An Informal Derivation of Ito’s Lemma

Matt Krems

August 11, 2008

1 Ito’s Equation

We start out by looking at an equation for the time derivative of a stochastic variable X(t). The form wewill look at is

X(t) = µ + σl(t) (1)where µ and σ are constants and l(t) is a fluctuating term with the following properties:

〈l(t)〉 = 0 (2)〈l(t)l(t′)〉 = δ(t− t′) (3)

These two properties mean the fluctuating term is “white noise”, i.e. zero ensemble average and δ-autocorrelation.We can now look at the value of the stochastic variable, X(t), at a slightly later time, t + ε, such thatX(t + ε) = X(t) + ∆X(t) where

∆X(t) =∫ t+ε

t

dt′X(t) =∫ t+ε

t

dt′ [µ + σl(t)] = εµ + σ

∫ t+ε

t

dt′l(t′) (4)

Consider now a Taylor expansion of an arbitrary function, f(X(t)):

f(X(t+ε)) = f(X(t)+∆X(t)) = f(X(t))+f ′(X(t))∆X(t)+12!

f ′′(X(t))[∆X(t)]2+13!

f ′′′[∆X(t)]3+ . . . (5)

The linear term, ∆X(t), has the average

〈∆X(t)〉 = 〈εµ〉+ σ

∫ t+ε

t

dt′〈l(t′)〉 ≈ εµ

where I use Eq.(2) and only keep terms of order ε or smaller. The average of the quadratic term, 〈[∆X(t)]2〉is

〈[∆X(t)]2〉 =∫ t+ε

t

dt1

∫ t+ε

t

dt2〈[µ + σl(t1)][µ + σl(t2)]〉

≈ ε2µ2 + σ2〈l(t1)l(t2)〉

Now the second term is of order ε due to the δ-function correlation. Thus we find that:

〈[∆X(t)]2〉 = εσ2 + h.o.t

The average of the cubic term gives higher order terms (h.o.t.) than ε and so on. Thus we find that in thelimit ε → 0:

〈f(X(t))〉 = 〈f ′(X(t))〉〈X(t)〉+12〈f ′′(X(t))〉 (6)

In fact, as it turns out, this is not just true for the average, but for the derivative f(X(t)) itself1, i.e. f(X(t))obeys the stochastic differential equation

f(X(t)) = f ′(X(t))X(t) +12f ′′(X(t)) (7)

We note that if X(t) was a deterministic variable, then the extra term 12f ′′(X(t)) would not be here. This

is a fundamental result and the primary difference of the Ito calculus.1see Section 18.13.3 of “Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets” by H.

Kleinert

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2 Ito’s Equation in Differential Form

We can write Eq.(1) in differential form:

dX(t) = µdt + σl(t)dt (8)

and definel(t)dt ≡ dW (t) (9)

where W (t) is a Wiener process corresponding to Brownian motion, which has the appropiate properties2.We arrive at

dX(t) = µdt + σdW (t). (10)

Now we can look at f(X(t)) which can be written as f(t, W ) due to Eq.(11). We can do a multi-variableTaylor expansion in the increment df = f(t)− f(t′) 3 where dt = t− t′ and keeps terms up to order dt:

df(t, W ) =∂f

∂tdt +

∂f

∂WdW +

12

∂2f

∂W∂WdWdW + . . . (11)

We can infer from the previous analysis that dW ∝√

dt such that dWdW is of order dt 4 and we get

df(t, W ) =(

∂f

∂t+

12

∂2f

∂W∂W

)dt +

∂f

∂WdW (12)

2This is an intentionally vague comment as I need to figure out exactly why Brownian motion is relevant here.3Since W is also a function of t, we can write the function f solely as a function of t.4There is indeed a hint of this but I need to see a more formal proof of this feature.

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