An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2-3 Haijun Li An Introduction to Stochastic Calculus Week 2-3 1 / 24
Outline 1 Brownian Motion Basic Properties Processes Related to Brownian Motion 2 Simulation of Brownian Sample Paths Simulation via the Functional Central Limit Theorem Simulation via Series Representations Haijun Li An Introduction to Stochastic Calculus Week 2-3 2 / 24
Definition A stochastic process B = (B t, t 0) is called a standard Brownian motion or a Wiener process if B 0 = 0, it has stationary, independent increments, for every t > 0, B t has a normal N(0, t) distribution, and it has continuous sample paths. Haijun Li An Introduction to Stochastic Calculus Week 2-3 3 / 24
3-D Brownian Motion Haijun Li An Introduction to Stochastic Calculus Week 2-3 4 / 24
A Fascinating History Brownian motion is named after the botanist Robert Brown who first observed, in the 1820s, the irregular motion of pollen grains immersed in water. By the end of the nineteenth century, the phenomenon was understood by means of kinetic theory as a result of molecular bombardment. In 1900, Louis Bachelier had employed it to model the stock market, where the analogue of molecular bombardment is the interplay of the myriad of individual market decisions that determine the market price. The construction in Bachelier s PhD thesis was in error but captured many properties of the process. In 1905, Albert Einstein, although unaware of the phenomenon and of previous research, predicted the existence of Brownian motion from purely theoretical consideration. Norbert Wiener (1923) was the first to put Brownian motion on a firm mathematical basis. Haijun Li An Introduction to Stochastic Calculus Week 2-3 5 / 24
Distributional Properties of Brownian Motion For any t > s, B t B s d = Bt s B 0 = B t s has an N(0, t s) distribution. That is, the larger the interval, the larger the fluctuations of B on this interval. µ B (t) := E(B t ) = 0, and for any t > s, 0 {}}{ C B (t, s) := E(B t B s ) E(B t )E(B s ) = E[((B t B s ) + B s )B s ] = E[(B t B s )B s ] + E(Bs 2 ) = E(B t B s )E(B s ) +s = min(s, t). }{{} 0 Brownian motion is a Gaussian process: its finite-dimensional distributions are multivariate Gaussian. Question: How irregular are Brownian sample paths? Haijun Li An Introduction to Stochastic Calculus Week 2-3 6 / 24
Self-Similarity A stochastic process X = (X t, t 0) is H-self-similar for some H > 0 if it satisfies the condition (s H X t1,..., s H X tn ) d = (X st1,..., X stn ) for every s > 0 and any choice of t i 0, i = 1,..., n. Self-similarity means that the properly scaled patterns of a sample path in any small or large time interval have a similar shape. Non-Differentiability of Self-Similar Processes For any H-self-similar process X with stationary increments and 0 < H < 1, X t X t0 lim sup =, at any fixed t 0. t t 0 t t 0 That is, sample paths of H-self-similar processes are nowhere differentiable with probability 1. Haijun Li An Introduction to Stochastic Calculus Week 2-3 7 / 24
Ethernet traffic burstiness at different time scales Haijun Li An Introduction to Stochastic Calculus Week 2-3 8 / 24
Path Properties of Brownian Motion Brownian motion is 0.5-self-similar. Its sample paths are nowhere differentiable. That is, any sample path changes its shape in the neighborhood of any time epoch in a completely non-predictable fashion (Wiener, Paley and Zygmund, 1930s). Unbounded Variation of Brownian Sample Paths n sup B ti (ω) B ti 1 (ω) =, for almost all ω, τ i=1 where the supremum is taken over all possible partitions τ : 0 = t 0 < < t n = T of any finite interval [0, T ]. The unbounded variation and non-differentiability of Brownian sample paths are major reasons for the failure of classical integration methods, when applied to these paths, and for the introduction of stochastic calculus. Haijun Li An Introduction to Stochastic Calculus Week 2-3 9 / 24
