Stochastic integration

The theory of stochastic integration deals with integrals and differential equations in stochastics . It generalizes the integral terms of Henri Léon Lebesgue and Thomas Jean Stieltjes to a broader set of integrators . There are stochastic processes with infinite variation , especially the Wiener process , approved as integrators. The theory of stochastic integration represents the basis of stochastic analysis , the applications of which are mostly concerned with the investigation of stochastic differential equations .

Integral terms after Itō and Stratonowitsch

Let two (not necessarily independent ) real-valued stochastic processes on a common probability space . The Itō integral (after Itō Kiyoshi ) from to above the interval is the name given to the random variable ${\displaystyle (X_{t}),(Y_{t}),t\in [a,b]}$ ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle [a,b]}$

${\displaystyle I:=\int _{a}^{b}X_{t-}\,\mathrm {d} Y_{t}:=\lim _{n\to \infty }\sum _{i=1}^{n}X_{a+(i-1)h}(Y_{a+ih}-Y_{a+(i-1)h}),\quad h={\frac {b-a}{n}}.}$

The corresponding Stratonowitsch integral (after Ruslan Leontjewitsch Stratonowitsch ) is calculated for the same choice of as ${\displaystyle h}$

${\displaystyle S:=\int _{a}^{b}X_{t}\circ \mathrm {d} Y_{t}:=\lim _{n\to \infty }\sum _{i=1}^{n}{\frac {1}{2}}(X_{a+(i-1)h}+X_{a+ih})(Y_{a+ih}-Y_{a+(i-1)h}).}$

With the Itō integral, the integrand is always evaluated at the beginning of the interval, with Stratonowitsch the start and end values ​​are averaged. With ordinary ( Riemann or Lebesgue ) integrals of deterministic (not random) and sufficiently smooth (e.g. continuous ) functions, this has no influence on the result, but in the stochastic case the following applies: If and are not independent, this can actually lead to different values lead (see example below). ${\displaystyle X}$ ${\displaystyle h}$ ${\displaystyle X}$ ${\displaystyle Y}$

As a class of possible integrators , semimartingals are admitted in the most general formulation ; the integrands are predictable processes . ${\displaystyle Y}$ ${\displaystyle X}$

A Brownian motion and the integral of ${\displaystyle B_{s}}$ ${\displaystyle B_{s}\,\mathrm {d} B_{s}}$

example

Be a (standard) Wiener process . The Itō integral is to be calculated . For the sake of brevity, write and use the identity ${\displaystyle (W_{t}),t>0}$ ${\displaystyle \int _{0}^{T}W_{t}\,\mathrm {d} W_{t}}$ ${\displaystyle B_{i}:=W_{iT/n},\Delta B_{i}:=B_{i+1}-B_{i}}$

${\displaystyle B_{i+1}^{2}-B_{i}^{2}=(B_{i+1}-B_{i})^{2}+2B_{i}(B_{i+1}-B_{i}),}$

so one obtains from the above integration rule

${\displaystyle {\begin{aligned}I&=\lim _{n\to \infty }\sum _{i=0}^{n-1}B_{i}(B_{i+1}-B_{i})\\&=\lim _{n\to \infty }\left({\frac {1}{2}}\sum _{i=0}^{n-1}(B_{i+1}^{2}-B_{i}^{2})-{\frac {1}{2}}\sum _{i=0}^{n-1}(B_{i+1}-B_{i})^{2}\right)\\&={\frac {1}{2}}\lim _{n\to \infty }\sum _{i=0}^{n-1}(B_{i+1}^{2}-B_{i}^{2})-{\frac {1}{2}}\lim _{n\to \infty }\sum _{i=0}^{n-1}(\Delta B_{i})^{2}\\&={\frac {1}{2}}\lim _{n\to \infty }\left(B_{n}^{2}-B_{0}^{2}\right)-{\frac {T}{2}}\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n-1}\left({\sqrt {\frac {n}{T}}}\Delta B_{i}\right)^{2}.\end{aligned}}}$

