This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Stochastic Calculus The Normal Distribution Preliminaries: Normal Random Variables Definition: A random variable Z with values in R is said to be normally distributed with mean and variance 2 > 0 if Z has density 1 2 2 p(z) = e (z) /2 . 2 In this case, we write Z N(, 2 ). Z is said to be a standard normal RV if Z N(0, 1). Remark: As 2 0, the distribution of Z concentrates on the point . Thus we say that Z is a degenerate normal RV with mean and variance 0 and write Z N(, 0) if P(Z = ) = 1.
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 1 / 42 Stochastic Calculus The Normal Distribution Properties of Normal Random Variables Affine transformations of normal RVs are normal: if Z N(, 2 ) and W = aZ + b, then W N(a + b, a2 2 ) Sums of independent normal RVs are normally distributed: if Z1 and Z2 are independent and Zi N(i , i2 ), then
2 2 Z1 + Z2 N(1 + 2 , 1 + 2 ). Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 2 / 42 Stochastic Calculus The Normal Distribution Multivariate Normal Random Variables Definition: A random variable Z = (Z1 , , Zd ) with values in Rd is said to be normally distributed if for every vector b Rd , the realvalued random variable W = b Z = b1 Z1 + + bd Zd is normally distributed. The distribution of Z is completely determined by its mean = (1 , , d ) and covariance matrix = (ij ), where i = EZi and we write Z N(, ). and ij = Cov (Zi , Zj ), Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 3 / 42 Stochastic Calculus The Normal Distribution Properties of Multivariate Normal Random Variables If is nonsingular, then Z has a density 1 1 p(z) = exp  (z  )T 1 (z  ) . 2 (2)d det() W N(A + B, AAT ) If Z1 and Z2 are independent and Zi N(i , i ), then Z1 + Z2 N(1 + 2 , 1 + 2 ). If Z N(, ) and W = AZ + B, where A Rnd and B Rn , then Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 4 / 42 Stochastic Calculus The Normal Distribution The Central Limit Theorem The following theorem explains why the normal distribution is of special interest. Theorem: Suppose that X1 , X2 , are independent realvalued RVs with E[Xi ] = and Var(Xi ) = 2 , and let Sn = X1 + + Xn . Then, for every z R,
n lim P Sn  n z n = P(Z z), where Z N(0, 1) is a standard normal random variable.
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 5 / 42 Stochastic Calculus The Normal Distribution Convergence in Distribution Definition: Suppose that Z and Z1 , Z2 , are realvalued RVs with cumulative distribution functions (cdf's) Fn (x) = P(Zn x) and F (x) = P(Z x). We say that the sequence (Zn ) converges in distribution to Z and 1 write Zn Z if Fn (x) F (x) as n at every point x at which F is continuous. CLT: Since the cdf of a nondegenerate normal RV is continuous at all points, the CLT asserts that Sn  n Z N(0, 1). n
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 6 / 42 Stochastic Calculus The Normal Distribution Convergence in Distribution: Alternative Definition Definition: Suppose that Z and Z1 , Z2 , are RVs with values in a metric space (E , d). We say that the sequence (Zn ) converges in 1 distribution to Z and write Zn Z if for every bounded continuous function f : E R,
n lim E[f (Zn )] = E[f (Z )]. Remark: If (E , d) is R with the Euclidean metric, this definition is equivalent to the one on the previous slide. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 7 / 42 Stochastic Calculus The Normal Distribution Multivariate Central Limit Theorem Theorem: Suppose that X1 , X2 , is a sequence of independent Rd valued RVs with mean vector and covariance matrix , and let Sn = X1 + + Xn . Then Zn = n1/2 Sn  n Z where Z N(0, ) is a ddimensional normal RV. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 8 / 42 Stochastic Calculus Brownian Motion Brownian Motion and Random Walks Suppose that X1 , X2 , are independent identicallydistributed realvalued RVs with EX1 = 0 and Var(X1 ) = 1. Given m, n 1, let t = m/n and define
(n) Wt m nt 1 1/2 1 Xk = t Xk n nt k=1 k=1 If we let n with t fixed, the CLT shows that t 1/2 Wt and so Wt
(n) (n) Z N(0, 1) t 1/2 Z N(0, t). Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 9 / 42 Stochastic Calculus Brownian Motion Similarly, if nt, ns are both integers, then Wt+s  Wt
(n) (n) = n(t+s) nt 1 1 Xk  Xk n n k=1 k=1 = 1 Xk n
k=nt+1 1/2 n(t+s) = s 1 ns n(t+s) k=nt+1 Xk s 1/2 Z N(0, s). Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 10 / 42 Stochastic Calculus Brownian Motion Likewise, if t1 t2 , then the random variables
1 1 (n) Wt1 = Xk n nt k=1 and
(n) Wt2  (n) Wt1 1 = n k=nt1 +1 nt2 Xk are independent and so Wt1 , Wt2  Wt1
(n) (n) (n) (Z1 , Z2 ) where Z1 , Z2 are independent mean zero normal random variables with variances t1 and t2  t1 , respectively.
