Itô calculus

Itō calculus, named after Kiyoshi Itō, treats mathematical operations on stochastic processes. Its most important concept is the Itō stochastic integral.

Definition

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 W : [0, T] × Ω → R is a Wiener process and that X : [0, T] × Ω → R is a stochastic process adapted to the natural filtration of the Wiener process. Then the Itō integral of X with respect to W is a random variable

$\int _{a}^{b}X_{t}\,\mathrm {d} W_{t}:\Omega \to \mathbb {R} ,$

defined to be the L² limit of

$\sum _{i=0}^{k-1}X_{t_{i}}\left(W_{t_{i+1}}-W_{t_{i}}\right)$

as the mesh of the partition 0 = t₀ < t₁ < ... < t_k = T of [0, T] tends to 0 (in the style of a Riemann-Stieltjes integral).

Technically speaking, the construction is first performed on a class of "elementary processes" and then extended to the closure of this class in the L² norm. The collection of all Itō integrable processes is sometimes denoted L²(W).

A crucial fact about this integral is Itō's lemma, which allows one to compare classical and stochastic integrals and compute the variance of an Itō integral.

Generalization: integration with respect to a martingale

The procedure used to define the Itō integral works for more general stochastic processes than the Wiener process W, and can be used to define the stochastic integral of any adapted process with respect to any martingale.

Let M : [0, T] × Ω → R be a real-valued martingale with respect to its natural filtration

${\mathcal {F}}_{t}^{M}:=\sigma \left\{M_{s}^{-1}(A)\left|A\in \mathrm {Borel} (\mathbb {R} ),0\leq s\leq t\right.\right\},$

i.e.

$\mathbb {E} (M_{t}|{\mathcal {F}}_{s}^{M})=M_{s}.$

Now let X : [0, T] × Ω → R be a stochastic process adapted to the filtration ${\mathcal {F}}_{t}^{M}$. Then the Itō integral of X with respect to M, denoted

$\int _{a}^{b}X_{t}\,\mathrm {d} M_{t},$

is defined to be the L² limit of

$\sum _{i=0}^{k-1}X_{t_{i}}\left(M_{t_{i+1}}-M_{t_{i}}\right)$

as the mesh of the partition 0 = t₀ < t₁ < ... < t_k = T of [0, T] tends to 0. The collection of all processes X for which the Itō integral with respect to M is defined is sometimes denoted L²(M).

The definition can be further extended to all processes such that

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by a localisation argument.

Other approaches

The Stratonovich integral is another way to define stochastic integrals. Its derivation rule is simpler than Ito's lemma.

In the definition of the Stratonovich integral, the same limiting procedure is used except for choosing the value of the process X at the midpoint of each subinterval instead of the left-hand end-point: i.e.

$X_{(t_{i+1}+t_{i})/2}$ in place of $X_{t_{i}}.$

Conversion between Itō and Stratonovich integrals may be performed using the formula

$\int _{0}^{T}\sigma (t,X_{t})\circ \mathrm {d} W_{t}={\frac {1}{2}}\int _{0}^{T}\sigma '(t,X_{t})\sigma (t,X_{t})\,\mathrm {d} t+\int _{0}^{T}\sigma (t,X_{t})\,\mathrm {d} W_{t},$

where $X$ is some process, $\sigma '(t,x):={\frac {\partial \sigma }{\partial x}}(t,x)$, and $\int _{0}^{T}\sigma (t,X_{t})\circ \mathrm {d} W_{t}$ denotes the Stratonovich integral.

Further extensions of Itō calculus: stochastic derivative

Itō calculus, as ground-breaking and remarkable as it is, for over 60 years has only been an integral calculus: there was no explicit pathwise differentiation theory behind it. However, in 2004 (published in 2006) Hassan Allouba defined the derivative of a given semimartingale S with respect to Brownian motion using covariation: for

$S_{t}=S_{0}+V_{t}+M_{t},$

where V is a process of bounded variation and M is a local martingale, the derivative of S with respect to B is defined to be

$\mathbb {D} _{B_{t}}S_{t}={\frac {\mathrm {d} \langle S,B\rangle _{t}}{\mathrm {d} \langle B,B\rangle _{t}}}={\frac {\mathrm {d} \langle S,B\rangle _{t}}{\mathrm {d} t}},$

where the covariation of Brownian motion is just the quadratic variation.

This stochastic derivative turns out to have many of the properties of the usual derivative of elementary calculus. The main difference is that where an indefinite integral (anti-derivative) in the usual sense is determined only up to an additive constant of integration, an indefinite integral in this stochastic calculus is determined only up to a process bounded variation (which could be a function of deterministic variables). These processes are the "constants" in Stochastic Calculus and Differentiation. To see why this is the case, observe that when we take the covariation of something deterministic and something that is random, the covariation vanishes: i.e., for any continuous, deterministic function f with n derivatives,

$\mathbb {D} _{B_{t}}f(t)={\frac {\mathrm {d} \langle f,B\rangle _{t}}{\mathrm {d} t}}=0.$

There is a version of the fundamental theorem of calculus for this derivative/integral pair:

$\mathbb {D} _{B_{t}}\int _{0}^{t}X_{s}\mathrm {d} B_{s}=X_{t}$

and

$\int _{0}^{t}\mathbb {D} _{B_{s}}S_{s}\mathrm {d} B_{s}=S_{t}-S_{0}-V_{t}.$

Furthermore, this stochastic calculus has stochastic versions of many other theorems of elementary calculus.

• the chain rule:

$\mathbb {D} _{B_{t}}f(S_{t})=f'(S_{t})\mathbb {D} _{B_{t}}S_{t}.$

• the summation rule:

$\mathbb {D} _{B_{t}}\left(aS_{t}^{(1)}\pm bS_{t}^{(2)}\right)=a\mathbb {D} _{B_{t}}S_{t}^{(1)}\pm b\mathbb {D} _{B_{t}}S_{t}^{(2)}.$

• the product rule:

$\mathbb {D} _{B_{t}}\left(S_{t}^{(1)}S_{t}^{(2)}\right)=S_{t}^{(2)}\mathbb {D} _{B_{t}}S_{t}^{(1)}+S_{t}^{(1)}\mathbb {D} _{B_{t}}S_{t}^{(2)}.$

• the quotient rule:

$\mathbb {D} _{B_{t}}\left({\frac {S_{t}^{(1)}}{S_{t}^{(2)}}}\right)={\frac {S_{t}^{(2)}\mathbb {D} _{B_{t}}S_{t}^{(1)}-S_{t}^{(1)}\mathbb {D} _{B_{t}}S_{t}^{(2)}}{[S_{t}^{(2)}]^{2}}}.$

There are also corresponding stochastic versions of Rolle's theorem and the mean value theorem.

See also

• Mathematical Finance Programming in TI-Basic, which implements Ito calculus for TI-calculators.

Reference

• Allouba, Hassan (2006). "A Differentiation Theory for Itô's Calculus". Stochastic Analysis and Applications. 24: 367–380. DOI 10.1080/07362990500522411.

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer. ISBN 3-540-04758-1.