What's the relationship between quadratic covariation and covariance in two Itô process?

Question

Suppose that the latent log-price of two arbitrary assets $X_{t}=\left(X_{t}^{(1)}, X_{t}^{(2)}\right)$ follows a continuous Itô process $$ \begin{aligned} d X_{t}^{(1)} &:=\mu_{t}^{(1)} d t+\sigma_{t}^{(1)} d W_{t}^{(1)} \\ d X_{t}^{(2)} &:=\mu_{t}^{(2)} d t+\sigma_{t}^{(2)} d W_{t}^{(2)} \end{aligned} $$ where $\mu_{t}^{(1)}, \mu_{t}^{(2)}, \sigma_{t}^{(1)}, \sigma_{t}^{(2)}$ are random processes, and $W_{t}^{(1)}$ and $W_{t}^{(2)}$ are standard Brownian motions, with (random) high-frequency correlation

1. $d\left\langle W^{(1)}, W^{(2)}\right\rangle_{t}=\rho_{t} d t .$
2. $Cov(dW^{(1)}_t,dW_t^{(2)})=\rho_tdt$

Do the two conditions represent the same thing? If not, which condition is used most, Condition 1 I suppose, right?

Answer 1

The bottom line is that the difference between instantaneous covariance and instantaneous covariation is that the former is a property of the joint distribution of $W^{1}$ and $W^{2}$ evaluated in $t$, while the latter is the result of calling the Law of Large Numbers on a realized joint path of $W^{1}$ and $W^{2}$. Such call is justified by the fact that the Brownian motion has no natural step size and is thus possible to consider the limit of the number of steps going to infinite.

In order to convey the difference between the two, one could write:

• for the instantaneous covariance:

$$ \text{Cov}(dW_t^{1}, dW_t^{2})=\lim_{\epsilon \rightarrow 0}\int_{\mathcal{F}(t)} \{ \int_t^{t+\epsilon}[(W_{t+\epsilon}^{1}(\omega)-W_t^{1}(\omega))(W_{t+\epsilon}^{2}(\omega)-W_t^{2}(\omega))] \} d\mathbb{P}(\omega)=\rho(t)dt $$

• for the instantaneous covariation, considering the educated guess that

$$ P\{ \langle W^{1}, W^{2} \rangle [T]=\int_0^T \rho(t) dt \} = 1, $$

since the Brownian motions can be arbitrarily scalable, informally we write

$$ d\langle W^{1}, W^{2} \rangle_t = dW_t^{1}dW_t^{2} = \rho(t) dt, $$

meaning that the two processes accumulate covariation at rate $\rho(t)$ per unit of time.

In other words, given the parameters of the joint distribution of $W^{1}$ and $W^{2}$, namely the covariance, or the instantaneous covariance, when the joint distribution is conditional to time, one can define the covariation and the instantaneous covariation. Basically, such is the pipeline.

Answer 2

Not sure how much you still care but do not get sucked into the stochastic differential notation dX. it isn't an actual random variable with a distribution per se. So the second notation is not sensible at all. If you wish to formalize the concept of instantaneous covariance of two variable at one instant. Namely cov ($x_t,y_t$) would be the better and simpler way to write it.

As for NicholasLP limiting definition of your second identity, which I dismissed for not being well defined, he defined it with the limit outside the covariance function. Tmi have no recollection of the covariance to be a continuous function si that definition does not give you the term in your second expression.

So to recap:

1. I have never seen assumption 2 in the literature. And it does not make sense at all.
2. The previous answer's attempt at defining it is incorrect. So no need to worry about how those two differ.