1) Using Jensen’s Inequality to argue that
i) $|E[X]| \leq E[|X|]$
ii) $|E[X]|^{2} \leq E\left[|X|^{2}\right]$,
provided that all the expectations involved exist. (A challenge: if you claim that a function is “convex”, you must prove it!)
Assume that a process $\left{X_{n}: n=1,2, \cdots\right}$ satisfies the following: (i) $\sup {n} E\left[\left|X{n}\right|\right]<\infty$, and (ii) $\forall \varepsilon>0$ $\exists \delta>0$, such that
$$
E\left{\left|X_{n}\right|: A\right}<\varepsilon, \quad \text { whenever } P(A)<\delta $$ Show that $\lim {K \rightarrow \infty} \sup {n} E\left{\left|X_{n}\right| 1_{\left{\left|X_{n}\right|>K\right}}\right}=0 .$ Namely $\left{X_{n}\right}$ is uniformly integrable.
Show that if $\left{X_{n}\right}$ is uniformly integrable, then
(i) $\sup {n} E\left(\left[X{n}\right]^{+}\right)<\infty$
(ii) If we assume further that $\left{X_{n}\right}$ is a martingale, then $\lim {n} X{n}=X_{\infty}$ exists (by upcrossing theorem), and $E\left|X_{\infty}\right|<\infty$ (Hint: use Fatou).
1) Using Jensen’s Inequality to argue that
i) $|E[X]| \leq E[|X|]$
ii) $|E[X]|^{2} \leq E\left[|X|^{2}\right]$,
Assume that $Z$ is a random variable with $E|Z|<\infty$ and $\left{\mathcal{G}{n}\right}$ is a filtration. Show that $M{n}:=E\left{Z \mid \mathcal{G}{n}\right}$ is an U.I. $\left{\mathcal{G}{n}\right}$-martingale.
Let $\left{B_{t}: t \geq 0\right}$ be a standard Brownian motion. For any $t \geq 0$ and $n \geq 0$, define
$$
Z_{n}=\sum_{i=1}^{2^{n}}\left(B_{i t / 2^{n}}-B_{(i-1) t / 2^{n}}\right)^{2}-t
$$
Calculate $E Z_{n}$ and $E\left[Z_{n}\right]^{2}$
Let $\left{B_{t}: t \geq 0\right}$ be a standard Brownian motion. For any $t \geq 0$ and $n \geq 0$, define
$$
Let $M_{t}=\int_{0}^{t} h(s) d B_{s}, N_{t}=\int_{0}^{t} g(s) d B_{s}, t \geq 0$, where $h, g \in L_{\mathbf{F}}^{2}([0, T] \times \Omega)$. Define $\langle M, N\rangle_{t}:=$ $\int_{0}^{t} h(s) g(s) d s, t \geq 0$. Show that $M_{t} N_{t}-\langle M, N\rangle_{t}$ is a martingale. (Hint: Recall $\left.\langle M\rangle_{t}=\int_{0}^{t}|h(s)|^{2} d s .\right)$
8) Assume that $X$ is a solution to the following SDE:
$$
d X_{t}=b\left(t, X_{t}\right) d t+\sigma\left(t, X_{t}\right) d B_{t}, \quad X_{0}=x, \quad t \geq 0
$$
and $u=u(t, x)$ is a classical $\left(C^{1,2}\right)$ solution to the PDE:
$$
\left{\begin{array}{l}
0=u_{t}+\frac{1}{2} \sigma^{2}(t, x) u_{x x}+b(t, x) u_{x}+c(t) u+f(t, x) \
u(T, x)=g(x)
\end{array}\right.
$$
Argue that $u(t, x)=E\left{e^{\int_{t}^{T} c(s) d s} g\left(X_{T}\right)+\int_{t}^{T} e^{\int_{t}^{s} c(r) d r} f\left(s, X_{s}\right) d s \mid X_{t}=x\right}$
