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1. Stochastic Process 2. Poisson HETERGENEITY IN QUANTITATIVE MACROECONOMICS @ TSE OCTOBER 17, 2016 STOCHASTIC CALCULUS BASICS SANG YOON (TIM)LEE Very simple notes (need to add references). It is NOT meant to be a substitute for a real course in stochastic calculus, just listing heuristic derivations of the stuff most oftenly used in economics. Ito calculus is a lot more than only dealing with Poisson jumps and Wiener processes. Some abuses of notation included without clarification. 1. Stochastic Process A stochastic process is a collection of random variables (measurable functions) fXt : t 2 Tg , Xt : W ! S, ordered in t (time), along with a measurable space (S, S). The probability space (W, F, P) denotes, respectively, the “state space" (set of all possible histories), the s-algebra that contains all possible sets (Borel sets) of histories induced by W, and the probability mea- sure over F. The space (S, S) contains the range of the function Xt : W 7! S and its corresponding s-algebra. For example, for most of our applications Xt 2 R or R+. If Xt is measurable, any process induces a measure Pt that we can construct using the original probability space. This calls for the notion of a filtration: a weakly increasing collection of Borel sets on S, fFt, t 2 Tg, s.t. for all s < t 2 T, Fs ⊂ Ft ⊂ F. The process X is adapted to the filtration fFtgt2T if Xt is Ft-measurable. This just means that for any Xt, I can compute the probability only using Ft and not all of F. Hence, a well defined stochastic process is always adapted to its natural filtration n −1 o Ft = s Xs (A) : s ≤ t, A 2 S . This just means that for any history of Xt up to time t, all possibly realizable trajectories can be mapped backed into a subset of Ft, so that I can compute its probability for all points up to time t. This generates an induced probability measure over X. EXAMPLE 1 Let W = [0, 1]¥. Then any w 2 W is just a coordinate on the infinite dimensional unit cube. If we let Xt : W 7! S denote the t-th coordinate, S is just the unit interval [0, 1]. If we construct, say, the probability measure so that P = P1 × · · · × P¥, where each Pt is the uniform distribution, Xt is i.i.d. uniform. 2. Poisson (Jump) Process Let Nt be the random variable equal to the number of “hits" up to time t. The (adapted) state space is R[0,t] and range is all right-continuous paths that increase by 1. Now define 1 the probability measure over w as Poisson: (l(t − s])n P fN − N = ng = · exp(−l(t − s]) t s n! where l is the rate of arrival. This is what is usually called the Poisson process. (Not to be confused with what we use more often in economics: Xt is a Compound Poisson Process (CPP) if if it changes to some value at rate lt, studied below. In fact this is a new random variable in which Xt changes to some value if Nt > Ns for all s < t, and you could redefine the probability space to the histories of Nt rather than R[0,t]. This is the set of all right-continuous paths that increase by 1.) More typically, the Poisson process is defined as a counting process: DEFINITION 1 A continuous stochastic process Nt is Poisson if 1.N t is a counting process: Z+ (a)N t lives in (Z+, 2 ), for all t ≥ 0, (b)N s ≤ Nt for all s ≤ t, (c) lims#t Ns ≤ lims"t Ns for all t ≥ 0; that is, no hit can happen simultaneously. 2.N 0 = 0 a.s., 3. N is a stochastic process with stationary, independent increments The two definitions are equivalent; there are many other definitions as well but I refer you to the internet. It is easier to show that the earlier definition implies the counting process; by definition, increments are independent. The probability of getting 0, 1, or 2 or more hits in a time interval dt > 0 is P(Nt+dt − Nt = 0) = exp(−ldt) ≈ 1 − ldt + o(dt) 2 2 P(Nt+dt − Nt = 1) = ldt · exp(−ldt) ≈ ldt − l dt + o(dt) ≈ ldt 2 −ldt P(Nt+dt − Nt ≥ 2) = (ldt) · e /2 + o(dt) = o(dt). Clearly, the actual probability that something happens in any interval dt (and (t, t + dt], since the increments are independent) is 0. Conversely, one way to make sense of the counting process is to realize that stationarity implies E[N(T)/T] = lim N(T)/T = l T!