.

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