Integral representation of martingales in the Brownian excursion filtration.

*(English)*Zbl 0635.60057
Probabilités XX, Proc. Sémin., Strasbourg 1984/85, Lect. Notes Math. 1204, 465-502 (1986).

[For the entire collection see Zbl 0593.00014.]

Let \(B_ t\) be a Brownian motion from zero and let \(L(x,t)\) be its local time. Let \({\mathcal E}_ x\) be the excursion field, i.e. the \(\sigma\)-field generated by the excursions of \(B_ t\) below x. These fields are closely linked to the Ray-Knight theorems and to the behavior of local time in the space variable.

D. Williams [Semin. Probab. XIII, Univ. Strasbourg 1977/78, Lect. Notes Math. 721, 490-494 (1979; Zbl 0422.60058)] settled part of a conjecture of the reviewer: all \({\mathcal E}_ x\) martingales are continuous. The reviewer [Stochastic processes, Semin. Evanston/Ill. 1982, Prog. Probab. Stat. 5, 237-302 (1983; Zbl 0524.60074)] settled the other half of the conjecture, namely that continuity could be proved via a martingale representation theorem, by showing that local time can be regarded as a stochastic measure L(dx dt) on the plane and that each square-integrable martingale \(\{M_ x,x\geq x_ 0\}\) can be written in the form \[ M_ x=M_{x_ 0}+\int^{\infty}_{0}\int^{x}_{- \infty}f(y,t)L(dy dt) \] for certain \({\mathcal E}_ x\)-adapted f. Unfortunately, neither William’s nor the reviewer’s proofs were complete. Each depended on some high-order formulas which were simply too complicated to prove in detail.

The purpose of this article is to give a rigorous proof of the continuity of \({\mathcal E}_ x\) martingales via the integral representation route. The author first makes a time-change in the x,t-plane - the time-change depends on the level x - and maps \(L(x,t)\) into a process \(\tilde L(x,t)\) which has the property that for each fixed t, \(L(x,t)+x^-\) is a supermartingale. The key step is to find the explicit form of its Meyer decomposition.

Once this is done the author defines a “parameterized stochastic integral” and shows that Williams’ CMO martingales can be written in terms of it. The CMO martingales form a dense set, so the martingales representation and continuity both follow.

[Remarks: The paper omits a detailed proof that the distributional measure \((d/dt)\int^{x}_{a}\tilde L^ 2(y,t)dy\) of Lemma 4.2 is in fact a bicontinuous function, but it is true.

Let \(B_ t\) be a Brownian motion from zero and let \(L(x,t)\) be its local time. Let \({\mathcal E}_ x\) be the excursion field, i.e. the \(\sigma\)-field generated by the excursions of \(B_ t\) below x. These fields are closely linked to the Ray-Knight theorems and to the behavior of local time in the space variable.

D. Williams [Semin. Probab. XIII, Univ. Strasbourg 1977/78, Lect. Notes Math. 721, 490-494 (1979; Zbl 0422.60058)] settled part of a conjecture of the reviewer: all \({\mathcal E}_ x\) martingales are continuous. The reviewer [Stochastic processes, Semin. Evanston/Ill. 1982, Prog. Probab. Stat. 5, 237-302 (1983; Zbl 0524.60074)] settled the other half of the conjecture, namely that continuity could be proved via a martingale representation theorem, by showing that local time can be regarded as a stochastic measure L(dx dt) on the plane and that each square-integrable martingale \(\{M_ x,x\geq x_ 0\}\) can be written in the form \[ M_ x=M_{x_ 0}+\int^{\infty}_{0}\int^{x}_{- \infty}f(y,t)L(dy dt) \] for certain \({\mathcal E}_ x\)-adapted f. Unfortunately, neither William’s nor the reviewer’s proofs were complete. Each depended on some high-order formulas which were simply too complicated to prove in detail.

The purpose of this article is to give a rigorous proof of the continuity of \({\mathcal E}_ x\) martingales via the integral representation route. The author first makes a time-change in the x,t-plane - the time-change depends on the level x - and maps \(L(x,t)\) into a process \(\tilde L(x,t)\) which has the property that for each fixed t, \(L(x,t)+x^-\) is a supermartingale. The key step is to find the explicit form of its Meyer decomposition.

Once this is done the author defines a “parameterized stochastic integral” and shows that Williams’ CMO martingales can be written in terms of it. The CMO martingales form a dense set, so the martingales representation and continuity both follow.

[Remarks: The paper omits a detailed proof that the distributional measure \((d/dt)\int^{x}_{a}\tilde L^ 2(y,t)dy\) of Lemma 4.2 is in fact a bicontinuous function, but it is true.

Reviewer: J.Walsh