Stochastic integrals

Prove (10.5) in Proposition 10.3.

Setup: let $M$ be a square integrable martingale and suppose

where each $K_{j}$ is $\mathcal{F}_{a_{j}}$ measurable and bounded, and the intervals $\left(a_{j}, b_{j}\right]$ satisfy $a_{1}<b_{1} \leq a_{2}<b_{2} \leq \cdots \leq a_{J}<b_{J}$. Define

Proposition 10.3: With $H$ as in (10.3) and $N$ defined by (10.4), $N_{t}$ is a continuous martingale,

and

Check the consistency of the first two extensions of the definition of stochastic integrals.

Show that if $M$ is a continuous square integrable martingale, and $T$ a finite stopping time, then

Show that if $N_{t}=\int_{0}^{t} H_{s} d M_{s}$ where $M$ is a continuous square integrable martingale, $H$ is predictable, and $\mathbb {E}\left[\int_{0}^{\infty } H_{s}^{2} d\langle M\rangle_{s}\right]<\infty$, and $L_{t}=\int_{0}^{t} K_{s} d N_{s}$, where $K$ is predictable and $\mathbb {E}\left[\int_{0}^{\infty } K_{s}^{2} d\langle N\rangle_{s}\right]<\infty$, then

Show that if $M, H$, and $N$ are as in Exercise 10.4, then $\langle M, N\rangle_{t}=\int_{0}^{t} H_{s} d\langle M\rangle_{s}$.

Hint: Derive a formula for $\langle N+M\rangle_{t}$ from the fact that

Suppose that $M$ and $L$ are square integrable martingales, $H$ is predictable and satisfies (10.2), and $N_{t}=\int_{0}^{t} H_{s} d M_{s}$. Show that

Sometimes the stochastic integral of $H$ with respect to $M$ is defined to be the square integrable martingale $N$ for which (10.13) holds for all square integrable martingales $L$.

Show that if $M$ and $N$ are square integrable martingales with continuous paths, then

Hint: Imitate an appropriate proof of the Cauchy-Schwarz inequality. This result is a special case of the inequality of Kunita-Watanabe.