Question

Let \(M_{1}\) and \(M_{2}\) be the martingales associated with the components of the multivariate counting process \(N=\left(N_{1}, N_{2}\right)\) with continuous compensators. Show that $$ \left\langle M_{1}, M_{2}\right\rangle=\left[M_{1}, M_{2}\right]=0. $$

Short answer

Based on the solution above, the cross-covariation process or cross-bracket process, denoted as \(\langle M_{1}, M_{2}\rangle\) or \([M_{1}, M_{2}]\), for two martingales \(M_1\) and \(M_2\) associated with the components of a multivariate counting process \(N = (N_1, N_2)\) is equal to 0. This conclusion is reached after analyzing the infinitesimal increments of \(M_1\) and \(M_2\) and their cross-covariation/cross-bracket process, and then showing that each term of the expanded product results in a value of 0.

Step-by-step solution

Recall the definition of a martingale and the associated continuous compensators

Recall that a martingale is a stochastic process with the property that its expected value at any future time is equal to its current value, given its past values. Let \(A_{1}(t)\) and \(A_{2}(t)\) be the continuous compensators of \(N_{1}\) and \(N_{2}\), respectively. We have $$ M_{1}(t) = N_{1}(t) - A_{1}(t)\quad \text{ and }\quad M_{2}(t) = N_{2}(t) - A_{2}(t). $$

Analyze \(dM_{1}\) and \(dM_{2}\)

Consider the infinitesimal increment of \(M_1\) and \(M_2\): $$ dM_{1}(t) = dN_{1}(t) - dA_{1}(t)\quad \text{ and }\quad dM_{2}(t) = dN_{2}(t) - dA_{2}(t). $$

Determine the cross-covariation/cross-bracket process

The cross-covariation/cross-bracket process is defined as the following: $$ \langle M_{1}, M_{2}\rangle (t) = [M_{1}, M_{2}](t) = \sum_{s \leq t} dM_{1}(s)dM_{2}(s). $$ Substitute the expressions for \(dM_1(t)\) and \(dM_2(t)\) from Step 2 into the definition: $$ \langle M_{1}, M_{2}\rangle (t) = \sum_{s \leq t} \left( dN_{1}(s) - dA_{1}(s) \right) \left( dN_{2}(s) - dA_{2}(s) \right). $$

Show that \(\langle M_{1}, M_{2}\rangle = [M_{1}, M_{2}] = 0\)

Now, we expand the product and analyze each term: - The term \(\sum_{s \leq t} dN_{1}(s)dN_{2}(s)\) is equal to 0 since the increments of the counting processes \(N_1\) and \(N_2\) are disjoint. - The terms \(\sum_{s \leq t} dN_{1}(s)dA_{2}(s)\) and \(\sum_{s \leq t} dA_{1}(s)dN_{2}(s)\) are both 0 since the increments of the counting processes are uncorrelated with their compensators. - The term \(\sum_{s \leq t} dA_{1}(s)dA_{2}(s)\) is a non-decreasing function of time. Therefore, we have $$ \langle M_{1}, M_{2}\rangle (t) = [M_{1}, M_{2}](t) = 0, $$ which completes the proof.