"Frontmatter". In: Analysis of Financial Time Series

ITO’S LEMMA 2276.3.2 Stochastic DifferentiationTurn next to the case in which G is a differentiable function of x t and t, andx t is anIto’s process. The Taylor expansion becomesG = ∂G∂xx +∂G∂tA discretized version of the Ito’sprocessist + 1 ∂ 2 G2 ∂x 2 (x)2 + ∂2 G∂x∂t x t + 1 ∂ 2 G2 ∂t 2 (t)2 +···. (6.3)x = µt + σɛ √ t, (6.4)where, for simplicity, we omit the arguments of µ and σ ,andx = x t+t − x t .From Eq. (6.4), we have(x) 2 = µ 2 (t) 2 + σ 2 ɛ 2 t + 2µσ ɛ(t) 3/2 = σ 2 ɛ 2 t + H(t), (6.5)where H(t) denotes higher order terms of t. This result shows that (x) 2 containsa term of order t, which cannot be ignored when we take the limit as t → 0.However, the first term in the right-hand side of Eq. (6.5) has some nice properties:E(σ 2 ɛ 2 t) = σ 2 t,Var(σ 2 ɛ 2 t) = E[σ 4 ɛ 4 (t) 2 ]−[E(σ 2 ɛ 2 t)] 2 = 2σ 4 (t) 2 ,where we use E(ɛ 4 ) = 3 for a standard normal random variable. These two propertiesshow that σ 2 ɛ 2 t converges to a nonstochastic quantity σ 2 t as t → 0.Consequently, from Eq. (6.5), we have(x) 2 → σ 2 dt as t → 0.Plugging the prior result into Eq. (6.3) and using the Ito’s equation of x t in Eq. (6.2),we obtaindG = ∂G∂x=dx +∂G∂tdt + 1 ∂ 2 G2 ∂x 2 σ 2 dt()∂G∂x µ + ∂G + 1 ∂ 2 G∂t 2 ∂x 2 σ 2 dt + ∂G∂x σ dw t,which is the well-known Ito’s lemma in Stochastic Calculus.Recall that we suppressed the argument (x t , t) from the drift and volatility termsµ and σ in the derivation of Ito’s lemma. To avoid any possible confusion in thefuture, we restate the lemma as follows.