Construction and basic properties of the Itô integral

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

Suppose $W$ is a Brownian motion and $H$ is adapted, bounded, and right continuous. Let $t \geq 0$. Show

converges in probability to $H_{t}$.

Let $W$ be a one-dimensional Brownian motion with filtration $\left\{ \mathcal{F}_{t}\right\}$ and let $f(r, s)$ be a deterministic function. Define the multiple stochastic integral by

provided

and similarly for higher-order multiple stochastic integrals.

By $I_{t}(f) = \int_{(0, t]} f_{s}(\omega ) d W_{s}(\omega )$, $t \in [0, T]$, we denote the Itô stochastic integral with respect to a Wiener process $W = \left\{ W_{t}, t \geqslant 0\right\}$ of a nonanticipating (i.e., adapted to the filtration and measurable in the pair of variables $(t, \omega )$) random function $f = f_{t}(\omega )$ such that $\mathbb {E}\left[\int_{0}^{T} f_{t}^{2}(\omega ) d t\right] < \infty$.

Let $\tau$ be a Markov time such that $\tau (\omega ) \leqslant T$ for all $\omega \in \Omega$ ($T$ is a positive constant). Prove that $I_{\tau }(f) = I_{T}\left(f \mathbf{1}_{(0, \tau ]}\right)$, where $I_{\tau }(f): = I_{\tau (\omega )}(f)$.

Definition 10. We say that a function $f$ belongs to the class $J_{1}$ if $f:[0, \infty ) \times \Omega \rightarrow \mathbb {R}$ is progressively measurable and

The idea of constructing the integral $I_{t}(f)$ for a function $f \in J_{1}$ is to choose a sequence of functions $f_{n}, n \in \mathbb {N}$, for which the integral $I_{t}\left(f_{n}\right)$ has already been constructed and such that

From this one derives that the sequence $\left\{ I_{t}\left(f_{n}\right)\right\}_{n \in \mathbb {N}}$ is Cauchy in probability. Consequently, there exists a random variable, denoted $I_{t}(f)$, such that $I_{t}\left(f_{n}\right) \stackrel{\mathbb {P}}{\rightarrow } I_{t}(f), n \rightarrow \infty$. The limit $I_{t}(f)$ is still written as $\int_{(0, t]} f(s, \omega ) d W_{s}$ or $(f \cdot W)_{t}$.

Prove that the sequence $\left\{ f_{n}, n \in \mathbb {N}\right\}$ described above in (85) exists. Prove that if $f \in J_{1}$, then there exists an a.s. continuous modification of the process $I_{t}(f), t \geqslant 0$.

Let $\psi_{1}$ and $\psi_{2}$ be predictable step functions, and show that

whenever both sides exist.

Let $\psi$ be an adapted process with $\left\| \psi \right\| < \infty$. Show that $\left\| I(\psi )\right\|_{2} = \left\| \psi \right\|$.

Let $\tau$ be a stopping time with respect to the natural filtration of the Wiener process $\left(W_{t}, t \geqslant 0\right)$, with $\mathbb {E}\left[\tau \right]<+\infty$. Prove that

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

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 $\mathrm{BM}^{1},\left(Y_{t}\right)_{t \geqslant 0}$ 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 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.

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 $M \in \mathcal{M}_{T, \text{ loc }}^{c}$ and $f(t, \omega )$ be an adapted càdlàg (right continuous, finite left limits) process and $t \in [0, T]$.

Let $M \in \mathcal{M}_{T, \text{ loc }}^{c}$ and $f_{n}$ be a sequence in $\mathcal{L}_{T}^{0}(M)$. If $f_{n} \rightarrow f$ in ucp - sense, then $f \in \mathcal{L}_{T}^{0}(M)$.