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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 $t$.

Itô's lemma expresses the differential of a function $f(W,t)$ as $df(W,t) = \left[f_t + \frac{1}{2} f_{WW}\right] dt + f_W dW$. This formula includes a second-order partial derivative term with respect to $W$ multiplied by $dt$, which arises from the non-zero quadratic variation of Brownian motion.

Conditions: $f$ is a sufficiently smooth function of $W$ and $t$.; $W$ is a standard Brownian motion.

Itô's lemma expresses the differential of a function $f(W,t)$ as $df(W,t) = \left[f_t + \frac{1}{2} f_{WW}\right] dt + f_W dW$. This formula includes a second-order partial derivative term with respect to $W$ multiplied by $dt$, which arises from the non-zero quadratic variation of Brownian motion.

Conditions: $f$ is a sufficiently smooth function of $W$ and $t$.; $W$ is a standard Brownian motion.

A stochastic process $S$ is defined as following geometric Brownian motion if it satisfies the stochastic differential equation $dS = \mu S\,dt + \sigma S\,dW$, where $\mu$ and $\sigma$ are constants representing the drift and volatility, and $W$ is a standard Brownian motion.

Conditions: $\mu$ and $\sigma$ are constants.; $W$ is a standard Brownian motion.

A stochastic process $S$ is defined as following geometric Brownian motion if it satisfies the stochastic differential equation $dS = \mu S\,dt + \sigma S\,dW$, where $\mu$ and $\sigma$ are constants representing the drift and volatility, and $W$ is a standard Brownian motion.

Conditions: $\mu$ and $\sigma$ are constants.; $W$ is a standard Brownian motion.