Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion (Wiener process). It has important applications in mathematical finance and stochastic differential equations.The central concept is the Itō stochastic integral:where "X" is a Brownian motion or, more generally, a semimartingale.The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. In particular, it is not differentiable at any point and has infinite variation over every time interval. As a result, the integral cannot be defined in the usual way (see Riemann-Stieltjes integral).The main insight is that the integral can be defined as long as the integrand "H" is adapted, which means that its value at time "t" can only depend on information available up until this time.
The prices of stocks and other traded financial assets can be modeled by stochastic processes such as Brownian motion or, more often, geometric Brownian motion (see Black-Scholes). Then, the Itō stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount "Ht" of the stock at time "t". In this situation, the condition that "H" is adapted corresponds to the necessary restriction that the trading strategy can only make use of the available information at any time. This prevents the possibility of unlimited gains through high frequency trading: buying the stock just before each uptick in the market and selling before each downtick. Similarly, the condition that "H" is adapted implies that the stochastic integral will not diverge when calculated as a limit of Riemann sums.
Important results of Itō calculus include the integration by parts formula and Itō's lemma, which is a change of variables formula. These differ from the formulas of standard calculus, due to quadratic variation terms.
Notation
The integral of a process "H" with respect to another process "X" up until a time "t" is written as:This is itself a stochastic process with time parameter "t", which is also written as "H" · "X". Alternatively, the integral is often written in differential form "dY = H dX", which is equivalent to "Y - Y0 = H · X".As Itō calculus is concerned with continuous-time stochastic processes, it is assumed that there is an underlying filtered probability space.:The sigma algebra "Ft" represents the information available up until time "t", and a process "X" is adapted if "Xt" is "Ft"-measurable. A Brownian motion "B" is understood to be an "Ft"-Brownian motion, which is just a standard Brownian motion with the property that "Bt+s - Bt" is independent of "Ft" for all "s,t ≥ 0".
Integration with respect to Brownian motion
The Itō integral can be defined in a manner similar to the Riemann-Stieltjes integral, that is as a limit in probability of Riemann sums; such a limit does not necessarily exist pathwise. Suppose that "B" is a Wiener process (Brownian motion) and that "H" is a left-continuous, adapted and locally bounded process.If π"n" is a sequence of partitions of [0,"t"] with mesh going to zero, then the Itō integral of "H" with respect to "B" up to time "t" is a random variable
:
It can be shown that this limit converges in probability.
For some applications, such as martingale representation theorems and local times, the integral is needed for processes that are not continuous. The predictable processes form the smallest class which is closed under taking limits of sequences and contains all adapted left continuous processes.If "H" is any predictable process such that ∫0t "H2 ds" < ∞ for every "t" ≥ "0" then the integral of "H" with respect to "B" can be defined, and "H" is said to be "B"-integrable.Any such process can be approximated by a sequence "Hn" of left-continuous, adapted and locally bounded processes, in the sense that
:
in probability. Then, the Itō integral is
:
where, again, the limit can be shown to converge in probability.The stochastic integral satisfies the Itō isometry:which holds when "H" is bounded or, more generally, when the integral on the right hand side is finite.
Itō processes
An Itō process is defined to be an adapted stochastic process which can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time,:Here, "B" is a Brownian motion and it is required that σ is a predictable "B"-integrable process, and μ is predictable and (Lebesgue) integrable. That is,: