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Chapter 6 Brownian Motion and Beyond

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Chapter 6 Brownian Motion and Beyond Chapter 6 Brownian motion and beyond Brownian motion is probably the most important stochastic process in probability theory and applications. Some of its basic properties are developed in the first half of this chapter, together with a brief discussion of martingales. Based on Brownian motion, the stochastic integrals and stochastic differential equations are introduced, including an application in option pricing in a single stock market. This topic is more advanced than the rest of these notes. An effort is made to present a complete description without being bogged down in advanced theory. For a more complete treatment, the reader is referred to a standard text on stochastic differential equations such as [7]. 6.1 Brownian motion Some motivation: Brownian motion has been used to describe the erratic motion of a pollen particle suspended in a liquid that is caused by the bombardment of surrounding molecules. Viewing the motion in one dimension, one may imagine a particle moving along the real line R in a continuous random path. It is reasonable to assume that the displacements of the particle over non-overlapping time intervals are independent and the distribution of each displacement depends only on how long the interval is. For any time t, let the interval [0; t] be divided into many small intervals of equal length, then the displacement over [0; t] is the sum of the iid displacements over the small intervals. Suggested by the CLT, the position of the particle at time t should be a normal random variable. This leads to the following formal definition. Definition of Brownian motion: Let Z(t) be a real-valued process indexed by continuous time t ≥ 0. It will be called a BM (Brownian motion) if (i) its paths are continuous; 143 (ii) it has independent and stationary increments in the sense that for any s < t, Z(t)−Z(s) is independent of the process Z up to time s and its distribution depends only on t − s; (iii) Z(t) has a normal distribution for each t > 0. In fact, by the theorem below, which is not proved here but see [5, Theorem 13.4], (iii) is essentially a consequence of (i) and (ii). Theorem 6.1 Let Z(t) be a real-valued process satisfying (i) and (ii). Assume for some s > 0, Z(s) is not a constant. Then Z(t) is a BM. Drift and variance (coefficients): Let Z(t) be a BM. Then Z(1) is normal with mean µ and variance σ2. By (ii), it is easy to see that for any s < t, Z(t) − Z(s) is normal of mean µ(t − s) and variance σ2(t − s). The constants µ and σ2 in Theorem 1 will be called, respectively, the drift and variance (coefficients) of the BM. The distribution of the BM Z(t) as a process is completely determined by the initial position Z(0), the drift µ, and the variance σ2. In particular, if Z(0) = z (a constant), then for each fixed t > 0, the distribution of Z(t) is normal of mean z + µt and variance σ2t. Scaling properties of BM: It is easy to show that if Z(t) is a BM with drift µ and variance σ2, then for any constants a and b > 0, aZ(bt) is a BM with drift abµ and variance a2bσ2. In particular, −Z(t) is a BM with drift −µ and same variance σ2. Time and space shift: Let Z(t) be a BM. It is easy to show that for any fixed T > 0, the time-shifted process ZT (t) = Z(T + t) − Z(T ) is a BM with the same drift and variance, and zero initial position. Moreover, ZT (t) is independent of process Z(t) up to time T . It can be shown that this property holds also when the constant time shift T is replaced by a stopping time τ, that is, if τ is a stopping time of the BM Z(t), then under the conditional probability P (· j τ < 1), Zτ (t) = Z(τ + t) − Z(τ) is a BM with the same drift and variance as Z(t), and zero initial position, which is independent of Z(t) up to τ. It is easy to see that the BM Z(t) also possesses a space shift property: For any real z, Z(t) + z is a BM with the same drift and variance, but a different starting position Z(0) + z. Standard Brownian motion: An SBM (standard Brownian motion) is a BM B(t) with B(0) = 0, drift µ = 0, and variance σ2 = 1. Note that if Z(t) is a BM with arbitrary drift µ and variance σ2, then Z(t) = Z(0) + σB(t) + µt, (6.1) where B(t) = [Z(t) − Z(0) − µt]/σ is an SBM independent of Z(0). 