Stochastic integrals: L²-Theory

Let $\left(M_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ and $\left(N_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ be $L^{2}$ martingales; then $\left(M_{n} N_{n} - \langle M, N\rangle_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ is a martingale.

Let $\left(M_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ and $\left(N_{n}, \mathscr {F}_{n}\right)_{n \geqslant 0}$ be $L^{2}$ martingales. Show that $\left|\langle M, N\rangle_{n}\right| \leqslant \sqrt{\left\langle M\right\rangle_{n}} \sqrt{\left\langle N\right\rangle_{n}}$.

Let $f \in \mathcal{S}_{T}$ be a simple process and $M \in \mathcal{M}_{T}^{2, c}$. Show that the definition of the stochastic integral $\int_{0}^{T} f(s) d M_{S}$ (cf. Definition 15.9) does not depend on the particular representation of the simple process (15.12).

Show that $\left\| M\right\|_{\mathcal{M}_{T}^{2}}: = \left(\mathbb {E}\left[\sup_{s \leqslant T}\left|M_{s}\right|^{2}\right]\right)^{\frac{1}{2}}$ is a norm in the family $\mathcal{M}_{T}^{2}$ of equivalence classes of $L^{2}$ martingales ( $M$ and $\widetilde{M}$ are said to be equivalent if, and only if, $\sup_{0 \leqslant t \leqslant T}\left|M_{t} - \widetilde{M}_{t}\right| = 0$ a.s.).

Show that the limit (15.19) does not depend on the approximating sequence.

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $\tau$ a stopping time. Find $\left\langle B^{\tau }\right\rangle_{t}$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $f, g \in \mathcal{L}_{T}^{2}$. Show that

Let $\left(B_{t}\right)_{t \geqslant 0}$ be $\mathrm{BM}^{1}$ and $\left(f_{n}\right)_{n \geqslant 1}$ a sequence in $\mathcal{L}_{T}^{2}$ such that $f_{n} \rightarrow f$ in $L^{2}\left(\lambda_{T} \otimes \mathbb {P}\right)$. Then $f_{n} \bullet B \rightarrow f \bullet B$ in $\mathcal{M}_{T}^{2, c}$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $f \in L^{2}\left(\lambda_{T} \otimes \mathbb {P}\right.$ ) for some $T > 0$. Assume that the limit $\lim_{\epsilon \rightarrow 0} f(\epsilon ) = f(0)$ exists in $L^{2}(\mathbb {P})$. Show that

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1},\left(Y_{t}\right)_{t \geqslant 0}$ and adapted process such that $Y_{t} Y_{s} = Y_{t}$ for all $s \leqslant t$ and $\left(X_{t}\right)_{t \geqslant 0} \in \mathcal{L}_{T}^{2}$. Show that

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Show that $(0, \infty ) \ni s \mapsto f(s, \omega ): = \; \mathbb {1}_{(0, \infty )}\left(B_{s}\right)$ is continuous in $L^{2}(\mathbb {P})$.

Use the fact that any continuous, square - integrable martingale with bounded variation paths is constant (cf. Proposition 17.2) to show the following: $\left\langle f \cdot B\right\rangle_{t}: = \int_{0}^{t}\left|f(s)\right|^{2} d s$ is the unique continuous and increasing process such that $(f \cdot B)^{2} - f^{2} \cdot \left\langle B\right\rangle$ is a martingale.

The quadratic covariation of two continuous $L^{2}$ martingales $M, N \in \mathcal{M}_{T}^{2, c}$ is defined as in the discrete case by the polarization formula (15.8).

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $f, g \in \mathcal{L}_{T}^{2}$. Show that $\langle f \bullet B, g \bullet B\rangle_{t} = \int_{0}^{t} f(s) g(s) d s$ for all $t \in [0, T]$.

Use Theorem 15.15.c) to show that the stochastic integrals for the right and left continuous simple processes $f(t, \omega ): = \sum_{j = 1}^{n} \phi_{j - 1}(\omega ) \; \mathbb {1}_{\left[s_{j - 1}, s_{j}\right)}(t)$ and $g(t, \omega ): = \sum_{j = 1}^{n} \phi_{j - 1}(\omega ) \; \mathbb {1}_{\left(s_{j - 1}, s_{j}\right]}(t)$ coincide.

Let $\left(X_{n}\right)_{n \geqslant 0}$ be a sequence of r. v. in $L^{2}(\mathbb {P})$. Show that $L^{2} - \lim_{n \rightarrow \infty } X_{n} = X$ implies $L^{1} - \lim_{n \rightarrow \infty } X_{n}^{2} = X^{2}$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Show that $\int_{0}^{T} B_{s}^{2} d B_{s} = \frac{1}{3} B_{T}^{3} - \int_{0}^{T} B_{s} d s, T > 0$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $f \in \operatorname {BV}[0, T], T < \infty$, a non - random function. Prove that $\int_{0}^{T} f(s) d B_{s} = f(T) B_{T} - \int_{0}^{T} B_{s} d f(s)$ (the latter integral is understood as the mean - square limit of Riemann - Stieltjes sums). Conclude from this that the Itô integral extends the Paley - Wiener - Zygmund integral, cf. Paragraph 13.5.

Adapt the proof of Proposition 15.18 and prove that we also have

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a one - dimensional Brownian motion, $T < \infty$ and $\Pi_{n}, n \geqslant 1$, a sequence of partitions of $[0, T]: 0 = t_{n, 0} < t_{n, 1} < \cdots < t_{n, k(n)} = T$. Assume that $\lim_{n \rightarrow \infty }\left|\Pi_{n}\right| = 0$ and define intermediate points $\theta_{n, k}^{\alpha }: = t_{n, k}+\alpha \left(t_{n, k+1} - t_{n, k}\right)$ for some fixed $0 \leqslant \alpha \leqslant 1$. Show that the Riemann - Stieltjes sums

converge in $L^{2}(\mathbb {P})$ and find the limit $L_{T}(\alpha )$. For which $\alpha$ is $L_{t}(\alpha ), 0 \leqslant t \leqslant T \mathrm{a}$ martingale?

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a one - dimensional Brownian motion and $\tau$ a stopping time. Show that $f(s, \omega ): = \; \mathbb {1}_{[0, T \wedge \tau (\omega ))}(s), 0 \leqslant s \leqslant T < \infty$ is progressively measurable. Use the very definition of the Itô integral to calculate $\int_{0}^{T} f(s) d B_{s}$ and compare the result with the localization principle of Theorem 15.15.

Show that

is a $\sigma$ - algebra.

Show that every right or left continuous adapted process is $\mathscr {P}$ measurable.

Let $\tau$ be a stopping for a $\mathrm{BM}^{1}\left(B_{t}\right)_{t \geqslant 0}$ and define the stochastic interval $(0, \tau \rrbracket : = \left\{ (s, \omega ): 0 < s \leqslant \tau (\omega )\right\}$. Show, using the white noise approach from Section 15.6 that $N_{t}\left(\; \mathbb {1}_{(0, \tau \rrbracket )}\right) = B_{\tau \wedge t}$.