I want to show that the stochastic process
$$
S_t^i = S_0^i \exp\left( \int_0^t \left(\mu_s^i – \frac{1}{2} \sum_{j=1}^m (\sigma_s)^{ij} \right)^2 d s + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j \right)
$$
solves the stochastic differential equation
$$
dS_t^i = \mu_t^i S_t^i dt + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j.
$$
I just want to plug in this expression and thereby verifying that it solves the SDE, just as in the non-stochastic case. For this I first recognize that the SDE is a shorthand for
$$
S_t^i = S_0^i+ \int_0^t \mu_s^i S_s^i ds + \int_0^t \sum_{j=1}^m \sigma_s^{ij} S_s^i ~ dW_s^j.
$$
Now first evaluating the integral
$$
\int_0^t \sum_{j=1}^m \sigma_s^{ij} S_s^i ~ dW_s^j
$$
Here I have no idea how to start, I know I must apply Ito's formula in some clever way, but when I look it up it is always written with some mysterios $d X_i \cdot d X_j$ for the second order terms, and remarks like "these are evaluated according to the rules $dB_i dB_j = \delta_{ij} dt, dt dB_i = dB_i dt = 0$" which I cannot make sense of, so I have no idea how to apply Ito's formula here to solve the stochastic integral? Any help?
Best Answer
To keep notation simple, we write $S_t$ instead of $S_t^i$, $\sigma_t^j$ instead of $\sigma_{t}^{ij}$ and so on. That's okay, because $i$ is a fixed number throughout this calculation.
Suppose $(X_t)_{t \geq 0}$ is an Itô process of the form
$$dX_t = b(t) \, dt + \eta(t) \, dW_t$$
where $\eta = (\eta_1,\ldots,\eta_m)$ and $(W_t)_{t \geq 0}$ is an $m$-dimensional Brownian motion. Then Itô's formula states that
$$\begin{align*} f(X_t)-f(X_0) &= \int_0^t f'(X_s) \, dX_s + \frac{1}{2} \int_0^t f''(X_s) \, d\langle X \rangle_s \end{align*}$$
for $f \in C^2$ where
$$\begin{align*} \int_0^t f'(X_s) \, dX_s &:= \int_0^t f'(X_s) \eta_s \, dW_s + \int_0^t f'(X_s) b(s) \, ds \\ d\langle X \rangle_s &:= \sum_{j=1}^m \eta_j^2(s) \, ds. \end{align*}$$
In your setting, we have $$b(t) = \left(\mu_t - \frac{1}{2} \sum_{j=1}^m \sigma_t^j \right)^2 \qquad \eta(t) := (\sigma_t^1,\ldots,\sigma_t^m).$$
Applying Itô's formula for $f(x) := \exp(x)$ shows that $(S_t)_{t \geq 0}$ is a solution to the given SDE.