# Category Archives: Ito Formula

## SDE Example: quadratic geometric BM

Show that the solution $$X_t$$ of

$dX_t=X_t^2 dt + X_t dB_t$

where $$X_0=1$$ and $$B_t$$  is a standard Brownian motion has the representation

$X_t = \exp\Big( \int_0^t X_s ds -\frac12 t + B_t\Big)$

## Practice with Ito and Integration by parts

Define

$X_t =X_0 + \int_0^t B_s dB_s$

where $$B_t$$ is a standard Brownian Motion. Show that $$X_t$$ can also be written

$X_t=X_0 + \frac12 (B^2_t -t)$

## BDG Inequality

Consider $$I(t)$$ defined by $I(t)=\int_0^t \sigma(s,\omega)dB(s,\omega)$ where $$\sigma$$ is adapted and $$|\sigma(t,\omega)| \leq K$$ for all $$t$$ with probability one. Inspired by   problem “Homogeneous Martingales and Hermite Polynomials”  Let us set
\begin{align*}Y(t,\omega)=I(t)^4 – 6 I(t)^2\langle I \rangle(t) + 3 \langle I \rangle(t)^2 \ .\end{align*}

1. Quote  the problem “Ito Moments” to show that $$\mathbb{E}\{ |Y(t)|^2\} < \infty$$ for all $$t$$. Then  verify that $$Y_t$$ is  a martingale.
2. Show that $\mathbb{E}\{ I(t)^4 \} \leq 6 \mathbb{E} \big\{ \{I(t)^2\langle I \rangle(t) \big\}$
3. Recall the Cauchy-Schwartz inequality. In our language it states that
\begin{align*}
\mathbb{E} \{AB\} \leq (\mathbb{E}\{A^2\})^{1/2} (\mathbb{E}\{B^2\})^{1/2}
\end{align*}
Combine this with the previous inequality to show that\begin{align*}\mathbb{E}\{ I(t)^4 \} \leq 36 \mathbb{E} \big\{\langle I \rangle(t)^2 \big\} \end{align*}
4. We know that  $$I^4$$ is a submartingale (because $$x \mapsto x^4$$ is convex). Use the Kolmogorov-Doob inequality and all that we have just derived to show that
\begin{align*}
\mathbb{P}\left\{ \sup_{0\leq s \leq T}|I(s)|^4 \geq \lambda \right\} \leq ( \text{const}) \frac{ \mathbb{E}\left( \int_0^T \sigma(s,\omega)^2 ds\right)^2 }{\lambda}
\end{align*}

## Ito Variation of Constants

For  functions $$f(x)$$ and$$g(x)$$ and constant $$\beta>0$$,  define $$X_t$$ as the solution to the following SDE
$dX_t = – \beta X_t dt + h(X_t)dt + g(X_t) dW_t$

where $$W_t$$ is a standard Brownian Motion.

1.  Show that $$X_t$$ can be written as
$X_t = e^{-\beta t} X_0 + \int_0^{t} e^{-\beta (t-s)} h(X_s) ds + \int_0^{t} e^{-\beta (t-s)} g(X_s) dW_s$
See exercise:  Ornstein–Uhlenbeck process for guidance.
2. Assuming that $$|h(x)| < K$$  and $$|g(x)|<K$$, show that  there exists a constant $$C(X_0)$$ so that
$\mathbf E [|X_t|] < C(X_0)$
for all $$t >0$$. It might be convenient the remember the Cauchy–Schwarz inequality.
3. * Assuming that $$|h(x)| < K$$  and $$|g(x)|<K$$, show that  for any integer $$p >0$$ there exists a constant $$C(p,X_0)$$ so that
$\mathbf E [|X_t|^{2p}] < C(p,X_0)$
for all $$t >0$$. See exercise: Ito Moments for guidance.

## Ornstein–Uhlenbeck process

For $$\alpha \in \mathbf R$$ and $$\beta >0$$,  Define $$X_t$$ as the solution to the following SDE
$dX_t = – \beta X_t dt + \alpha dW_t$

where $$W_t$$ is a standard Brownian Motion.

1.  Find $$d(e^{\beta t} X_t)$$ using Ito’s Formula.
2. Use the calculation of   $$d(e^{\beta t} X_t)$$ to show that
\begin{align}  X_t = e^{-\beta t} X_0 + \alpha \int_0^t e^{-\beta(t-s)} dW_s\end{align}
3. Conclude that $$X_t$$ is Gaussian process (see exercise: Gaussian Ito Integrals ). Find its mean and variance at time $$t$$.
4. * Let $$h(t)$$ and $$g(t)$$ be  deterministic functions of time and let $$Y_t$$ solve
$dY_t = – \beta Y_t dt + h(t)dt+ \alpha g(t) dW_t$
show find a formula analogous to part 2 above for $$Y_t$$ and conclude that $$Y_t$$ is still Gaussian. Find it mean and Variance.

## Ito Integration by parts

Recall that if $$u(t)$$ and $$v(t)$$ are deterministic functions which are once differentiable then the classic integration by parts formula states that
$\int_0^t u(s) (\frac{dv}{ds})(s)\,ds = u(t)v(t) – u(0)v(0) – \int_0^t v(s) (\frac{du}{ds})(s)\,ds$

As is suggested by the formal relations

$(\frac{dv}{ds})(s)\,ds=dv(s) \qquad\text{and}\qquad (\frac{du}{ds})(s)\, ds=du(s)$

this can be rearranged  to state

$u(t)v(t)- u(0)v(0)= \int_0^t u(s) dv(s) + \int_0^t v(s) du(s)$

which holds for more general Riemann–Stieltjes integrals. Now consider two Ito processes $$X_t$$ and $$Y_t$$ given by

$dX_t=b_s ds + \sigma_s dW_t \qquad\text{and}\qquad dY_t=f_s ds + g_s dW_t$

where $$W_t$$ is a standard Brownian Motion. Derive the “Integration by Parts formula” for Ito calculus by applying Ito’s formula to $$X_tY_t$$. Compare this the the classical formula given above.