Brownian Bridge Let B = (B t, t 0) denote Brownian Motion. The process X = (X t, 0 t 1) defined by X t := B t tb 1, 0 t 1 satisfies that X 0 = X 1 = 0. This process is called the (standard) Brownian bridge. Since multivariate normal distributions are closed under linear transforms, the finite-dimensional distributions of X are Gaussian. The Brownian bridge is characterized by two functions µ X (t) = 0 and C X (t, s) = min(t, s) ts, for all s, t [0, 1]. The Brownian bridge appears as the limit process of the normalized empirical distribution function of a sample of iid uniform U(0, 1) random variables. Haijun Li An Introduction to Stochastic Calculus Week 2-3 10 / 24
Brownian Bridge Haijun Li An Introduction to Stochastic Calculus Week 2-3 11 / 24
Brownian Motion with Drift Let B = (B t, t 0) denote Brownian Motion. The process X := (µt + σb t, t 0), for constants σ > 0 and µ R, is called Brownian motion with (linear) drift. X is a Gaussian process with expectation and covariance functions µ X (t) = µt, C X (t, s) = σ 2 min(t, s), s, t 0. Haijun Li An Introduction to Stochastic Calculus Week 2-3 12 / 24
Brownian Motion with Drift Haijun Li An Introduction to Stochastic Calculus Week 2-3 13 / 24
Geometric Brownian Motion The process X = (e µt+σb t, t 0), for constants σ > 0 and µ R, is called geometric Brownian motion. Since E(e tz ) = e t2 /2 for an N(0, 1) random variable Z, it follows from the self-similarity of Brownian motion that µ X (t) = e µt E(e σb t ) = e µt E(e σt1/2 B 1 ) = e (µ+0.5σ2 )t. Since B t B s and B s are independent for any s t, and B t B s d = Bt s, then C X (t, s) = e (µ+0.5σ2 )(t+s) (e σ2t 1). In particular, σ 2 X (t) = e(2µ+σ2 )t (e σ2t 1). Geometric Brownian Motion is used to model stock prices in the Black-Scholes model (Black, Scholes and Merton 1973) and is the most widely used model of stock price behavior. Haijun Li An Introduction to Stochastic Calculus Week 2-3 14 / 24
Geometric Brownian Motion Figure : Two Sample Paths Haijun Li An Introduction to Stochastic Calculus Week 2-3 15 / 24
Central Limit Theorem Consider a sequence {Y 1, Y 2,..., } of iid non-degenerate random variables with mean µ Y = EY 1 and variance σ 2 Y = var(y 1) > 0. Define the partial sums: R 0 := 0, R n := n i=1 Y i, n 1. Central Limit Theorem (CLT) If Y 1 has finite variance, then the sequence (R n ) obeys the CLT via the following uniform convergence: ( ) Rn E(R n ) P [var(r n )] 1/2 x Φ(x) 0, as n, sup x R where Φ(x) denotes the distribution of the standard normal distribution. That is, for large sample size n, the distribution of [R n E(R n )]/[var(r n )] 1/2 is approximately standard normal. Haijun Li An Introduction to Stochastic Calculus Week 2-3 16 / 24
Functional Approximation Let (Y i ) be a sequence of iid random variables with mean µ Y = EY 1 and variance σy 2 = var(y 1) > 0. Consider the process S n = (S n (t), t [0, 1]) with continuous sample paths on [0, 1], { (σ 2 S n (t) = Y n) 1/2 (R i i µ Y ), if t = i/n, i = 0,..., n linearly interpolated, otherwise. Example: If Y i s are iid N(0, 1), consider the restriction of the process S n on the points i/n: S n (i/n) = n 1/2 ik=1 Y k, i = 0,..., n. S n (0) = 0. S n has independent increments: for any 0 i 1 i m n, S n (i 2 /n) S n (i 1 /n),..., S n (i m /n) S n (i m 1 /n) are independent. For any 0 i n, S n (i/n) has a normal N(0, i/n) distribution. S n and Brownian motion B on [0, 1], when restricted to the points i/n, have very much the same properties. Haijun Li An Introduction to Stochastic Calculus Week 2-3 17 / 24
