(H \cdot M)_{t} = \sum_{k = 0}^{n - 1} Z_{k} (M_{t \land t_{k + 1}} - M_{t \land t_{k}}) .

It is constant after

t_{n}

, so the symbol

(H \cdot M)_{\infty}

means its value there. Put

Δ_{k} M = M_{t_{k + 1}} - M_{t_{k}}

. If

k < ℓ

, then

Z_{k} Δ_{k} M Z_{ℓ}

F_{t_{ℓ}}

-measurable, and the conditional expectation of

Δ_{ℓ} M

given that sigma-algebra is zero. Boundedness of the coefficients and square-integrability of

M

justify the expectations, so all cross terms vanish:

E [(H \cdot M)_{\infty}^{2}] = \sum_{k} E [Z_{k}^{2} (Δ_{k} M)^{2}] .

For each diagonal term, the martingale increment identity also gives

E [Z_{k}^{2} (Δ_{k} M)^{2}] = E [Z_{k}^{2} (M_{t_{k + 1}}^{2} - M_{t_{k}}^{2})] .

Indeed, the omitted term

2 Z_{k}^{2} M_{t_{k}} Δ_{k} M

has expectation zero. Apply the assumed martingale property of

M^{2} - [M]

to replace this difference of squares by the corresponding quadratic variation increment. Thus

E [(H \cdot M)_{\infty}^{2}] = k \sum E [Z_{k}^{2} ([M]_{t_{k + 1}} - [M]_{t_{k}})] = E [(H^{2} \cdot [M])_{\infty}] .

The intervals

(t_{k}, t_{k + 1}]

in the Lebesgue–Stieltjes integral exactly match these increments, including possible jumps of

M

 Ancestors (10)

 Discussion (0) New discussion

There are no discussions about this article yet.

There are currently no matching articles.