Lecture 8: Quadratic Variation

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Lecture 8 Fall 2013 9/30/2013 Quadratic variation property of Brownian motion Content. 1. Unbounded variation of a Brownian motion. 2. Bounded quadratic variation of a Brownian motion. 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 |f (tk ) − f (tk−1 )|, LV (f ) £ sup 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 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. Specif­ ically, P(∀ t ≥ 0 : lim sup | h→0 B(t + h) − B(t) | = ∞) = 1. h 1 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 implies P(∪M ≥1 A(M, T )) = 0 which is what we need. 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 An ⊂ An+1 (1) A(M, T ) ⊂ ∪n An . (2) and Find k = max{j : Yk = max{|B( j n ≤ t}. Define k+2 k+1 k+1 k 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 |B( k+2 k+1 k+2 k+1 ) − B( )| ≤ |B( ) − B(t)| + |B(t) − B( )| n n n n 2 1 ≤ 2M + 2M n n 6M ≤ . n The other two differences are analyzed similarly. 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 ) = 2 0. Combining this with (1), we conclude P(An ) = 0. Combining with (2), 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 incre­ ments of a Brownian motion are independent and identically distributed, then X P(Bn ) ≤ P(Yk ≤ 6M/n) 0≤k≤T n 3 2 2 1 1 ≤ T nP(max{|B( ) − B( )|, |B( ) − B( )|, |B( ) − B(0)|} ≤ 6M/n) n n n n n 1 (3) = T n[P(|B( )| ≤ 6M/n)]3 . n 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 (3) is, ignoring constants, O(n(1/ n)3 ) = √ O(1/ n) and thus converges to zero as n → ∞. We proved limn P(Bn ) = 0. 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 observa­ tion 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 X 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 2. Consider an arbitrary sequence of partitions Πi , i = 1, 2, . . .. Sup­ pose limi→∞ Δ(Πi ) = 0. Then lim E[(Q(Πi , B) − T )2 ] = 0. i→∞ 3 (4) Suppose in addition limi→∞ i2 Δ(Πi ) = 0 (that is the resolution Δ(Πi ) con­ verges to zero faster than 1/i2 ). Then almost surely Q(Πi , B) → T. (5) In words, the standard Brownian motion has almost surely finite quadratic vari­ ation which is equal to T . 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 or integration by parts, and we skip the details. 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 X Q(Πi ) − T = θi . 1≤i≤n Now consider the second moment of this difference X E(Q(Πi ) − T )2 = E(B(ti ) − B(ti−1 ))4 1≤i≤n X −2 E(B(ti ) − B(ti−1 ))2 (ti − ti−1 ) + 1≤i≤n X (ti − ti−1 )2 . 1≤i≤n Using the E[Z 4 ] = 3 property, this expression becomes X X X (ti − ti−1 )2 + (ti − ti−1 )2 3(ti − ti−1 )2 − 2 1≤i≤n =2 X 1≤i≤n 1≤i≤n (ti − ti−1 )2 1≤i≤n ≤ 2Δ(Πi ) X (ti − ti−1 ) 1≤i≤n = 2Δ(Πi )T. Now if limi Δ(Πi ) = 0, then the bound converges to zero as well. This estab­ lishes the first part of the theorem. To prove the second part identify a sequence Ei → 0 such that Δ(Πi ) = Ei /i2 . By assumption, such a sequence exists. By Markov’s inequality, this is 4 bounded by P((Q(Πi ) − T )2 > 2Ei ) ≤ E(Q(Πi ) − T )2 2Δ(Πi )T T ≤ = 2 2Ei 2Ei i (6) Since i iT2 < ∞, then the sum of probabilities in (6) is finite. Then apply­ ing the Borel-Cantelli Lemma, the probability that (Q(Πi ) − T )2 > 2Ei for infinitely many i is zero. Since Ei → 0, this exactly means that almost surely, limi Q(Πi ) = T . 3 Additional reading materials • Sections 6.11 and 6.12 of Resnick’s [1] chapter 6 in the book. References [1] S. Resnick, Adventures in stochastic processes, Birkhuser Boston, Inc., 1992. 5