Optional and predictable brackets for counting-process martingales

Work on the finite horizon $[0,T]$ of Simple counting-process intensity and compensator contract. For càdlàg (right-continuous with left limits) finite-variation processes $X,Y$ starting at zero, with $\mathrm{\Delta }{X}_s:=X_s-X_{s-}$ and $\mathrm{\Delta }{Y}_s:=Y_s-Y_{s-}$, define their optional quadratic covariation by $[X,Y{]}_t:={\sum }_{0<s\le t}\mathrm{\Delta }{X}_s\mathrm{\Delta }{Y}_s,$ and write $[X]:=[X,X]$. The series is absolutely convergent because each path has finite variation. This is the jump-sum convention for optional brackets used on this page.

For a square-integrable model martingale $M=N-A$ from Simple counting-process intensity and compensator contract, a continuous predictable bracket $⟨M⟩$ is a continuous, predictable, increasing process $B$ with $B_0=0$, $\mathbb{E}[B_T]<\mathrm{\infty }$, and $M^2-B$ a martingale, when such a process exists. For two square-integrable model martingales $M=N-A$ and $M^{\mathrm{\prime }}=N^{\mathrm{\prime }}-A^{\mathrm{\prime }}$ from that contract, a continuous predictable cross bracket $⟨M,M^{\mathrm{\prime }}⟩$ is a continuous, predictable finite-variation process $C$ with $C_0=0$, $\mathbb{E}\mathrm{Var}(C{)}_T<\mathrm{\infty }$, and $[M,M^{\mathrm{\prime }}]-C$ a martingale, when such a process exists. The uniqueness lemma below fixes the meaning of “the” bracket in this continuous compensator class. No general bracket-existence theorem is asserted. The Axiom of Choice is inherited through the model martingale's conditional-expectation interface; this definition makes no further choice.

Tian states the continuous-compensator bracket identities for counting process martingales and explicitly refers their general proofs elsewhere. Bass gives the jump-square convention and product formula for semimartingales with jumps. The local results below prove the particular existence and identities required by this page.