Quadratic Variation Property of Brownian Motion

Advanced Stochastic Processes. LECTURE 7 Quadratic variation property of Brownian motion Lecture outline • Unbounded variation of a Brownian motion. • Bounded quadratic variation of a Brownian motion. 7.1. Unbounded variation of a Brownian motion Any sequence of values 0 < t0 < t1 < · · · < tn < T is called a partition Π = Π(t0 , . . . , tn ) of an interval [0, T ]. Given a continuous function f : [0, T ] → R its total variation is defined to be � LV (f ) � sup |f (tk ) − f (tk−1 )|, Π 1≤k≤n where the supremum is taken over all possible partitions Π of the interval [0, T ] for all n. A function f is defined to have bounded variation if its total variation is finite. Theorem 7.1. Almost surely no path of a Brownian motion has bounded variation for every T ≥ 0. Namely, for every T P(ω : LV (B(ω)) < ∞) = 0. The main tool is to use the following result from real analysis, which we do not prove: if a function f has bounded variation on [0, T ] then it is differentiable almost everywhere on [0, T ]. We will now show that quite the opposite is true. Proposition 1. Brownian motion is almost surely nowhere differentiable. Specifically, P(∀ t ≥ 0 : lim sup | h→0 B(t + h) − B(t) | = ∞) = 1. h 1 2 Proof. Fix T > 0, M > 0 and consider A(M, T ) ⊂ C[0, ∞) – the set of all paths ω ∈ C[0, ∞) such that there exists at least one point t ∈ [0, T ] such that lim sup | h→ 0 B(t + h) − B(t) | ≤ M. h We claim that P(A(M, T )) = 0. This would imply that Brownian motion is almost surely nowhere differentiable in [0, T ]. Then we take a union of the sets A(M, T ) with increasing T and conclude that B is almost surely nowhere differentiable on [0, ∞). If ω ∈ A(M, T ), then there exists t ∈ [0, T ] and n such that |B(s) − B(t)| ≤ 2M |s − t| for all s ∈ (t − n2 , t + n2 ). Now define An ⊂ C[0, ∞) to be the set of all paths ω such that for some t ∈ [0, T ] |B(s) − B(t)| ≤ 2M |s − t| for all s ∈ (t − n2 , t + n2 ). Then (7.2) An ⊂ An+1 and (7.3) A(M, T ) ⊂ ∪n An . Find k = max{j : j n ≤ t}. Define k+2 k k+1 k+1 k k−1 ) − B( )|, |B( ) − B( )|, |B( ) − B( )|}. n n n n n n In other words, consider the maximum increment of the Brownian motion over these three short intervals. We claim that Yk ≤ 6M/n for every path ω ∈ An . To prove the bound required bound on Yk we first consider k+2 k+1 k+2 k+1 )| ≤ |B( ) − B(t)| + |B(t) − B( ) − B( |B( )| n n n n 2 1 ≤ 2M + 2M n n 6M ≤ . n The other two differences are analyzed similarly. Yk = max{|B( Now consider event Bn which is the set of all paths ω such that Yk (ω) ≤ 6M/n for some 0 ≤ k ≤ T n. We showed that An ⊂ Bn . We claim that limn P(Bn ) = 0. Combining this with (7.2), we conclude P(An ) = 0. Combining with (7.3), this will imply that P(A(M, T )) = 0 and we will be done. Now to obtain the required bound on P(Bn ) we note that, since the increments of a Brownian motion are independent and identically distributed, then � P(Yk ) P(Bn ) ≤ 0≤k≤T n (7.4) 3 2 2 1 1 ≤ T nP(max{|B( ) − B( )|, |B( ) − B( )|, |B( ) − B(0)|} ≤ 6M/n) n n n n n 1 = T n[P(|B( )| ≤ 6M/n)]3 . n LECTURE 7. QUADRATIC VARIATION 3 Finally, we just analyze this probability. We have √ 1 P(|B( )| ≤ 6M/n) = P(|B(1)| ≤ 6M/ n). n √ Since