Justification for substituting "Itô differentials"

Question

I'm reading Shreve's Stochastic Calculus for Finance, Volume II. In it, he uses the stochastic differential notation. For example, he may write

$$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\mathrm{d}t\tag{ 1}$$

to mean (formally)

$$\int_{t=s_0}^{t=s_1}\mathrm{d}X(t) = \int_{t=s_0}^{t=s_1}\sigma(t)\mathrm{d}W(t)+\int_{t=s_0}^{t=s_1}\alpha(t)\mathrm{d}t\text{.}\tag{2}$$

This is fine and makes sense. The problem is that he will often substitute the LHS of $(1)$ for the RHS of $(1)$ in other stochastic differential equations. For example, given the function $S(t)$, he may use $(1)$ to justify the equality

$$S(t) \mathrm{d}X(t) = S(t)\sigma(t)\mathrm{d}W(t)+S(t)\alpha(t)\mathrm{d}t\text{,}\tag{3}$$

which is equivalent to

$$\int_{t=s_0}^{t=s_1} S(t) \mathrm{d}X(t) = \int_{t=s_0}^{t=s_1} S(t)\sigma(t)\mathrm{d}W(t)+\int_{t=s_0}^{t=s_1} S(t)\alpha(t)\mathrm{d}t\text{.}\tag{4}$$

This trick appears to be purely symbolic. Intuitively, it makes sense. However, I am having trouble proving that you can make this substitution. Is there a proof that you can make these kinds of substitutions (in this example or in general)?

Answer 1

I don't see where in the book Shreve justifies (3) directly from (1). But he would have almost surely referred to the discretized versions of the "differentials". That is:

$$ X(t_{i+1})-X(t_{i}) \approx \sigma(t_i)(W(t_{i+1})-W(t_{i}))+\alpha(t_{i})(t_{i+1}-t_{i}) $$

implies $$ S(t_{i})(X(t_{i+1})-X(t_{i})) \approx S(t_{i})\sigma(t_i)(W(t_{i+1})-W(t_{i}))+S(t_{i})\alpha(t_i)(t_{i+1}-t_{i}). $$

We would then take summations and limits (needed in an integration mechanism) on both sides and justify why a stochastic integral wrt to an Ito process integrator can be defined as in (4), which is Definition 4.4.5 in the book, given after Shreve had already defined stochastic integrals wrt to $W(t)$ and $t$ integrators.