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308 CHAPTER 10. TARGET-ZONE MODELSall variables except interest rates are in logarithms. The time derivativeof a function x(t) is denoted with the ‘dot’ notation, úx(t) =dx(t)/dt.In order to work with these models, you need some background instochastic calculus.10.1 Fundamentals of Stochastic CalculusLet x(t) be a continuous-time deterministic process that grows at theconstant rate, η such that, dx(t) =ηdt. LetG(x(t),t)besomepossiblytime-dependent continuous and differentiable function of x(t). Fromcalculus, you know that the total differential of G isdG = ∂G∂x dx(t)+∂G dt. (10.1)∂tIf x(t) is a continuous-time stochastic process, however, the formulafor the total differential (10.1) doesn’t work and needs to be modiÞed.In particular, we will be working with a continuous-time stochasticprocess x(t) called a diffusion process where the growth rate of x(t)randomly deviates from η,dx(t) =ηdt + σdz(t). (10.2)ηdt is the expected change in x conditional on information available att, σdz(t) isanerrortermandσ is a scale factor. z(t) is called a Wienerprocess or Brownian motion and it evolves according to,z(t) =u √ t, (10.3)where u iid ∼ N(0, 1). At each instant, z(t) is hit by an independent drawu from the standard normal distribution. InÞnitesimal changes in z(t)can be thought of asdz(t) =z(t + dt) − z(t) =u t+dt√t + dt − ut√t =ũ√dt, (10.4)where u t+dt√t + dt ∼ N(0,t+ dt) andut√t ∼ N(0,t)deÞne the newrandom variable ũ ∼ N(0, 1). 1 The diffusion process is the continuoustimeanalog of the random walk with drift η. Sampling the diffusion1 Since E[u t+dt√t + dt−ut√t] = 0, and Var[ut+dt√t + dt−ut√t]=t+dt−t = dt,u t+dt√t + dt − ut√t deÞnes a new random variable, ũ√dt, whereũiid∼ N(0, 1).