Stochastic Integral - Exercises

Exercise 1

$G,H \in \mathcal{L}_{loc}$ and $M,N \in \mathcal{M}_{loc}$ that it holds

$$ \begin{aligned} \langle \int_{}^{} GdM\rangle & = \int_{}^{} G^2_sd\langle M\rangle_s\ \langle \int_{}^{} G dM,\int_{}^{} H dN\rangle & = \int_{}^{} G_sd \langle M,\int_{}^{} H dN\rangle_s\ &= \int_{}^{} G_sH_s d\langle M,N\rangle_s

\end{aligned} $$

Show using Doob-Meyer decomposition (Start with $G$, $H$ in $\mathcal{S}$) the chain rule

\[\int_{}^{} Gd\left( \int_{}^{} H d M \right)=\int_{}^{} GH dM\]
Let $M \in \mathcal{M}_c^2$. We know from the Doob-Meyer decomposition that

\[M^2=\tilde{M}+\langle M\rangle\]

for some martingale $\tilde{M}$. Show using Ito's formula that

\[\tilde{M}=2\int_{}^{} M_s dM_s\]

Exercise 2

Show using Ito's Formula that the process $X$ given by

\[X_t=e^{t/2}\cos\left( W_t \right)\]

is a martingale.
Consider the following stochastic differential equation

\[dX=-\theta X dt+\sigma dW, \quad X_0=x\]

for $\theta, \sigma >0$. This stochastic differential equation describes exponential convergence to $0$. Show that it has a solution by considering the process $U_t=e^{\theta t}X_t$.
Using the previous strategy, provide an explicit expression for the OU process which is a mean reverting process:

\[dX = \kappa \left( \theta - X \right)dt + \sigma dW, \quad X_0 = x\]

Exercise 3

stopping time

\[\tau = \inf \left\{ t\colon W_t \not \in (a,b) \right\}\]

which is clearly such that $0< \tau <\infty$, almost surly (however not uniformly bounded). We are interested into some exponential moments of $\tau$, namely, want to show that

\[ \label{eq:01} E\left[ \exp\left( \frac{\theta^2 \tau}{2} \right) \right] = \frac{\cos\left( \frac{a-b}{2} \theta \right)}{\cos\left( \frac{a+b}{2} \theta \right)}, \quad 0\leq \theta < \frac{\pi}{a+b} \]

To this aim, we define the process:

\[X_t = \exp\left( \frac{1}{2} \theta^2 t \right) \cos \left( \theta\left( W_t -\frac{b-a}{2} \right) \right)\]

Using Ito, show that $X$ is a local martingale.
Show that it is a martingale.
Using the fact that $\cos$ is strictly positive on $(-\frac{\pi}{2}, \frac{\pi}{2})$ show that $X^\tau$ is a strict positive martingale.
Using Fatou, show that

\[E\left[ \exp\left( \frac{\theta^2 \tau}{2} \right) \right] < \frac{\cos\left( \frac{a-b}{2} \theta \right)}{\cos\left( \frac{a+b}{2} \theta \right)}<\infty\]
Deduce that $\sup_{t} X^\tau_t$ is in $L^1$, and using dominated convergence theorem, deduce the equality [eq:01]

Exercise 4

\[h_n(x)=(-1)^n \exp\left( \frac{x^2}{2} \right) \frac{d^n}{dx^n}\exp\left( -\frac{x^2}{2} \right)\]

and

\[ H_n(x,y)= \begin{cases} y^{n/2} h_n\left( \frac{x}{y} \right)& \text{for }y>0\\ x^n & \text{for }y=0 \end{cases} \]

Show that 1. $h_n$ are the polynomials that satisfies the identity $\sum_{n\geq 0} u^n h_n(x)/ n!= \exp(ux-u^2/2)$. 2. $\sum_{n\geq 0} u^n H_n(x,y)/ n!= \exp(ux-(yu^2)/2)$. 3. Compute $H_n$ explicitly for $n=0,1,2,3,4$. for a continuous local martingale $M$ we define

\[L^{(n)}=H_n(M,\langle M\rangle)\]

show that 1. $L^{(n)}$ is a continuous local martingale for every $n$; 2. it holds

$$L^{(n)}_t= n! \int_{0}^{t}\int_{0}^{s_1} \ldots \int_{0}^{s_{n-1}}dM_{s_n}\ldots dM_{s_1}$$

Compute $L^{(n)}$ for $n=0,1,2,3,4$.

Exercise 5

time horizon $T<\infty$. 1. Show that $\langle B\rangle_t = t$. We consider the space $\mathcal{L}^2:=\mathcal{L}^2(P\otimes dT)$ of those processes $H=(H_t)_{0\leq t\leq T}$ which are progressive and such that

\[E\left[ \int_{0}^{T}H_t^2 d\langle B\rangle_t \right]=E\left[ \int_{0}^{T}H_t^2dt \right]<\infty.\]

For a fixed time horizon $T<\infty$, define the process

\[H^n_t=\sum_{k=1}^nB_{t_{k-1}^n}1_{]t_{k-1}^n,t_k^n]}(t), \quad 0\leq t\leq T\]

where $t_k^n=kT/n$, $k=0,\ldots, n$. 1. Though $B_{t_{k-1}^n}$ is $\mathcal{F}_{t_{k-1}^n}$-measurable, it is not uniformly bounded and therefore not element of $\mathcal{S}$ as given in the lecture. Show however that it belongs to $\mathcal{L}^2$. 2. Show that $H^n\to B$ in $\mathcal{L}^2$ -- for the $L^2$-norm. In particular, $B \in \mathcal{L}^2$. 3. Show that there exists a random variable $I_T\in \mathcal{L}^2$ such that

$$(H^n\bullet B)_T=\sum_{k=1}^nH^n_{t_k^n}\left( B_{t_k^n}-B_{t_{k-1}^n} \right)=\sum_{k=1}^nB_{t_{k-1}^n}\left( B_{t_{k}^n}-B_{t_{k-1}^n} \right)$$

converges in $\mathcal{L}^2$ to $I_T$.
We denote this random variable the stochastic integral of $B$, that is

$$I_T:=\int_{0}^{T}B_t dB_t$$

Using the relation $b(a-b)=(a^2-b^2-(a-b)^2)/2$, show using the approximation above that

\[\int_{0}^{T}B_tdB_t=\frac{1}{2}\left(B_T^2-T \right)\]