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 11 / 42 Stochastic Calculus Brownian Motion It follows that Wt1 , Wt2
(n) (n) = (Z1 , Z1 + Z2 ) (Wt1 , Wt2 ) (n) (n) (n) (n) Wt1 , Wt1 + (Wt2  Wt1 ) where (Wt1 , Wt2 ) is a bivariate normal random variable with mean vector 0 and covariance matrix given by 11 = Var(Z1 ) = t1 22 = Var(Z1 + Z2 ) = t2 12 = Cov(Z1 , Z1 + Z2 ) = t1 = t1 t2 . Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 12 / 42 Stochastic Calculus Brownian Motion In general, given 0 t1 t2 tm , we have (n) (n) (n) (n) (n) Wt1 , Wt2  Wt1 , , Wtm  Wtm1 (Z1 , Z2 , , Zm ) where Z1 , , Zm are independent mean zero normal RVs and Var(Zi ) = ti  ti1 . Consequently, Wt1 , Wt2 , , Wtm
(n) (n) (n) (Wt1 , Wt2 , , Wtm ) where (Wt1 , , Wtm ) is an mdimensional normal RV with E [Wti ] = 0 Cov Wti , Wtj = ti t j . Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 13 / 42 Stochastic Calculus Brownian Motion Next, let us define a sequence of random continuous functions W (n) : [0, T ] R by piecewise linear interpolation:
(n) Wt t + (1  m+1 = n t . n = (n) t Wm/n (n) t )W(m+1)/n , t m m+1 , n n Here we will treat each random function W (n) as a random variable taking values in the metric space C[0, T ] equipped with the sup norm    sup (t)  (t).
0tT Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 14 / 42 Stochastic Calculus Brownian Motion Donsker's Invariance Principle Theorem: As n , W (n) W where W is a C[0, T ]valued RV with the following properties: W0 = 0. If t0 < < tm , then Wt0 , Wt1  Wt0 , , Wtm  Wtm1 are independent. If t, s 0, Wt+s  Wt N(0, s). Remark: The invariance principle refers to the fact that the limiting distribution does not depend on the distribution of the individual Xi 's. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 15 / 42 Stochastic Calculus Brownian Motion Brownian Motion Definition: A realvalued continuoustime stochastic process B = (Bt ; t 0) is said to be a onedimensional Brownian motion if B has independent increments: if t0 tn , then Bt0 , Bt1  Bt0 , , Btn  Btn1 are independent. B has stationary Gaussian increments: Bt+s  Bt N(0, s). B has continuous sample paths: with probability one, t Bt is a continuous map. If B0 = 0, then B is said to be a standard onedimensional Brownian motion. Brownian motions are also called Wiener processes.
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 16 / 42 Stochastic Calculus Brownian Motion Some Properties of Brownian Motion Path regularity: With probability one, the map t Bt is not Lipschitz continuous (and therefore nondifferentiable): Bt+h  Bt < = 0. P t 0 : lim sup h h0 Thus Brownian paths are far from smooth. However, it can be shown that they have finite quadratic variation: sup
P n i=1 Bti  Bti1 2 t a.s. where the sup is taken over all partitions 0 t1 tn = t of [0, t].
