Chapter 6Itos Stochastic Calculus

6.1 Introduction

When Bachelier first applied Wiener process on modeling the fluctuation of assetprices, the price of an asset at time t, Xt , has an infinitesimal increment dXt propor-tional to the increment dWt of the Wiener process, i.e.,

dXt = dWt ,

where is a positive constant. As a result, an asset with initial price X(0) = x worths

Xt = x+Wt

at time t. This model suffers from one serious flaw: for any t > 0 the price Xt can benegative with non-zero probability (but actual stock prices are never negative). Totackle this problem, successors assumed that the relative price dXt/Xt of an asset isproportional to dWt , i.e.,

dXt = Xt dWt . (6.1)

Although this equation looks like a differential equation, traditional methods areno longer applicable because the paths of Wt are not differentiable (Theorem 5.2).A way around the obstacle was found in the 1940s by Ito, who gave a rigorousmeaning to (6.1) by writing it as

Xt = x+ t

0Xs dWs, (6.2)

where the integral with respect to Wt on the right-hand side is called the Ito stochas-tic integral. In this section we discuss the definition and properties of stochasticintegral, and explore the applications to pricing financial products.

107

108 6 Itos Stochastic Calculus

6.2 Ito Stochastic Integral

6.2.1 Motivation

Motivated by (6.2), we construct the Ito stochastic integral in the form t

0 f (s)dWsfor some stochastic process/random function f (s) ( f (s,), to be precise). We followan approach similar to constructing Riemann integral, i.e., define the integral by thelimit of the discretized version

n1

i=0

f (si)(Wti+1 Wti) , (6.3)

where si [ti, ti+1]. The major differences between Riemann and Ito integrals are1. Riemann integration results in a real number, but Ito integration results in a

random variable (since Wt is random). Thus, while defining Riemann integralinvolves convergence of real numbers, defining Ito integral in (6.3) requiresconvergence of random variables, which is considerably more difficult.

2. If a Riemann integral exists, then si can be an arbitrary point in [ti, ti+1] since theupper and lower Riemann sum converge. However, in Ito integral, the limit willbe different depending on the choice of si. This is due to the non-zero quadraticvariation of the Brownian motion Wt . See Exercise 6.1.

To avoid the ambiguity in 2), the definition of stochastic integrals will fix thechoice si = ti for each i in the approximating sum (6.3). The choice si = ti is naturalif we regard f (t) as the trading strategy and Wt as the stock price: For the i+ 1-thperiod [ti, ti+1], the trading strategy should only depend on the information up totime ti. Hence, f (ti) units are invested and a profit of f (ti)(Wti+1 Wti) is made.Therefore, n1i=0 f (ti)(Wti+1 Wti) represents the total profit.

Remark 6.1. (Previsible) To be precise, in defining stochastic integral we requirethe integrand f (t) to be previsible or predictable, i.e., f (t) is Ft for all t whereFt s

6.2 Ito Stochastic Integral 109

E( T

0| f (t)|2 dt

)< .

Let M 2 be the class of stochastic processes f (t) such that f (t)M 2T for any T > 0.Recall that a random variable X is in L 2, or X L 2, if E|X |2 < . Both M 2and L 2 are related to the existence of second moment, but M 2 is for a stochasticprocess and L 2 is for a random variable.

Since the Ito integral is a random variable and the integrand is a random function(stochastic process), we need to define a measure of length and a mode of conver-gence in defining the integral by a limit of a discretization in (6.3).

Definition 6.2. (L 2 and M 2T Norms) For a random variable X and a stochasticprocess f f (t), the L 2 and M 2T norm are given respectively by

XL 2 =E(X2) and fM 2T =

E( T

0| f (t)|2 dt

). (6.4)

Definition 6.3. (L 2, M 2T and M 2 Convergences) A sequence of random vari-ables {Xn} converges in L 2 to X if

XnXL 2 =

E(|XnX |2) 0 .

as n . A sequence of random functions / stochastic processes { fn(t)} con-verges in M 2T to f if for any T ,

fn fM 2T =

E( T

0| fn(t) f (t)|2 dt

) 0 .

Similarly, { fn(t)} converges in M 2 to f if { fn(t)} converges to f in M 2T for all T .

Using Definitions 6.1 and 6.3, we define the Ito Integral on the class of stochasticprocess M 2 as follows:

Definition 6.4. (Ito Integral) For any T > 0 and any stochastic process f M 2,the stochastic integral of f on [0,T ] is defined by

IT ( f ) = limn

n1

j=0

f (t j)(W (t j+1)W (t j)

),

where (0 = t0 < t1 < < tn1 < tn = T ) is any partition of [0,T ] with max j |t jt j1| 0 as n . We write IT ( f ) as

IT ( f ) = T

0f (t)dWt ,

(or

T0

f dW).

110 6 Itos Stochastic Calculus

The following theorem justifies the above definition of Ito integral.

Theorem 6.1. (Existence and Uniqueness of Ito Integral) Suppose that a functionf M 2 satisfies the following assumptions: For all t 0,

A1)f (t) is almost surely continuous, i.e., P(lim0 | f (t + ) f (t)|= 0) = 1,A2)f (t) is adapted to the filtration {Ft}, where Ft = ({Ws,s t}).

