# Tag Archives: JCM_math545_HW4_S14

## Quadratic Variation of Ito Integrals

Given a stochastic process  $$f_t$$ and $$g_t$$ adapted to a filtration $$\mathcal F_t$$ satisfying

$\int_0^T\mathbf E f_t^2 dt < \infty\quad\text{and}\quad \int_0^T\mathbf E g_t^2 dt < \infty$

define

$M_t =\int_0^t f_s dW_s \quad \text{and}\quad N_t =\int_0^t g_s dW_s$

for some standard Brownian Motion also adapted to the  filtration $$\mathcal F_t$$ . Though it is not necessary, assume that  there exists a $$K>0$$ so that  $$|f_t|$$ and $$|g_t|$$  are less than some $$K$$  for all $$t$$ almost surely.

Let $$\{ t_i^{(n)} : i=0,\dots,N(n)\}$$ be sequence of partitions of $$[0,T]$$ of the form

$0 =t_0^{(n)} < t_1^{(n)} <\cdots<t_N^{(n)}=T$

such that

$\lim_{n \rightarrow \infty} \sup_i |t_{i+1}^{(n)} – t_i^{(n)}| = 0$

Defining

$V_n[M]=\sum_{i=1}^{N(n)} \big(M_{t_i} -M_{t_{i-1}}\big)^2$

and

$Q_n[M,N]= \sum_{i=1}^{N(n)} \big(M_{t_i} -M_{t_{i-1}}\big)\big(N_{t_i} -N_{t_{i-1}}\big)$

Clearly $$V_n[M]= Q_n[M,M]$$. Show  that the following points hold.

1. The “polarization equality” holds:
$4 Q_n[M,N] =V_n[M+N] -V_n[M-N]$
Hence it is enough to understand the limit of $$n \rightarrow \infty$$ of $$Q_n$$ or $$V_n$$.
2. $\mathbf E V_n[M]= \int_0^T \mathbf E f_t^2 dt$
3. * $$V_n[M]\rightarrow \int_0^T f_t^2 dt$$ as $$n \rightarrow \infty$$ in $$L^2$$. That is to say
$\lim_{n \rightarrow \infty}\mathbf E \Big[ \big( V_n[M] – \int_0^T f_t^2 dt \big)^2 \Big]=0$
This limit is called the Quadratic Variation of the Martingale $$M$$.
4. Using the results above, show that $$Q_n[M,N]\rightarrow \int_0^T f_t g_t dt$$ as $$n \rightarrow \infty$$ in $$L^2$$. This is called the cross-quadratic variation of $$M$$ and $$N$$.
5. * Prove by direct calculation that  in the spirit of 3) from above that   $$Q_n[M,N]\rightarrow \int_0^T f_t g_t dt$$ as $$n \rightarrow \infty$$ in $$L^2$$.

In this context, one writes $$\langle M \rangle_T$$ for the limit of the $$V_n[M]$$  which is called the quadratic variation process of $$M_T$$. Similarly  one writes  $$\langle M,N \rangle_T$$ for the  limit of $$Q_n[M,N]$$  which is called the cross-quadratic variation process of $$M_T$$ and $$N_T$$. Clearly $$\langle M \rangle_T = \langle M,M \rangle_T$$ and $$\langle M+N,M \rangle_T = \langle M, M+N\rangle_T= \langle M \rangle_T + \langle M, N\rangle_T$$.

## Covariance of Ito Integrals

Let $$f_t$$ and $$f_t$$ be two stochastic processes adapted to a filtration $$\mathcal F_t$$ such that

$\int_0^\infty \mathbf E (f_t^2) dt < \infty \qquad \text{and} \qquad \int_0^\infty \mathbf E (g_t^2) dt < \infty$

Let $$W_t$$ be a standard brownian motion  also adapted to the filtration $$\mathcal F_t$$ and define the stochastic processes

$X_t =\int_0^t f_s dW_s \qquad \text{and} \qquad Y_t=\int_0^t g_s dW_s$

Calculate the following:

1. $$\mathbf E (X_t X_s )$$
2. $$\mathbf E (X_t Y_t )$$
Hint: You know how to compute $$\mathbf E (X_t^2 )$$ and $$\mathbf E (Y_t^2 )$$. Use the fact that $$(a+b)^2 = a^2 +2ab + b^2$$ to answer the question. Simplify the result to get a compact expression for the answer.
3. Show that if $$f_t=\sin(2\pi t)$$ and $$g_t=\cos(2\pi t)$$ then $$X_1$$ and $$Y_1$$ are independent random variables.(Hint: use the result here  to deduce that $$X_1$$ and $$Y_1$$ are mean zero gaussian random variables. Now use the above results to show that the covariance of $$X_1$$ and $$Y_1$$ is zero. Combining these two facts implies that the random variables are independent.)

## Gaussian Ito Integrals

In this problem, we will show that the Ito integral of a deterministic function is a Gaussian Random Variable.

Let $$\phi$$ be deterministic elementary functions. In other words there exists a  sequence of  real numbers $$\{c_k : k=1,2,\dots,N\}$$ so that

$\sum_{k=1}^\infty c_k^2 < \infty$

and there exists a partition

$0=t_0 < t_1< t_2 <\cdots<t_N=T$

so that

$\phi(t) = \sum_{k=1}^N c_k \mathbf{1}_{[t_{k-1},t_k)}(t)$

1. Show that if $$W(t)$$ is a standard brownian motion then the Ito integral
$\int_0^T \phi(t) dW(t)$
is a Gaussian random variable with mean zero and variance
$\int_0^T \phi(t)^2 dt$
2. * Let $$f\colon [0,T] \rightarrow \mathbf R$$ be a deterministic function such that
$\int_0^T f(t)^2 dt < \infty$
Then it can be shown that there exists a sequence of  deterministic elementary functions $$\phi_n$$ as above such that
$\int_0^T (f(t)-\phi_n(t))^2 dt \rightarrow 0\qquad\text{as}\qquad n \rightarrow \infty$
Assuming this fact, let $$\psi_n$$ be the characteristic function of the random variable
$\int_0^T \phi_n(t) dW(t)$
Show that for all $$\lambda \in \mathbf R$$, show that
$\lim_{n \rightarrow \infty} \psi_n(\lambda) = \exp \Big( -\frac{\lambda^2}2 \big( \int_0^T f(t)^2 dt \big) \Big)$
Then use the the convergence result here to conclude that
$\int_0^T f(t) dW(t)$
is a Gaussian Random Variable with mean zero and variance
$\int_0^T f(t)^2 dt$
by identifying the limit of the characteristic functions above.
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