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 17 / 42 Stochastic Calculus Brownian Motion Markov property: B is a Markov process: for all t, s 0 and every Borel set A R, P Bt+s A(Bu ; 0 u t) = P Bt+s ABt In other words, conditional on Bt , the future values Bt+s are independent of the past values Bu for u < t. Furthermore, the transition probabilities of Brownian motion are known explicitly: P Bt+s ABt = x = P x + (Bt+s  Bt ) A 1 2 e (y x) /2s dy . = A s 2 Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 18 / 42 Stochastic Calculus Brownian Motion Connection with the heat equation: The transition density of Brownian motion 1 2 p(t; x, y ) = e (xy ) /2t t 2 is the fundamental solution to the heat equation: t p(t; x, y ) = 1 yy p(t; x, y ) 2 p(0; x, y ) = x (y ). Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 19 / 42 Stochastic Calculus Brownian Motion Infinitesimal Generators Definition: Suppose that X = (Xt ; t 0) is a continuoustime Markov process with values in a metric space (E , d). The infinitesimal generator of X is the operator G defined by the limit Gf (x) = lim 1 Ex [f (Xt )]  f (x) . t0 t The domain of G is the subset of Cb (E ) containing those functions f for which the above limit exists for every x E and convergence is uniform over E . Notation: We write Ex [Xt ] for E[Xt X0 = x]. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 20 / 42 Stochastic Calculus Brownian Motion Infinitesimal Generator of Brownian Motion Let f be a compactlysupported function with three continuous derivatives and let Z N(0, 1). Then Gf (x) = lim = = = = 1 Ex [f (Bt )]  f (x) t0 t 1 lim E[f (x + tZ )]  f (x) t0 t 1 1 2 lim f (x + tz)  f (x) e z /2 dz t0 t 2  1 1 1 2 2 lim f (x) tz + f (x)tz + tR(t; z) e z /2 dz t0 t 2 2  1 f (x). 2
APM 530  Lecture 9 Fall 2010 21 / 42 Jay Taylor (ASU) Stochastic Calculus Brownian Motion Suppose that X = (Xt ; t 0) is a continuoustime Markov process with values in (E , d) and generator G . If f Cb (E ) and u(t; x) Ex [f (Xt )] then u is a solution to the following initial value problem: t u(t; x) = Gu(t; x) u(0; x) = f (x). Example: If Xt = Bt is Brownian motion, then u(t; x) = Ex [f (Bt )] solves the heat equation 1 ut (t; x) = uxx (t; x) 2 with initial data u(0; x) = f (x). Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 22 / 42 Stochastic Calculus Diffusion Processes Stochastic Difference Equations Suppose that b, a : R R are differentiable functions with a 0 and let tN be a sequence of positive numbers that tends to 0. Consider the following stochastic difference equation: ~ (N) = Xn + b Xn ~ (N) ~ (N) tN + a Xn ~ (N) tN Zn , Xn+1 where Z1 , Z2 , is a sequence of identicallydistributed random variables with E[Z1 ] = 0 and Var(Z1 ) = 1. ~ Interpretation: We can think of X (N) as a biased random walk in which both the mean and the variance of each step depend on the current location of the process.
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 23 / 42 ~ (N) = x0 X0 Stochastic Calculus Diffusion Processes We can again use linear interpolation of the solution to the difference equation to construct a sequence of continuoustime processes X (N) :
(N) Xt t ~ (N) ~ (N) = t Xm + (1  t ) Xm+1 , m+1 = N t . N t m m+1 , N N In particular, by fixing T > 0, we can regard X (N) as a pathvalued random variable with values in C[0, T ]. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 24 / 42 Stochastic Calculus Diffusion Processes We are interested in the limiting behavior of X (N) as tN 0. We first consider two special cases. If a 0, then X (N) is just the Euler approximation to the solution of the initial value problem: X = b(X ) with X (0) = x0 and so X (N) converges uniformly to X on compact intervals [0, T ]. If b(x) 0 and a(x) 2 , then X (N) x0 + B, where B is a standard Brownian motion. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 25 / 42 Stochastic Calculus Diffusion Processes Our next theorem can be seen as an extension of Donsker's invariance principle for stochastic difference equations. Theorem: There is a Markov process X with continuous sample paths such that X (N) X in C[0, T ]. Furthermore, the generator of X is the secondorder differential operator 1 Gf (x) = a(x)f (x) + b(x)f (x) 2 and the transition density of X is the fundamental solution to the following parabolic partial differential equation: t p(t; x, y ) = 1 yy a(y )p(t; x, y )  y b(y )p(t; x, y ) 2 p(0; x, y ) = x (y ). Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 26 / 42 Stochastic Calculus Diffusion Processes Diffusion Processes Definition: A realvalued continuoustime Markov process X = (Xt ; t 0) is said to be a diffusion process if X has continuous sample paths. The infinitesimal generator of X is a secondorder differential operator of the form 1 Gf (x) = a(x)f (x) + b(x)f (x). 2 Example: Brownian motion is a diffusion process. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 27 / 42 Stochastic Calculus Diffusion Processes The functions b(x) and a(x) are the infinitesimal drift and infinitesimal variance coefficients of X and can be calculated using the formulas: 1 b(x) = lim Ex Xt  x t0 t 1 a(x) = lim Ex (Xt  x)2 t0 t Furthermore, Ex Xt+t  Xt = b(x)t + o(t) Varx Xt+t  Xt = a(x)t + o(t) E (Xt+t  Xt )n = o(t), n 3.
