Ito process

An Ito Process is a type of stochastic process described by Japanese mathematician Kiyoshi Ito, which can be written as the sum of the integral of a process over time and of another process over a Brownian Motion.
Those processes are the base of Stochastic Integration, and are therefore widely used in Financial Mathematics and Stochastic Calculus.

Description of the Ito Processes

Let $(\Omega ,{\mathcal {F}},\mathbb {F} ,\mathbb {P} )$ be a probability space with a filtration $\mathbb {F} =({\mathcal {F}}_{t})_{t\geq 0}$ that we consider as complete (that is to say, all sets which measure is equal to zero are contained in ${\mathcal {F}}_{0}$ ). Let also be $B=(B_{t}^{1},\dots ,B_{t}^{d})_{t\geq 0}$ a d-dimensional $\mathbb {F}$ - Standard Brownian Motion.
Then we call Ito Process all process $(X_{t})_{t\geq 0}$ that can be written like :

$X_{t}=X_{0}+\int _{0}^{t}K_{s}\mathrm {ds} +\sum _{j=1}^{d}\int _{0}^{t}H_{s}^{j}\mathrm {dB} _{s}^{j}$

Where :

$X_{0}$ is ${\mathcal {F}}_{0}$ measurable
$(K_{t})_{t\geq 0}$ is a progressively measurable process such as $\forall t\geq 0,\;\int _{0}^{t}|K_{s}|{\textrm {ds}}<+\infty$ almost surely.
$(H_{t}^{i})_{t\geq 0,\ i\in [1\dots d]}$ is progressively measurable and such as $\forall i\in [1\dots d],\;\forall t\geq 0,\;\int _{0}^{t}(H_{s}^{i})^{2}\mathrm {ds} <+\infty$ almost surely.

We note then ${\mathcal {I}}$ the set of Ito Processes. We can also note that all Ito Processes are continuous and adapted to the filtration $\mathbb {F}$ . We can also write the Ito Process under a 'differential form' :

$dX_{t}=K_{t}dt+\sum _{j=1}^{d}H_{t}^{j}dB_{t}^{j}$

Using the fact that the brownian part is a local martingal, and that all continuous local martingal with finite variations equal to zero in zero is indistinguishible of the null process, we can show that this decomposition is unique (except for indistinguishibility) for each Ito Process.

Stochastic Integral with respect to an Ito Process

Let $X$ be an Ito Process. We can define the set of processes ${\mathcal {L}}(X)$ that we can integer with respect to $X$ :

${\mathcal {L}}(X)=\{(Y_{t})_{t\geq 0}\mathrm {progressively} \;\mathrm {measurable} \;|\;\forall t\geq 0,\;\int _{0}^{t}|Y_{s}||K_{s}|{\textrm {ds}}+\sum _{j=1}^{d}\int _{0}^{t}(Y_{s}H_{s}^{j})^{2}{\textrm {dB}}_{s}^{j}<+\infty \;{\textrm {almost}}\;\mathrm {surely} \}$

We can then write :

$\int _{0}^{t}Y_{s}{\textrm {dX}}_{s}=Y_{0}+\int _{0}^{t}Y_{s}K_{s}\mathrm {ds} +\sum _{j=1}^{d}\int _{0}^{t}Y_{s}H_{s}^{j}\mathrm {dB} _{s}^{j}$

Stability of the Ito Processes Set

It is important to note that, $\forall X\in {\mathcal {I}},\ {\mathcal {I}}\in {\mathcal {L}}(X)$ . Which means that any Ito Process can be integrated with respect to any other Ito Process. Moreover, the Stochastic Integral with respect to an Ito Process is still an Ito Process. This exceptional stability is one of the reasons of the wide use of Ito Processes. The other reason is the Ito Formula.

Quadratic Variation and Ito's Formula

Quadratic Variation of an Ito Process

Let $X\in {\mathcal {I}}$ . The construction of the Stochastic Integral makes the usual formula for deterministic functions $f(t)^{2}=f(0)^{2}+2\int _{0}^{t}f(s)\mathrm {df} (s)$ wrong for the Ito Processes. We then define the quadratic variation as the process $<X,X>_{t}$ :

$<\!X,X\!>_{t}=X_{t}^{2}-X_{0}^{2}-2\int _{0}^{t}X_{s}\mathrm {dX} _{s}$

This process is adapted, continuous, equal to zero in zero, and its trajectories are almost surely growing.
We can define the same way the covariation of two Ito Processes :

$<\!X,Y\!>_{t}=X_{t}Y_{t}-X_{0}Y_{0}-\int _{0}^{t}X_{s}\mathrm {dY} _{s}-\int _{0}^{t}Y_{s}\mathrm {dX} _{s}$

Which is also adapted, continuous, equal to zero in zero, and has finite variations.
We can then note the following properties of the quadratic variation :

If $X$ or $Y$ has finite variations : $<\!X,Y\!>_{t}=0$
If $M,N\in {\mathcal {I}}$ are local martingals
$<\!M,N\!>_{t}$

is the only adapted, continuous, equal to zero in zero process with finite variations such as #; $MN-<\!M,N\!>$ : is a local martingal.
If we have
$X\in {\mathcal {I}}$

$X_{t}=X_{0}+\int _{0}^{t}H_{s}^{0}\mathrm {ds} +\sum _{j=1}^{d}\int _{0}^{t}H_{s}^{j}\mathrm {dB} _{s}^{j}$

$Y\in {\mathcal {I}}$

$Y_{t}=Y_{0}+\int _{0}^{t}K_{s}^{0}\mathrm {ds} +\sum _{j=1}^{d}\int _{0}^{t}K_{s}^{j}\mathrm {dB} _{s}^{j}$

Then

$<\!X,Y\!>_{t}=\sum _{j=1}^{d}\int _{0}^{t}H_{s}^{j}K_{s}^{j}\mathrm {dB} _{s}^{j}$