The Reflection Principle; The Distribution of the Maximum - Brownian Motion with Drift

Advanced Stochastic Processes. LECTURE 6 The Reflection Principle. The Distribution of the Maximum. Brownian motion with drift Lecture outline • Quick intro to stopping times • Reflection principle • Brownian motion with drift 6.1. Technical preliminary: stopping times Stopping times are loosely speaking ”rules” by which we interrupt the process without looking at the process after it was interrupted. For example ”sell your stock the first time it hits $20 per share” is a stopping rule. Whereas, ”sell your stock one day before it hits $20 per share” is not a stopping rule, since we do not know the day (if any) when it hits this price. Given a stochastic process {Xt }t≥0 with t ∈ Z+ or t ∈ �+ , a a random variable T is called a stopping time if for every time the event T ≤ t is completely determined by the history {Xs }0≤s≤t . This is not a formal definition. The formal definition will be given later when we study filtration. Then we will give the definition in terms of the underlying (Ω, F, P). For now, though, let us just adopt this loose definition. 6.2. The Reflection principle. The distribution of the maximum The goal of this section is to obtain the distribution of M (t) � sup B(s) 0≤s≤t for any given t. Surprisingly the resulting expression is very simple and follows from one of the key properties of the Brownian motion – reflection principle. 1 2 Given a > 0, define Ta = inf{t : B(t) = a} – the first time when Brownian motion hits level a. When no such time exists we define Ta = ∞, although we now show that it is finite almost surely. Proposition 1. Ta < ∞ almost surely. Proof. Note that if B hits some level b ≥ a almost surely, then by continuity and since B(0) = 0, it hits level a almost surely. Therefore, it suffices to prove that lim supt B(t) = ∞ almost surely. This in its own order will follow from lim supn B(n) = ∞ almost surely. � Problem 1. Prove that lim supn |B(n)| = ∞ almost surely. HINT: Use the independent increments property of the Brownian motion and continuity property: P(∩Aj ) = limj P(Aj ) for monotone decreasing sequences of events Aj , which we established earlier. The differential property of the Brownian motion suggests that (6.1) B(Ta + s) − B(Ta ) = B(Ta + s) − a is also a Brownian motion, independent from B(t), t ≤ Ta . The only issue here is that Ta is a random instance and the differential property was established for fixed times t. Turns out (we do not prove this) the differential property also holds for a random time Ta since it is a stopping time and is finite almost surely. The first is an immediate consequence of its definition: we can determine whether Ta ≤ t by checking looking at the path B(u), 0 ≤ u ≤ t. The almost sure finiteness follows from Proposition 1. The property (6.1) is called the strong independent increments property of the Brownian motion. Theorem 6.2 (The reflection principle). Given a standard Brownian motion B(t), for every a≥0 � ∞ x2 1 e− 2t dx. (6.3) P(M (t) ≥ a) = 2P(B(t) ≥ a) = 2 √ 2πt a Proof. We have P(B(t) ≥ a) = P(B(t) ≥ a, M (t) ≥ a) + P(B(t) ≥ a, M (t) < a). Note, however, that P(B(t) ≥ a, M (t) < a) = 0 since M (t) ≥ B(t). Now P(B(t) ≥ a, M (t) ≥ a) = P(B(t) ≥ a|M (t) ≥ a)P(M (t) ≥ a) = P(B(t) ≥ a|Ta ≤ t)P(M (t) ≥ a). We have that B(Ta + s) − a is a Brownian motion. Conditioned on Ta ≤ t, 1 2 since the Brownian motion satisfies P(B(t) ≥ 0) = 1/2 for every t. Applying this identity, we obtain 1 P(B(t) ≥ a) = P(M (t) ≥ a). 2 This establishes the required identity (6.3). � P(B(t) ≥ a|Ta ≤ t) = P(B(Ta + (t − Ta )) − a ≥ 0|Ta ≤ t) = LECTURE 6. REFLECTION PRINCIPLE 3 We now establish the joint probability distribution of M (t) and B(t). Proposition 2. For every a > 0, y ≥ 0 (6.4) P(M (t) ≥ a, B(t) ≤ a − y) = P(B(t) > a + y). Proof. We have P(B(t) > a + y) = P(B(t) > a + y, M (t) ≥ a) + P(B(t) > a + y, M (t) < a) = P(B(t) > a + y, M (t) ≥ a) = P(B(Ta + (t − Ta )) − a > y |M (t) ≥ a)P(M (t) ≥ a). But since B(Ta + (t − Ta )) − a, by differential property is also a Brownian motion, then, by symmetry P(B(Ta + (t − Ta )) − a > y |M (t) ≥ a) = P(B(Ta + (t − Ta )) − a < −y |M (t) ≥ a) = P(B(t) < a − y |M (t) ≥ a). We conclude P(B(t) > a + y) = P(B(t) < a − y |M (t) ≥ a)P(M (t) ≥ a) = P(B(t) < a − y, M (t) ≥ a). � We now compute the Laplace transform of the hitting time Ta . Proposition 3. For every λ > 0 √ E[e−λTa ] = e− 2λa . Proof. We first compute the density of Ta . We have � ∞ x2 2 a P(Ta ≤ t) = P(M (t) ≥ a) = 2P(B(t) ≥ a) = √ e− 2t dx = 2(1 − N ( √ )). t 2πt a By differentiating with respect to t we obtain that the density of Ta is given as a 1 − a2 e 2t . 3 √ t 2 2π Therefore � ∞ a2 a −λTa E[e ]= e−λt 3 √ e− 2t dt. t 2 2π 0 Computing this integral is a boring exercise (Why do we not integrate over the negative part?) √ − 2λa . � in calculus. We just state the result which is e 6.3. Brownian motion with drift So far we considered a Brownian motion which is characterized by zero mean and some variance parameter σ 2 . The standard Brownian motion is the special case σ = 1. There is a natural way to extend this process to a non-zero mean process by considering Bµ (t) = µt + B(t), given a Brownian motion B(t). Some properties of Bµ (t) follow immediately. For example given s < t, the increments Bµ (t) − Bµ (s) have mean µ(t − s) and variance σ 2 (t − s). 4 Also, by the Time Reversal property of B (see Lecture 5) we know that limt→∞ (1/t)B(t) = 0 almost surely. Therefore, almost surely Bµ (t) lim = µ. t→∞ t When µ < 0 this means that Mµ (∞) � supt≥0 Bµ (t) < ∞ almost surely. On the other hand Mµ (∞) ≥ 0 (why?). Our goal now is to compute the probability distribution of Mµ (∞). For simplicity assume that B is the standard Brownian motion. The general case is obtained by dividing Bµ (t) = µt + B(t) by σ and observing that now the driftless part B(t)/σ corresponds to the standard Brownian motion. Theorem 6.5. For every µ < 0, the distribution of Mµ (∞) is exponential with parameter 2|µ|. Namely, for every x ≥ 0 P(Mµ (∞) > x) = e−2|µ|x . The proof is in Section 6.8 of Resnick’s book in the course packet. The proof consists of two parts. We first show that the distribution of Mµ (∞) is exponential. Then we compute its parameter. 6.4. Additional reading materials • Sections 6.5 and 6.8 from Chapter 6 of Resnick’s book ”Adventures in Stochastic Pro­ cesses” in the course packet. • Sections 7.3 and 7.4 in Durrett [2]. • Billingsley [1], Section 9.