If one uses on the one hand that it holds, and on the other hand the property that iid is distributed (because of the independent , normally distributed increases of the Brownian motion), then follows the law of large numbers for the lower limit ${\displaystyle B_{0}=W_{0}=0,B_{n}=W_{T}}$ ${\displaystyle \left({\sqrt {\frac {n}{T}}}\Delta B_{i}\right)^{2}}$ ${\displaystyle \chi ^{2}}$

${\displaystyle I={\frac {1}{2}}W_{T}^{2}-{\frac {T}{2}}.}$

In order to calculate the corresponding Stratonowitsch integral, one uses the continuity of the Brownian motion:

${\displaystyle {\begin{aligned}S&=\lim _{n\to \infty }\sum _{i=0}^{n-1}{\frac {1}{2}}(B_{i+1}+B_{i})(B_{i+1}-B_{i})\\&=\lim _{n\to \infty }\sum _{i=0}^{n-1}{\frac {1}{2}}(B_{i+1}^{2}-B_{i}^{2})\\&=\lim _{n\to \infty }{\frac {1}{2}}(B_{n}^{2}-B_{0}^{2})\\&={\frac {1}{2}}W_{T}^{2}\end{aligned}}}$

Itō and Stratonowitsch integral over the same process thus lead to different results, whereby the Stratonowitsch integral corresponds more to the intuitive idea from the usual (deterministic) integral calculus.

Martingale property

By far the most commonly used integrator is Brownian motion. The decisive advantage that the Stratonowitsch integral does not have and which ultimately led to the Itō integral becoming widely accepted as the standard is the following property: ${\displaystyle Y}$

Let be a constant expectation Lévy process , a non-anticipatory bounded function of and (i.e., for each is measurable in terms of the σ-algebra generated by the random variables ), then the process is ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle t}$ ${\displaystyle t>0}$ ${\displaystyle X_{t}}$ ${\displaystyle \sigma (Y_{s};s<t)}$ ${\displaystyle Y_{s},\,s<t}$

${\displaystyle t\mapsto \int _{0}^{t}X_{s}\,\mathrm {d} Y_{s}}$

a local martingale regarding the natural filtration of . Under additional constraints, the integral process is even a martingale . ${\displaystyle Y}$

Application: Itō process

Starting from Itōschen integral term, it is now possible to define a broad class of stochastic processes: Thus, a stochastic process with Ito process called when there is a Brownian motion with and stochastic processes , are with ${\displaystyle (X_{t})}$ ${\displaystyle t\geq 0}$ ${\displaystyle (W_{t})}$ ${\displaystyle t\geq 0}$ ${\displaystyle (a_{t}(X_{t},t))}$ ${\displaystyle (b_{t}(X_{t},t))}$

${\displaystyle X_{t}=X_{0}+\int _{0}^{t}a_{s}(X_{s},s)\,\mathrm {d} s+\int _{0}^{t}b_{s}(X_{s},s)\,\mathrm {d} W_{s}\,,}$

where it is assumed that the two integrals exist. In differential notation , this equation is called

${\displaystyle \mathrm {d} X_{t}=a_{t}(X_{t},t)\,\mathrm {d} t+b_{t}(X_{t},t)\,\mathrm {d} W_{t}}$

written down. An Itō process can therefore be viewed as a generalized Wiener process with random drift and volatility.

The predicate “ is an Itō process” thus becomes a stochastic counterpart to the concept of differentiability . Based on this, Itō himself defined the first stochastic differential equations . ${\displaystyle X}$

If the drift coefficient and the diffusion coefficient do not depend on time, one speaks of Itō diffusion - if they also depend on time, then there is a more general Itō process. ${\displaystyle a_{t}}$ ${\displaystyle b_{t}}$

Through numerous applications in mathematical modeling, especially in statistical physics and financial mathematics , the Itō calculus has meanwhile developed into an indispensable mathematical tool.

See also

• Discrete stochastic integral

literature

• J. Jacod, A. Shiryaev: Limit theorems for stochastic processes . Springer, Berlin.
• P. Protter: Stochastic integrals and differential equations . Springer, Berlin.

Individual evidence

1. ^ Hui-Hsiung Kuo: Introduction to Stochastic Integration. Springer, 2006, ISBN 978-0387-28720-1 , p. 102 ( limited preview in Google book search).