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 of Riemann sums. 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 (the expected value is always zero).

Both summation and multiplication of random variables are defined in probability theory. The summation involves a convolution of the probability density function (PDF) and multiplication is repeated summation.

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).

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 average to the values of the process $X$ at the left- and right-hand endpoints of each subinterval: i.e.

${\frac {X_{t_{i+1}}+X_{t_{i}}}{2}}$ 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 Itô Claculus: Stochastic Derivative

Itô Calculus as ground breaking and remarkable as it is, it is only an integral calculus. The way it is defined there was no real explicit pathwise differentiation theory behind it. In 2006, a paper was published in the Stochastic Analysis and Applications journal written by Kent State University professor, Hassan Allouba, entitled "A Differentiation Theory for Itô's Calculus". To define this integral he used covariation to produce the derivative of a given semimartingale with respect to Brownian Motion, ie

For $S_{t}=S_{0}+V_{t}+M_{t}$, where $V$ is a process of bounded variation and $M$ is a local martingale

$\mathbb {D} _{B_{t}}S_{t}={\frac {d<S,B>_{t}}{d<B,B>_{t}}}={\frac {d<S,B>_{t}}{dt}},$ where the covariation of Brownian Motion is just the Quadratic Variance.

This derivative turns out that it would behave just as one would expect. It is very similar to the calculus we are used to seeing in high schools and beginner calculus, although there are a few twists. With our standard calculus when we integrated we ended up with a constant of integration. With Stochastic Calculus, when we integrate we end up with processes of bounded variation (which can be functions of deterministic variables). These processes are the "constants" in Stochastic Calculus and Differentiation. When we take the covariation of something deterministic and something that is random, the covariation vanishes, ie:

Let $f(x)$ be a continuous, deterministic function with n derivatives.

$\mathbb {D} _{B_{t}}f(t)={\frac {d<f,B>_{t}}{dt}}=0$

The Fundamental Theorem of Stochastic Calculus states that:

$(i)\mathbb {D} _{B_{t}}\int _{0}^{t}X_{s}dB_{s}=X_{t}$

$(ii)\int _{0}^{t}\mathbb {D} _{B_{s}}S_{s}dB_{s}=S_{t}-S_{0}-V_{t}$

A few other suprises come along the way of Stochastic Differentiation Theory. All of our familiar friends from calculus show up as well.

Stochastic Chain Rule: $\mathbb {D} _{B_{t}}f(S_{t})=f'(S_{t})\mathbb {D} _{B_{t}}S_{t}$

Summation Rule: $\mathbb {D} _{B_{t}}(aS_{t}^{(1)}$ ± $bS_{t}^{(2)})=a\mathbb {D} _{B_{t}}S_{t}^{(1)}$ ± $b\mathbb {D} _{B_{t}}S_{t}^{(2)}$

Product Rule: $\mathbb {D} _{B_{t}}(S_{t}^{(1)}S_{t}^{(2)})=S_{t}^{(2)}\mathbb {D} _{B_{t}}S_{t}^{(1)}+S_{t}^{(1)}\mathbb {D} _{B_{t}}S_{t}^{(2)}$

Quotient Rule: $\mathbb {D} _{B_{t}}{\frac {S_{t}^{(1)}}{S_{t}^{(2)}}}={\frac {S_{t}^{(2)}\mathbb {D} _{B_{t}}S_{t}^{(1)}-S_{t}^{(1)}\mathbb {D} _{B_{t}}S_{t}^{(2)}}{[S_{t}^{(2)}]^{2}}}$

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.