B(1) which has the standard normal distribution, its density at any point is at most 1/ 2π, √ then we have that this probability is at√a most (2(6M )/ 2πn). We conclude that the expression √ in (7.4) is, ignoring constants, ≈ n(1/ n)3 = 1/ n and thus converges to zero as n → ∞. We proved limn P(Bn ) = 0. � 7.2. Bounded quadratic variation of a Brownian motion Even though Brownian motion is nowhere differentiable and has unbounded total variation, it turns out that it has bounded quadratic variation. This observation is the cornerstone of Ito calculus, which we will study later in this course. We again start with partitions Π = Π(t0 , . . . , tn ) of a fixed interval [0, T ], but now consider instead � Q(Π, B) � (B(tk ) − B(tk−1 ))2 . 1≤k≤n where, we make (without loss of generality) t0 = 0 and tn = T . For every partition Π define Δ(Π) = max |tk − tk−1 |. 1≤k≤n Theorem 7.5. Consider an arbitrary sequence of partitions Πi , i = 1, 2, . . .. Suppose limi→∞ Δ(Πi ) = 0. Then (7.6) lim E[(Q(Πi , B) − T )2 ] = 0. i→∞ Suppose in addition limi→∞ i2 Δ(Πi ) = 0 (that is the resolution Δ(Πi ) converges to zero faster than 1/i2 ). Then almost surely (7.7) Q(Πi , B) → T. In words, the standard Brownian motion has almost surely finite quadratic variation which is equal to T . Remark. The first part of the Theorem says that the convergence to T happens in L2 norm (to be defined). The second says that the converges happens almost surely. As it turns out, these different types of convergences (almost surely, in probability, in distribution, in expectation, in L2 norm) are not necessarily equivalent. We will spend a lecture studying these. Proof. We will use the following fact. Let Z be a standard Normal random variable. Then E[Z 4 ] = 3 (cute, isn’t it?). The proof can be obtained using Laplace transforms of Normal random variables and we skip the details (see the course packet for a derivation). Let θi = (B(ti ) − B(ti−1 ))2 − (ti − ti−1 ). Then, using the independent Gaussian increments property of Brownian motion, θi is a sequence of independent zero mean random variables. We have � θi . Q(Πi ) − T = 1≤i≤n 4 Now consider the second moment of this difference � E(B(ti ) − B(ti−1 ))4 E(Q(Πi ) − T )2 = 1≤i≤n −2 � E(B(ti ) − B(ti−1 ))2 (ti − ti−1 ) + 1≤i≤n � (ti − ti−1 )2 . 1≤i≤n 4 Using the E[Z ] = 3 property, this expression becomes � � � (ti − ti−1 )2 + (ti − ti−1 )2 3(ti − ti−1 )2 − 2 1≤i≤n =2 � 1≤i≤n 1≤i≤n (ti − ti−1 )2 1≤i≤n ≤ 2Δ(Πi ) � (ti − ti−1 ) 1≤i≤n = 2Δ(Πi )T. Now if limi Δ(Πi ) = 0, then the bound converges to zero as well. This establishes the first part of the theorem. To prove the second part identify a sequence �i → 0 such that Δ(Πi ) = �i /i2 . By assumption, such a sequence exists. By Markov’s inequality, this is bounded by E(Q(Πi ) − T )2 T 2Δ(Πi )T ≤ = 2 (7.8) P((Q(Πi ) − T )2 > 2�i ) ≤ 2�i 2�i i � T Since i i2 < ∞, then the sum of probabilities in (7.8) is finite. Then applying the Borel-Cantelli Lemma, the probability that (Q(Πi ) − T )2 > 2�i for infinitely many i is zero. Since �i → 0, this � exactly means that almost surely, limi Q(Πi ) = T . 7.3. Additional reading materials • Sections 6.11 and 6.12 of Resnick’s chapter 6 in the book (course packet).