Exercise 6

consider the symmetric random walk

\[S_0 = s, \quad \text{and}\quad S_t = \sum_{s=1}^t Y_s\]

where $Y_1, Y_2, \ldots$ are iid random variables with $P[Y_1=1]=P[Y_1 =-1] = p=1/2$. As for the filtration we set $\mathcal{F}_0 = \{\emptyset, \Omega\}$ and $\mathcal{F}_t = \sigma(S_s\colon s\leq t)$. We finally restrict ourselves to a finite horizon $T>0$. 1. Show that if $M = (M_t)_{t=0, \ldots, T}$ is a martingale, there exists functions $v_t : \mathbb{R}^{t+1} \to \mathbb{R}$ such that

$$M_t = v_t(S_0, \ldots, S_t) \quad\text{for }t =0, \ldots, T$$

Show that the functions $v_t$ satisfies the recursive formula

\[ \begin{cases} v_T(S_0, S_1, \ldots, S_T) = M_T\\ v_t\left(S_0, S_1, \ldots, S_t \right) = pv_{t+1}(S_0, S_1, \ldots, S_t, S_t + 1) + (1-p)v_{t+1}\left(S_0, S_1, \ldots, S_t, S_t -1 \right) \end{cases} \]
Show that there exists function $H_t: \mathbb{R}^{t} \to \mathbb{R}$ -- if $t=0$ then it is a constant -- such that

\[M_t = M_0 + \sum_{s=1}^t H_s(S_0, S_1, \ldots, S_{s-1})\Delta S_s\]

Write explicitly $H_t$ as a function of $v_t(S_0, S_1, \ldots, S_{t-1}, S_{t-1}\pm 1)$. In particular, $H_t(S_0, S_1, \ldots, S_{t-1})$ is a predictable process.
Show that if $M_T = h(S_T)$ for some function $h:\mathbb{R} \to \mathbb{R}$, then $M$ is a Markov process, and $v_t$ only depends on $S_t$ as well as $\Delta_{t+1}$. From the proof, you will see from the simple linear system of equations to be solved that you get two equations for exactly two states. If $Y$ were taking three values instead of two you would not be able to solve the system.

Exercise 7

For $s<t$, using Ito's Formula, show that the (complex) random variable $\xi = \exp(\imath u(W_t- W_s))$ is such that

\[\xi = e^{-\frac{u^2(t-s)}{2}} + \int_{s}^{t} iu e^{iu(W_r-W_s) + \frac{u^2(r-1}{2}}dW_r\]

Deduce that for every $\xi$ in the set

\[\mathcal{S}^{\Pi} = \mathrm{Span}\left\{ \prod_{k=1}^n \exp\left( iu_k\left( W_{t_k} - W_{t_k-1} \right) \right)\colon u_1, \ldots, u_n \in \mathbb{R} \right\}\]

for a partition $\Pi = \{0=t_0<t_1< \ldots< t_N = T\}$, the martingale representation theorem holds, and therefore on $\mathcal{S}:=\cup_\Pi \mathcal{S}^\Pi$.
Show that the set $\mathcal{D} = \cup_\Pi \mathcal{D}^\Pi$ where

\[ \mathcal{D}^\Pi = \left\{ f\left(\Delta W\right) \colon f:\mathbb{R}^n\to\mathbb{R}\text{ Borel-measurable and } f\left(\Delta W\right)\in L^2 \right\} \]

is dense in $L^2(\mathcal{F}_T)$. Hint: Consider the martingale $M_n = E[\xi|\mathcal{F}^n]$ where $\mathcal{F}^n = \sigma(W_{t_1} - W_{t_0}, \ldots, W_{t_n}-W_{t_{n-1}})$ where $\Pi^n\subseteq \Pi^{n+1}$ and $|\Pi^n|\to 0$.
A Hilbert space ingredient: Show that if $F$ is a closed subspace of $L^2$ and $E\subseteq F$ is a subspace such that $F\cap E^\perp=0$, then $\bar{E}=F$.
The set $\mathcal S$ is dense in $L^2$. For this we show that $\mathcal{S}^\Pi$ is dense in $\mathcal{D}^\Pi$. For $\xi = f(W_{t_1} - W_{t_0}, \ldots, W_{t_n} - W_{t_{n-1}})$ in $\mathcal{D}^\Pi$, suppose that $E[\xi Z] = 0$ for every $Z$ in $\mathcal{S}^\Pi$. Denote by $\psi$ the joint density of $(W_{t_1} - W_{t_0}, \ldots, W_{t_n}- W_{t_{n-1}})$. Derive that

\[\int_{\mathbb{R}n}^{} f(x_1, \ldots, x_n)\psi(x_1, \ldots, x_n) \prod_{k=1}^n \exp(iu_k x_k)dx_1, \ldots, dx_n = 0\]

for any $u_1, \ldots, u_n$ and deduce that $f\equiv 0$. Conclude and show uniqueness of the representation.