I was reading Aldous' famous 1997 paper (Aldous, David. "Brownian excursions, critical random graphs and the multiplicative coalescent." The Annals of Probability (1997): 812-854.)

In his description of the multiplicative coalescent, Aldous states in page $836$ that "Of course, our infinitesimal notation $E(\Delta(Y(t))|\mathcal{F}_{t})=A(t)\,dt$ is just an intuitive way of expressing the rigorous assertion that $M(t)=Y(t)-\int_{0}^{t}A(t)\,dt$ is a local martingale; similarly the notation $\text{var}(\Delta(Y(t))|\mathcal{F}_{t})=B(t)\,dt$ means that $M(t)^{2}-\int_{0}^{t}B(t)\,dt$ is a local martingale"

And then he goes on to compute the values for his process $Y(t)$. I don't really understand as to how he arrived at the expressions as given above. I have read about Doob-Meyer decomposition from Jean-Francois Le Gall's "Brownian motion and stochastic calculus". But I have never seen such an explicit expression for the compensators and the quadratic variation with relative ease. While I can perhaps swallow the claim without proof that computing $E(\Delta(Y(t))|\mathcal{F}_{t})=A(t)\,dt$ indeed gives us the compensator. I really am confounded by how he arrives at the expression for $\text{var}(\Delta(Y(t))|\mathcal{F}_{t})$ in page $838$ and $839$ and why it indeed equals the quadratic variation of the martingale.

Can anyone explain the above to me or suggest me a reference which will help me better understand this.

I was reading Aldous' famous 1997 paper (Aldous, David. "Brownian excursions, critical random graphs and the multiplicative coalescent." The Annals of Probability (1997): 812-854.)

In his description of the multiplicative coalescent, Aldous states in page $836$ that "Of course, our infinitesimal notation $E(\Delta(Y(t))|\mathcal{F}_{t})=A(t)\,dt$ is just an intuitive way of expressing the rigorous assertion that $M(t)=Y(t)-\int_{0}^{t}A(t)\,dt$ is a local martingale; similarly the notation $\text{var}(\Delta(Y(t))|\mathcal{F}_{t})=B(t)\,dt$ means that $M(t)^{2}-\int_{0}^{t}B(t)\,dt$ is a local martingale"

And then he goes on to compute the values for his process $Y(t)$. I don't really understand as to how he arrived at the expressions as given above. I have read about Doob-Meyer decomposition from Jean-Francois Le Gall's "Brownian motion and stochastic calculus". But I have never seen such an explicit expression for the compensators and the quadratic variation which he arrives at with relative ease. I suppose it is due to the the nature of the model. While I can perhaps swallow the claim without proof that computing $E(\Delta(Y(t))|\mathcal{F}_{t})=A(t)\,dt$ indeed gives us the compensator. I really am confounded by how he arrives at the expression for $\text{var}(\Delta(Y(t))|\mathcal{F}_{t})$ in page $838$ and $839$ and why it indeed equals the quadratic variation of the martingale.

Can anyone explain the above to me or suggest me a reference which will help me better understand this.

I was reading Aldous' famous 1997 paper (Aldous, David. "Brownian excursions, critical random graphs and the multiplicative coalescent." The Annals of Probability (1997): 812-854.)

In his description of the Multiplicative Coalescent, Aldous states in page $836$ that "Of course, our infinitesimal notation $E(\Delta(Y(t))|\mathcal{F}_{t})=A(t)\,dt$ is just an intuitive way of expressing the rigorous assertion that $M(t)=Y(t)-\int_{0}^{t}A(t)\,dt$ is a local martingale; similarly the notation $\text{var}(\Delta(Y(t))|\mathcal{F}_{t})=B(t)\,dt$ means that $M(t)^{2}-\int_{0}^{t}B(t)\,dt$ is a local martingale"

And then he goes on to compute the values for his process $Y(t)$. I don't really understand as to how he arrived at the expressions as given above. I have read about Doob-Meyer decomposition from Jean Francois LeGall's Brownian motion and Stochastic Calculus. But I have never seen such an explicit expression for the compensators and the quadratic variation with relative ease. While I can perhaps swallow the claim without proof that computing $E(\Delta(Y(t))|\mathcal{F}_{t})=A(t)\,dt$ indeed gives us the compensator. I really am confounded by how he arrives at the expression for $\text{var}(\Delta(Y(t))|\mathcal{F}_{t})$ in page $838$ and $839$ and why it indeed equals the quadratic variation of the martingale.

Can anyone explain the above to me or suggest me a reference which will help me better understand this.

I was reading Aldous' famous 1997 paper (Aldous, David. "Brownian excursions, critical random graphs and the multiplicative coalescent." The Annals of Probability (1997): 812-854.)

In his description of the multiplicative coalescent, Aldous states in page $836$ that "Of course, our infinitesimal notation $E(\Delta(Y(t))|\mathcal{F}_{t})=A(t)\,dt$ is just an intuitive way of expressing the rigorous assertion that $M(t)=Y(t)-\int_{0}^{t}A(t)\,dt$ is a local martingale; similarly the notation $\text{var}(\Delta(Y(t))|\mathcal{F}_{t})=B(t)\,dt$ means that $M(t)^{2}-\int_{0}^{t}B(t)\,dt$ is a local martingale"

And then he goes on to compute the values for his process $Y(t)$. I don't really understand as to how he arrived at the expressions as given above. I have read about Doob-Meyer decomposition from Jean-Francois Le Gall's "Brownian motion and stochastic calculus". But I have never seen such an explicit expression for the compensators and the quadratic variation with relative ease. While I can perhaps swallow the claim without proof that computing $E(\Delta(Y(t))|\mathcal{F}_{t})=A(t)\,dt$ indeed gives us the compensator. I really am confounded by how he arrives at the expression for $\text{var}(\Delta(Y(t))|\mathcal{F}_{t})$ in page $838$ and $839$ and why it indeed equals the quadratic variation of the martingale.

Can anyone explain the above to me or suggest me a reference which will help me better understand this.