¥ and instead of sending T to infinity, send the number of intervals dt in (0, T] to infinity to get that the expected number of hits in any given time interval is l: E[dNt] = E[N(dt)] ¥ = ldt = 0 · P(N(dt) = 0) + 1 · P(N(dt) = 1) + ∑ n · P(N(dt) = n) n=2 2 = P(N(dt) = 1) since two hits cannot occur at the same time. This is important later when we derive the stochastic HJB equation. 2.1 Compound Poisson Process Now define a jump process over the underlying Poisson process: Let Xt be a r.v. that is ga if Nt is even and gb if Nt is odd. Heuristically, 2 −ldt E[dXt] = 0 · exp(−ldt) + (gb − ga) · ldt · exp(−ldt) + 0 · (ldt) · exp /2 E[X˙ t] = lim (gb − ga) · l · exp(−ldt) = l(gb − ga) dt!0 More generally, let fZkgk≥1 be an i.i.d. ordered sequence of random variables with mea- sure Gz(z), independent of the Poisson process Nt. Let Xt be a continuous stochastic process that is a function of (Nt, Zk), and define Nt Xt = ∑ Zk. k=1 Then ( " ! #) ¥ n ¥ e−lt(lt)n EXt = ∑ E ∑ ZkjNt = n P(Nt = n) = ∑ n=1 k=1 n=1 n! " !# n ¥ e−lt(lt)n−1 = E Z = ltm = m · lt ∑ k Z ∑ ( − ) Z k=1 n=1 n 1 ! and Z t dXt = ZNt dNt = ZNt dNt 0 assuming X0 = 0. 2.2 Stochastic Integral with Poisson First consider a function f (Nt). The integral is easy to write as Nt Z t f (Nt) − f (0) = ∑ [ f (k) − f (k − 1)] = [ f (1 + Ns− ) − f (Ns− )] dNs k=1 0 Z t Z t = [ f (Ns) − f (Ns − 1)] dNs = [ f (Ns) − f (Ns− )] dNs, 0 0 3 where Ns− is the left limit of the Poisson process, and only one jump occurs in an dt by definition (or construction) of the Poisson process. For the compound process, recall that the waiting time for the kth hit of the Poisson process, Tk, is also a random variable s.t. that the event fTk > tg , fNt ≥ k − 1g; in particular this means that Tk − Tk−1 is an i.i.d. process by definition. For k = 1, the waiting time follows an exponential distribution. For k > 1, Z ¥ e−ls(ls)n−1 P(Tk > t) = l ds, (1) t (n − 1)! since P(Tk > t) = P(Tk > t ≥ Tk−1) + P(Tk−1 > t) Z ¥ e−ls(ls)n−2 = P(Nt = n − 1) + l ds t (n − 2)! e−lt(lt)n−1 Z ¥ e−ls(ls)n−2 = + l ds (n − 1)! t (n − 2)! and integration by parts leads to (1). Using waiting times, the stochastic integral of a function of a compound Poisson process can be written Nt Z t − − − − f (Yt) − f (0) = [ f (Y + Zk) − f (Y )] = [ f (Ys + ZNs ) − f (Ys )] dNs ∑ Tk Tk k=1 0 Z t = [ f (Ys) − f (Ys− )] dNs 0 Z t Z t = [ f (Ys) − f (Ys− )] (dNs − lds) + l [ f (Ys) − f (Ys− )] ds. 0 0 3. Wiener Process (Brownian Motion) DEFINITION 2 A Wiener process is defined by four properties: 1.W 0 = 0 a.s. 2. Independent increments: Wt − Ws is independent of Fs for all s ≤ t 3. Normality: Wt − Ws ∼ N (0, t − s) 4.W t is continuous a.s. We could spend the whole semester just talking about this, which we won’t. Basically, think of Brownian motion as a random walk in continuous time: the best predictor of dXt is 0, with Gaussian errors. So clearly, Wt is a particular type of a martingale (E[WtjFs] = Ws a.s., for all 0 ≤ s < t < ¥). 4 Most commonly you will encounter a Brownian motion with drift, a geometric Brow- nian motion, or a generic (Ito) diffusion process: dXt = mdt + sdWt, dXt = mXtdt + sXtdWt, dXt = m(Xt)dt + s(Xt)dWt the geometric Brownian motion simply gives dXt/Xt = d log Xt = mdt + sdWt, so it is the just a Brownian motion with drift in percentage points (or log-points, to be exact). In the Ito process, the instantaneous drift and variance depend on the current value of Xt and is related to the version of Ito’s Lemma that we will look at below. Before we move along, note that both the Poisson process and Brownian motion are Markov processes, but while the Brownian motion has a continuous time path a.s., the Poisson process has a discontinuous time path a.s. Also, Poisson was not a martingale, but dNt − ldt was. It will be useful to know the quadratic variation of the Brownian motion: we will use a particular formulation that exploits the CLT in discrete time: 2n−1 2 2 hWi ≡ E[Wt ] = lim [DiWt] t n!¥ ∑ i=1 D Wt ≡ W n − W n i ti+1 ti n n n where ti ≡ it/2 .
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