Itô calculus

Overview

Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance, in stochastic differential equations, and more recently even in machine learning. The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes:

Y

t

=

∫

0

t

H

s

d

X

s

,

{\displaystyle Y_{t}=\int _{0}^{t}H_{s}\,dX_{s},}

where H is a locally square-integrable process adapted to the filtration generated by X, which is a Brownian motion or, more generally, a semimartingale. The result of the integration is then another stochastic process. Concretely, the integral from 0 to any particular t is a random variable, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô stochastic integral amounts to an integral with respect to a function which is not differentiable at any point and has infinite variation over every time interval. The main insight is that the integral can be defined as long as the integrand H is adapted, which loosely speaking means that its value at time t can only depend on information available up until this time. Roughly speaking, one chooses a sequence of partitions of the interval from 0 to t and constructs Riemann sums. Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. It is crucial which point in each of the small intervals is used to compute the value of the function. The limit then is taken in probability as the mesh of the partition is going to zero. Numerous technical details have to be taken care of to show that this limit exists and is independent of the particular sequence of partitions. Typically, the left end of the interval is used. Important results of Itô calculus include the integration by parts formula and Itô's lemma, which is a change of variables formula. These differ from the formulas of standard calculus, due to quadratic variation terms. This can be contrasted to the Stratonovich integral as an alternative formulation; it does follow the chain rule, and does not require Itô's lemma. The two integral forms can be converted to one-another. The Stratonovich integral is obtained as the limiting form of a Riemann sum that employs the average of stochastic variable over each small timestep, whereas the Itô integral considers it only at the beginning. In mathematical finance, the described evaluation strategy of the integral is conceptualized as that we are first deciding what to do, then observing the change in the prices. The integrand is how much stock we hold, the integrator represents the movement of the prices, and the integral is how much money we have in total including what our stock is worth, at any given moment. 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 Itô 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 clairvoyance: 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.

Notation

The process Y defined before as

Y

t

=

∫

0

t

H

d X ≡

∫

0

t

H

s

d

X

s

,

{\displaystyle Y_{t}=\int _{0}^{t}H\,dX\equiv \int _{0}^{t}H_{s}\,dX_{s},}

is itself a stochastic process with time parameter t, which is also sometimes written as Y = H · X. Alternatively, the integral is often written in differential form dY = H dX, which is equivalent to Y − Y0 = H · X. As Itô calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying filtered probability space is given

Integration with respect to Brownian motion

The Itô 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 right-continuous (càdlàg), adapted and locally bounded process. If

{

π

n

}

{\displaystyle \{\pi _{n}\}}

is a sequence of partitions of [0, t] with mesh width going to zero, then the Itô integral of H with respect to B up to time t is a random variable

∫

0

t

H

d B =

lim

n → ∞

∑

[

t

i − 1

,

t

Itô processes

An Itô process is defined to be an adapted stochastic process that can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time,

X

t

=

X

0

+

∫

0

t

σ

s

d

B

s

+

∫

0

t

μ

Semimartingales as integrators

The Itô integral is defined with respect to a semimartingale X. These are processes which can be decomposed as X = M + A for a local martingale M and finite variation process A. Important examples of such processes include Brownian motion, which is a martingale, and Lévy processes. For a left continuous, locally bounded and adapted process H the integral H · X exists, and can be calculated as a limit of Riemann sums. Let πn be a sequence of partitions of [0, t] with mesh going to zero,

∫

0

t

H

d X =

lim

n → ∞

∑

t

i − 1

,

t

i

∈

π

n

H

t

i − 1

Properties

The following properties can be found in works such as Revuz & Yor (1999) and Rogers & Williams (2000):

The stochastic integral is a càdlàg process. Furthermore, it is a semimartingale. The discontinuities of the stochastic integral are given by the jumps of the integrator multiplied by the integrand. The jump of a càdlàg process at a time t is Xt − Xt−, and is often denoted by ΔXt. With this notation, Δ(H · X) = H ΔX. A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous. Associativity. Let J, K be predictable processes, and K be X-integrable. Then, J is K · X integrable if and only if JK is X-integrable, in which case