144 Markov property: A Brownian motion Z(t) has the Markov property in the following form: For any t > 0 and real number z, P [Ht j Gt;Z(t) = z] = P [Ht j Z(t) = z] = P [H0 j Z(0) = z]; (6.2) where Gt is an event determined by the BM before time t, such as Gt = [Z(t1) 2 I1;:::;Z(tk) 2 Ik] for 0 ≤ t1 < ··· < tk ≤ t and intervals I1;:::;Ik, Ht is an event determined by the BM after time t, such as Ht = [Z(t + h1) 2 J1;:::;Z(t + hm) 2 Jm] for 0 ≤ h1 < ··· < hm and intervals J1;:::;Jm, and H0 is the event Ht time shifted backward by t, that is, H0 = [Z(h1) 2 J1;:::;Z(hm) 2 Jm] when Ht is given above. To prove (6.2), by the independent and stationary increments of Z(t), P [Z(t + h1) 2 J1;:::;Z(t + hm) 2 Jm j Z(t1) 2 I1; :::;Z(tk) 2 Ik;Z(t) = z] = P [Z(t + h1) − Z(t) 2 J1 − z; : : : ; Z(t + hm) − Z(t) 2 Jm − z j Z(t1) 2 I1;:::;Z(tk) 2 Ik;Z(t) = z] = P [Z(t + h1) − Z(t) 2 J1 − z; : : : ; Z(t + hm) − Z(t) 2 Jm − z] (independent increments) = P [Z(h1) − Z(0) 2 J1 − z; : : : ; Z(hm) − Z(0) 2 Jm − z] (stationary increments) = P [Z(h1) 2 J1;:::;Z(hm) 2 Jm j Z(0) = z] = P [Z(t + h1) 2 J1;:::;Z(t + hm) 2 Jm j Z(t) = z]: Strong Markov property: A BM Z(t) in fact has the strong Markov property (not proved here) in the sense that the constant t in (6.2) may be replaced by a stopping time τ: P [Hτ j Gτ ;Z(τ) = z; τ < 1] = P [Hτ j Z(τ) = z; τ < 1] = P [H0 j Z(0) = z]: (6.3) Transition density: Let Z(t) be a BM with drift µ and variance σ2. Because given Z(0) = z, Z(t) has a normal distribution of mean z + µt and variance σ2t, we have, for any s > 0, t > 0, and interval J, 2 j 2 j P [Z(s + t) J Z(s) = z] = ZP [Z(t) J Z(0) = z] = pt(y − z)dy; (6.4) J 145 where 1 (y − µt)2 pt(y) = p exp[− ] (6.5) 2πσ2t 2σ2t is the pdf of the normal distribution of mean µt and variance σ2t. Because (6.4) is the probability of transition from point z to interval J in time t, pt(y − z) or pt(y) is called the transition density of the BM Z(t). Symbolically, we may write pt(y − z) = pdf of [Z(t) = y j Z(0) = z]: (6.6) p 2 For an SMB, the transition density is pt(y) = (1= 2πt) exp[−y =(2t)]. Distribution of Brownian motion: The finite dimensional distributions of the BM Z(t) are determined by the transition density. For example, for s < t, ≤ ≤ j ZP [Z(s) y; Z(t) z Z(0) = x] y = P [Z(t) ≤ z j Z(s) = u]ps(u − x)du Z−∞ y = P [Z(t − s) ≤ z j Z(0) = u]ps(u − x)du Z−∞ Z y z = ps(u − x)pt−s(v − u)dvdu: −∞ −∞ In general, for 0 < t1 < t2 < ··· < tn and real numbers z0; z1; z2; : : : ; zn, ≤ ≤ ≤ j ZP [Z(Zt1) z1Z;Z(t2) z2;:::;Z(tn) zn Z(0) = z0] z1 z2 zn = ··· pt (y1 − z0)pt −t (y2 − y1) −∞ −∞ −∞ 1 2 1 ··· − ··· ptn−tn−1 (yn yn−1)dyn dy2dy1: (6.7) Symbolically this may be written as joint pdf of [Z(t1) = z1;Z(t2) = z2;:::;Z(tn) = zn j Z(0) = z0] − − ··· − = pt1 (z1 z0)pt2−t1 (z2 z1) ptn−tn−1 (zn zn−1): (6.8) Exercise 6.1 Let Z(t) be a Brownian motion with Z(0) = z, drift µ, and variance σ2. Find E[Z(s)Z(t)] for s < t. Exercise 6.2 Let Z(t) be a BM with µ = 2 and σ2 = 9, and let τ be the first time t > 0 when Z(t) = 1. Find P [Z(τ + 1) < 0 j Z(0) = 0]. Exercise 6.3 Let B(t) be the SBM. For any real number a and 0 < t < 1, determine the conditional distribution of B(t) given B(1) = a. Note: The SBM B(t), 0 ≤ t ≤ 1, given B(1) = 0, is called a Brownian bridge process. 146 6.2 Standard Brownian motion and its maximum Reflection principle: Let B(t) be an SBM and let τa be its hitting time of point a > 0, which is the first time t > 0 when B(t) = a.
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