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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})} be a probability space with a filtration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}_0} ). Let also be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B=(B^1_t,\dots,B^d_t)_{t\geq 0}} a d-dimensional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}} - Standard Brownian Motion.
Then we call Ito Process all process Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_t)_{t\geq 0}} that can be written like :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_t = X_0 + \int_0^t K_s\mathrm{ds} + \sum_{j=1}^d\int_0^t H^j_s\mathrm{dB}_s^j}

Where :

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_0} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F}_0} measurable
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (K_t)_{t\geq 0}} is a progressively measurable process such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall t\geq 0,\; \int_0^t|K_s|\textrm{ds}<+\infty} almost surely.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H^i_t)_{t\geq 0,\ i\in[1\dots d]}} is progressively measurable and such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{I}} the set of Ito Processes. We can also note that all Ito Processes are continuous and adapted to the filtration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}} . We can also write the Ito Process under a 'differential form' :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dX_t = K_tdt + \sum_{j=1}^dH_t^jdB_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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} be an Ito Process. We can define the set of processes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}(X)} that we can integer with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X}  :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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_sH_s^j)^2\textrm{dB}_s^j < +\infty\; \textrm{almost}\; \mathrm{surely}\}}

We can then write :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^tY_s\textrm{dX}_s = Y_0 + \int_0^t Y_sK_s\mathrm{ds} + \sum_{j=1}^d\int_0^t Y_sH^j_s\mathrm{dB}_s^j}

Stability of the Ito Processes Set

Stability over addition

The sum of two Ito Processes is obviously another Ito Process

Stability over Integration

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <X,X>_t} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\!X,X\!>_t = X_t^2 - X_0^2 - 2\int_0^tX_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 :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\!X,Y\!>_t = X_tY_t - X_0Y_0 - \int_0^tX_s\mathrm{dY}_s - \int_0^tY_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 :

1. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} has finite variations : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\!X,Y\!>_t = 0}
2. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M,N\in\mathcal{I}} are local martingals

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\!M,N\!>_t}

   is the only adapted, continuous, equal to zero in zero process with finite variations such as

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle MN-<\!M,N\!>}

   is a local martingal.

3. If we have

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\in\mathcal{I}}

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_t = X_0 + \int_0^t H^0_s\mathrm{ds} + \sum_{j=1}^d\int_0^t H^j_s\mathrm{dB}_s^j}

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y\in\mathcal{I}}

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_t = Y_0 + \int_0^t K^0_s\mathrm{ds} + \sum_{j=1}^d\int_0^t K^j_s\mathrm{dB}_s^j}

   Then

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <\!X,Y\!>_t = \sum_{j=1}^d\int_0^t H^j_sK^j_s\mathrm{dB}_s^j}