Definition of cross variation of two processes

Question

I have a question about the following argument. I see in my book a claim that given 2 stochastic integrals :

\begin{align}X_1&:=\int_{0}^{t}f_s\mathsf dM_s\\ X_2&:=\int_{0}^{t}g_s\mathsf dN_s \end{align}

$N,M$ are some continuous local martingales then

$$\langle X_1,X_2\rangle_t=\int_{0}^{t}f_s g_s \mathsf d\langle M,N\rangle _s$$

What does the notation $ \mathsf d\langle M,N\rangle_s$ means in the sense of quadratic variation and how they got it here?

Thank you.

Answer 1

$\langle M,N \rangle$ is the cross variation of two processes. It's typically defined with the polarization identity used to calculate a vector inner product from a vector norm:

From Karatzas + Shreve 2nd Edition, 1.5.5, page 31:

For any two martingales $X, Y \in \mathscr{M}_2$, we define their cross-variation process $\langle X, Y \rangle$ by:

\begin{align*} \langle X, Y \rangle_t &\triangleq \frac{1}{4} \left[ \langle X + Y \rangle_t - \langle X - Y \rangle_t \right]; \quad 0 \le t < \infty \\ \end{align*}

Then the $d\langle M,N \rangle$ notation is the cross variation function being used as an integrator in a Lebesgue-Stieltjes or Riemann-Stieltjes integral.