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10.1. FUNDAMENTALS OF STOCHASTIC CALCULUS 309x(t) atdiscretepointsintimeyieldsx(t +1)− x(t) =Z t+1tZ t+1dx(s)Z t+1ds + σ= ηdz(s)tt| {z }z(t+1)−z(t)= η + σũ. (10.5)If x(t) follows the diffusion process (10.2), it turns out that the totaldifferential of G(x(t),t)isdG = ∂G∂x dx(t)+∂G ∂tdt +σ22∂ 2 Gdt. (10.6)∂x2 This result is known as Ito’s lemma. The next section gives a nonrigorousderivation of Ito’s lemma and can be skipped by uninterestedreaders.Ito’s LemmaConsider a random variable X with Þnite mean and variance, and apositive number θ > 0. Chebyshev’s inequality says that the probabilitythat X deviates from its mean by more than θ is bounded by its variancedivided by θ 2P{|X − E(X)| ≥ θ} ≤ Var(X) . (10.7)θ 2If z(t) follows the Wiener process (10.3), then E[dz(t)] = 0 andVar[dz(t) 2 ]=E[dz(t) 2 ] − [Edz(t)] 2 = dt. Apply Chebyshev’s inequalityto dz(t) 2 ,togetP {|[dz(t)] 2 − E[dz(t)] 2 | > θ} ≤ (dt)2θ 2 .Since dt is a fraction, as dt → 0, (dt) 2 goes to zero even faster thandt does. Thus the probability that dz(t) 2 deviates from its mean dtbecomes negligible over inÞnitesimal increments of time. This suggests