Then, for any T > 0, the Ito integral

IT ( f ) = T

0f (t)dWt

exists and is unique almost everywhere.

Proof. The proof proceeds in three steps:

1) Construct a sequence of adapted stochastic processes fn such that f fnM 2 0.

2) Show that IT ( fn) IT ( f )L 2 0.3) Show the a.s. uniqueness of the limit IT ( f ).

To begin,

1) First we find a sequence of M 2T functions f1, f2, . . . such that f fnM 2T =E( T

0 | fn(t) f (t)|2 dt) 0. Define

fn(t) =

{n k

nk1

nf (s)ds if t [ kn ,

k+1n ) for k = 1,2, . . . , [T n]1,

0 otherwise.(6.5)

Then by construction fn(t) is adapted to Ft and is a step function for each .(Function of this kind is called random step function. Since it is random, to beprecise, it should be denoted by fn(t) = fn(t,)). Note that

k+1n

kn

| fn(t)|2 dt = n

k

n

k1n

f (t)dt

2

k

n

k1n

| f (t)|2 dt ,a.s. (6.6)

by Cauchy Schwartz (CS) inequality. The above inequality holds almost surelyas CS is applied for each in f (t,).Next we show f fnM 2T = E(

T0 | fn(t) f (t)|2 dt) 0. By the a.s. continuity

of f (t), it can be shown that

limn

T0| fn(t) f (t)|2 dt = 0 a.s. (6.7)

Let Yn = T

0 | fn(t) f (t)|2 dt. Note that (6.7) means that limn Yna.s.= 0. Note

also that

6.2 Ito Stochastic Integral 111

Yn 2 T

0

(| f (t)|2 + | fn(t)|2

)dt 4

T0| f (t)|2 dt , Y ,

where the second inequality follows from summing over all k in (6.6). Since Yna.s.

0, |Yn| Y a.s. and EY < by the definition of M 2T , we can apply DominantConvergence Theorem (DCT) to obtain E(Yn) 0, i.e.,

f fnM 2T = E( T

0| fn(t) f (t)|2 dt

) 0 . (6.8)

2) To show the existence of Ito Integral, we need to show that IT ( fn) = T

0 fn(t)dWtconverges to an element in L 2. Since fn(t) is a step function taking constantvalues in each interval [ kn ,

k+1n ), we can write

IT ( fn) = k1

fn

(kn

)(Wk+1

nW k

n

).

Note that

IT ( fn)2L 2

= E

(k1

fn

(kn

)(Wk+1

nW k

n

))2

= E

(k1

j1

fn

(kn

)fn

(jn

)(Wk+1

nW k

n

)(W j+1

nW j

n

))

= k1

E

(fn

(kn

)2)E(

Wk+1nW k

n

)2(Independent increment and fn

(kn

)F k

n)

= k1

E

(fn

(kn

)2) 1n

(Stationary increment and E(W 2t ) = t)

= E T

0fn(t)2 dt (Riemann integral)

= fn2M 2T . (6.9)

Similarly, it can be shown that

IT ( fn) IT ( fm)2L 2 = fn fm2M 2T

. (6.10)

From (6.8), for any > 0, there is an N such that f fn2M 2T< 2 for all n > N.

Thus for n,m > N,

112 6 Itos Stochastic Calculus

IT ( fn) IT ( fm)2L 2 = fn fm2M 2T

fn f2M 2T + fm f2M 2T

(Triangular Inequality)

0 andtni =

iTn , set

6.3 Properties of the Stochastic Integral 113

fn(t) =n1

i=0

Wtni 1[tni ,tni+1)(t).

Then the sequence f1, f2, M 2T approximates f , since

E(

0| f (t) fn(t)|2 dt

)=

n1

i=0

tni+1tni

E(|Wt Wtni |

2)

dt

=n1

i=0

tni+1tni

(t tni ) dt

=12

n1

i=0

(tni+1 tni

)2=

12

T 2

n 0 as n .

From Theorem 6.1, limn IT ( fn) exists in L 2 sense. To find an explicit formulaof the limit, note from the equation a(ba) = 12 (b

2a2) 12 (ba)2 that

IT ( fn) =n1

i=0

Wtni

(Wtni+1 Wtni

)=

12

[n1

i=0

(W 2tni+1 W

2tni

)

n1

i=0

(Wtni+1 Wtni )2

]

=12

W 2T 12

n1

i=0

(Wtni+1 Wtni )2 1

2W 2T

12

T

in L 2 as n . The last convergence follows from the fact that the quadraticvariation of Wiener process is T . Therefore, we conclude that T

0Wt dWt = IT ( f ) =

12

W 2T 12

T .

ut

6.3 Properties of the Stochastic Integral

The basic properties of the Ito integral are summarized in the following theorem:

Theorem 6.2. The following properties hold for any f ,g M 2, any , R andany 06 s < T :

a) Linearity:

114 6 Itos Stochastic Calculus T0

( f (t)+g(t)) dW (t) = T

0f (t)dW (t)+

T0

g(t)