Why does solving $f_W=\sigma f$ introduce a function $\phi(t)$ rather than a constant?

By substituting the ansatz $S=f(W,t)$ into the SDE and applying Itô's lemma, the coefficients of $dW$ and $dt$ are matched. This yields the system of PDEs: $f_W = \sigma f$ and $f_t + \frac{1}{2} f_{WW} = \mu f$.

Conditions: $S = f(W,t)$ is the assumed solution form.; $dS = \mu S\,dt + \sigma S\,dW$ is the defining SDE.; Itô's lemma is applied to $f(W,t)$.

The symbol $\Delta u$ represents the Laplacian of the scalar field $u$. It is a compact notation for the sum of the unmixed second partial derivatives of $u$ with respect to all coordinate directions.

Conditions: $u$ is a scalar function of multiple variables.; The coordinates are orthogonal Cartesian coordinates (implied by the simple sum of second derivatives).; The notation generalizes the 2D case to $n$ dimensions.

The two-dimensional Laplace equation is defined as the sum of the second partial derivatives of a scalar function $u$ with respect to the spatial coordinates $x$ and $y$ equaling zero. It is presented on a rectangular domain defined by $0 < x < a$ and $0 < y < b$.

Conditions: The function $u$ is a scalar field depending on two independent spatial variables $x$ and $y$.; The domain is an open rectangle in the $xy$-plane.; The equation is homogeneous (equals zero).

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.

To find the total differential of $z = x^2y^2$, compute the partial derivatives with respect to x and y, then substitute them into the total differential formula. The result is $dz = 2$xy^$2 dx + 2x^2y dy$.

Conditions: The function is $z = x^2y^2$.; Standard rules of partial differentiation apply.