What would you like to understand?

The final explicit solution for the geometric Brownian motion process $S$ is $S = c_1 e^{\sigma W + \left(\mu - \frac{1}{2}\sigma^2\right)t}$, where $c_1$ is an integration constant determined by the initial condition.

Conditions: $\mu$ and $\sigma$ are constant.; The ansatz $S=f(W,t)$ is valid.; The ODE for $\phi(t)$ is solved.

The final explicit solution for the geometric Brownian motion process $S$ is $S = c_1 e^{\sigma W + \left(\mu - \frac{1}{2}\sigma^2\right)t}$, where $c_1$ is an integration constant determined by the initial condition.

Conditions: $\mu$ and $\sigma$ are constant.; The ansatz $S=f(W,t)$ is valid.; The ODE for $\phi(t)$ is solved.

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 $t$.