J ⋅ ( K ⋅ X ) = ( J K ) ⋅ X

{\displaystyle J\cdot (K\cdot X)=(JK)\cdot X}

Dominated convergence. Suppose that Hn → H and |Hn| ≤ J, where J is an X-integrable process. then Hn · X → H · X. Convergence is in probability at each time t. In fact, it converges uniformly on compact sets in probability. The stochastic integral commutes with the operation of taking quadratic covariations. If X and Y are semimartingales then any X-integrable process will also be [X, Y]-integrable, and [H · X, Y] = H · [X, Y]. A consequence of this is that the quadratic variation process of a stochastic integral is equal to an integral of a quadratic variation process,

[ H ⋅ X ] =

H

2

⋅ [ X ]

{\displaystyle [H\cdot X]=H^{2}\cdot [X]}

Integration by parts

As with ordinary calculus, integration by parts is an important result in stochastic calculus. The integration by parts formula for the Itô integral differs from the standard result due to the inclusion of a quadratic covariation term. This term comes from the fact that Itô calculus deals with processes with non-zero quadratic variation, which only occurs for infinite variation processes (such as Brownian motion). If X and Y are semimartingales then

X

t

Y

t

=

X

0

Y

0

+

∫

0

t

X

s −

d

Y

s

+

Itô's lemma

Itô's lemma is the version of the chain rule or change of variables formula which applies to the Itô integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous n-dimensional semimartingale X = (X1,...,Xn) and twice continuously differentiable function f from Rn to R, it states that f(X) is a semimartingale and,

d f (

X

t

) =

∑

i = 1

n

f

i

(

X

t

)

d

X

t

i

+

1 2

Local martingales

An important property of the Itô integral is that it preserves the local martingale property. If M is a local martingale and H is a locally bounded predictable process then H · M is also a local martingale. For integrands which are not locally bounded, there are examples where H · M is not a local martingale. However, this can only occur when M is not continuous. If M is a continuous local martingale then a predictable process H is M-integrable if and only if

∫

0

t

H

2

d [ M ] < ∞ ,

{\displaystyle \int _{0}^{t}H^{2}\,d[M]<\infty ,}

for each t, and H · M is always a local martingale. The most general statement for a discontinuous local martingale M is that if (H2 · [M])1/2 is locally integrable then H · M exists and is a local martingale.

Square integrable martingales

For bounded integrands, the Itô stochastic integral preserves the space of square integrable martingales, which is the set of càdlàg martingales M such that E[Mt2] is finite for all t. For any such square integrable martingale M, the quadratic variation process [M] is integrable, and the Itô isometry states that

E

[

( H ⋅

M

t

)

2

]

=

E

[

∫

0

t

H

2

d [ M ]

]

.

p-Integrable martingales

For any p > 1, and bounded predictable integrand, the stochastic integral preserves the space of p-integrable martingales. These are càdlàg martingales such that E(|Mt|p) is finite for all t. However, this is not always true in the case where p = 1. There are examples of integrals of bounded predictable processes with respect to martingales which are not themselves martingales. The maximum process of a càdlàg process M is written as M*t = sups ≤t |Ms|. For any p ≥ 1 and bounded predictable integrand, the stochastic integral preserves the space of càdlàg martingales M such that E[(M*t)p] is finite for all t. If p > 1 then this is the same as the space of p-integrable martingales, by Doob's inequalities. The Burkholder–Davis–Gundy inequalities state that, for any given p ≥ 1, there exist positive constants c, C that depend on p, but not M or on t such that

c

E

[

[ M

]

t

p 2

]

≤

E

[

(

M

t

∗

)

p

]

≤ C

Existence of the integral

Proofs that the Itô integral is well defined typically proceed by first looking at very simple integrands, such as piecewise constant, left continuous and adapted processes where the integral can be written explicitly. Such simple predictable processes are linear combinations of terms of the form Ht = A1{t > T} for stopping times T and FT-measurable random variables A, for which the integral is

H ⋅

X

t

≡

1

{ t > T }

A (

X

t

−

X

T

) .

{\displaystyle H\cdot X_{t}\equiv \mathbf {1} _{\{t>T\}}A(X_{t}-X_{T}).}

This is extended to all simple predictable processes by the linearity of H · X in H. For a Brownian motion B, the property that it has independent increments with zero mean and variance Var(Bt) = t can be used to prove the Itô isometry for simple predictable integrands,

E

[

( H ⋅

B