46 CHAPTER 1. Sample Paths of the Process S n, n = 2,..., 9. 0.0 0.2 0.4 0.6 0.8 1.O t Figure 1.3.11 Sample paths of the process S, for one sequence of realizations YI(w), Figure : 9 realizations of Y 1 (ω),..., Y 9 (ω)...., Yg(w) and n = 2,...,9. Haijun Li An Introduction to Stochastic Calculus Week 2-3 18 / 24
Functional Central Limit Theorem Let C[0, 1] denote the space of all continuous functions defined on [0, 1]. Donsker s Theorem If Y 1 has finite variance, then the process S n obeys the functional CLT: Eφ(S n ) Eφ(B), as n, for all bounded continuous functions φ : C[0, 1] R, where B is the Brownian motion on [0, 1]. The finite-dimensional distributions of S n converge to the corresponding finite-dimensional distributions of B: As n, P(S n (t 1 ) x 1,..., S n (t m ) x m ) P(B t1 x 1,..., B tm x m ), for all possible t i [0, 1], x i R. The max functional max 0 i n S n (t i ) converges in distribution to max 0 t 1 B t as n. Haijun Li An Introduction to Stochastic Calculus Week 2-3 19 / 24
Simulating a Brownian Sample Path Plot the paths of the processes S n, for sufficiently large n, and get a reasonable approximation to Brownian sample paths. Since Brownian motion appears as a distributional limit, completely different graphs for different values of n may appear for the same sequence of realizations Y i (ω)s. Simulating a Brownian Sample Path on [0, T ] Simulate one path of S n on [0, 1], then scale the time interval by the factor T and the sample path by the factor T 1/2. Haijun Li An Introduction to Stochastic Calculus Week 2-3 20 / 24
Figure 1.3.11 Sample paths of the process S, for one sequence of realizations Sample YI(w),..., Paths Yg(w) and of n = the 2,... Process,9. S n, for different n 3: n= 1000 0.0 0.2 0.4 0.6 0.8 1.o t Figure 1.3.12 Sample paths of the process S, for different n and the same sequence of realizations Figure YI (w),.. :. Realizations, YLOO,OOO (w). of Y 1 (ω),..., Y 100,000 (ω). Haijun Li An Introduction to Stochastic Calculus Week 2-3 21 / 24
Lévy-Ciesielski Representation Since Brownian sample paths are continuous functions, we can try to expand them in a series. However, the paths are random functions: for different ω, we obtain different path functions. This means that the coefficients of this series are random variables. Since the process is Gaussian, the coefficients must be Gaussian as well. Lévy-Ciesielski Expansion Brownian motion on [0, 1] can be represented in the form B t (ω) = n=1 t Z n (ω) φ n (x)dx, t [0, 1], 0 where Z n s are iid N(0, 1) random variables and (φ n ) is a complete orthonormal function system on [0, 1]. Haijun Li An Introduction to Stochastic Calculus Week 2-3 22 / 24
Paley-Wiener Representation There are infinitely many possible representations of Brownian motion. Let (Z n, n 0) be a sequence of iid N(0,1) random variables, then t B t (ω) = Z 0 (ω) (2π) 1/2 + 2 π 1/2 n=1 Z n (ω) sin(nt/2), t [0, 2π]. n This series converges for every t, and uniformly for t [0, 2π]. Simulating a Brownian Path via Paley-Wiener Expansion Calculate Z 0 (ω) t j (2π) 1/2 + 2 π 1/2 M n=1 Z n (ω) sin(nt j/2) n, t j = 2πj, for 0 j N. N The problem of choosing the right values for M and N is similar to the choice of the sample size n in the functional CLT. Haijun Li An Introduction to Stochastic Calculus Week 2-3 23 / 24
1 1 2 3 4 5 6 Paley-Wiener Expansion, N = 1, 000 I 0 1 2 3 1 5 6 I 1 1 2 3 4 5 6 1 0 1 2 3 4 5 6 I Figure 1.3.14 Simulation of one Brownian sample path from the discretization Figure : Left: M = 100 Right: M = 800 (1.24) of the Paley-Wiener representation with N = 1,000. Top left: all paths for M = 2,...,40. Top right: the path only for M = 40. Bottom left: A4 = 100. Bottom right: M = 800. Haijun Li An Introduction to Stochastic Calculus Week 2-3 24 / 24