Elements of Itô Stochastic Calculus

Let $0=t_{0}<t_{1}<\ldots <t_{n}$ and let random variables $f_{1}, \ldots , f_{n}$ be given, with $f_{i}$ being $\mathcal{F}_{t_i}$-measurable for all $i=1, \ldots , n$. Prove that then, for the step process $f(t, \omega )=\sum_{i=1}^{n} f_{i}(\omega ) I\left\{ t_{i-1}<t \leqslant t_{i}\right\}$, the process $\mathcal{J}_{t}(f)$ is a martingale with a.s. continuous trajectories.

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

Give an example of a stochastic differential equation with zero initial condition whose solution is not unique.

Solve the stochastic differential equations:

Prove that if $V(0)$ is independent of the Wiener process $\left(W_{t}, t \geqslant 0\right)$ and $V(0) \sim \mathcal{N}\left(0, \sigma^{2} /(2 a)\right)$, then the solution $V=\left(V_{t}, t \geqslant 0\right)$ of the Langevin equation $d V_{t}=-a V_{t} d t+\sigma d W_{t}$ is a Gaussian process with covariance function $\frac{\sigma^{2}}{2 a} e^{-a\left|s-t\right|}$.

Let $X_{t}=\left(X_{t}^{1}, X_{t}^{2}\right)$ be a two-dimensional process given by the stochastic differential equation

where $\left(W_{t}, t \geq 0\right)$ is a one-dimensional Wiener process.

Let $\left(X_{t}, t \geqslant 0\right)$ be a process from $L^{2}\left((0,+\infty ) \times \Omega , \mathcal{PRED}, \text{mes} \times \mathbb {P}\right)$. Prove that then the process

satisfies the relation $\mathbb {E}\left[Y_{t}\right] \leqslant 1$ for all $t>0$ (we assume that all necessary processes are integrable).

Let $\left(X_{t}, t \geqslant 0\right)$ be a bounded process from $L^{2}((0,+\infty ) \times \Omega , \mathcal{P R E D}$, mes $\times \mathbb {P})$, i.e., there exists $C>0$ such that $\left|X_{t}\right| \leqslant C$ for all $t \geqslant 0$. Prove that then the process

is a martingale.