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 28 / 42 Stochastic Calculus Stochastic Differential Equations Stochastic Differential Equations Informally, we can think of the diffusion process X as a solution to a stochastic differential equation Xt = b(Xt ) + where Wt is a white noise process a(Xt )Wt , EWt = 0 Cov(Wt , Ws ) = 0 (t  s). Remark: Although we would like to interpret Wt as the time derivative of a Brownian motion W , we know that W is nowhere differentiable. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 29 / 42 Stochastic Calculus Stochastic Differential Equations In fact, neither X nor W is differentiable. To get around this, rigorous treatments of stochastic calculus reformulate the SDE as a stochastic integral equation Xt = X0 + t 0 t b(Xs )ds + a(Xs )dWs .
0 However, this approach requires us to define the stochastic integral t Ys dWs 0 for a suitable class of processes Y = (Yt ; t 0). Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 30 / 42 Stochastic Calculus Stochastic Differential Equations Stochastic integrals can be defined in several ways. Here we briefly consider one approach that leads to the It^ integral. o Step 1: We will say that Y is a simple process if there are numbers 0 = t0 < t1 < < tn = T and bounded random variables 0 , , n such that Yt =
n1 k=0 k 1(ti ,ti+1 ] (t), and each variable k is measurable with respect to (Wu ; 0 u tk ) and some algebra independent of W . Interpretation: Y is piecewise constant and Yt does not depend on the increments Wt+s  Wt for s > 0.
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 31 / 42 Stochastic Calculus Stochastic Differential Equations Step 2: Suppose that Y is a simple process given by Yt = We then define T n1 k=0 k 1(ti ,ti+1 ] (t). Yt dWt = 0 n1 k=0 k (Wtk+1  Wtk ). Remark: Since Ytk = k , we are using the left endpoint approximation. Using (Ytk + Ytk+1 )/2 instead will lead to the Stratonovich integral. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 32 / 42 Stochastic Calculus Stochastic Differential Equations Step 3: If Y = (Yt ; t 0) is a process with rightcontinuous sample paths and Y (N) is a sequence of simple processes that converge to Y in the sense T (N) lim E (Yt  Yt )2 dt = 0,
N 0 then it can be shown that the sequence of stochastic integrals T 0 Yt (N) dWt is a Cauchy sequence in the space L2 (, F, P) of squareintegrable RVs. We then define the stochastic integral of Y with respect to W on [0, T ] to be the random variable Jay Taylor (ASU) T 0 Yt dWt L2  N 0 lim T Yt (N) dWt . APM 530  Lecture 9 Fall 2010 33 / 42 Stochastic Calculus Stochastic Differential Equations Properties of the It^ Integral o If Y is integrable with respect to W , then T E Ys dWs = 0 E 0 T Ys dWs 0 2 = T 0 2 It^'s formula: If F Cb , then o E Ys2 ds. T T 0 1 F (Ws )dWs = F (WT )  F (W0 )  2 F (s)ds. 0 Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 34 / 42 Stochastic Calculus Stochastic Differential Equations Example: The OrnsteinUhlenbeck Process The linear SDE dXt = Xt dt + dWt can be solved explicitly by using the integrating factor e t . The solution is known as an OrnsteinUhlenbeck process and can be expressed as a stochastic integral: t t Xt = X0 e + e (ts) dWs 0 t = X0 e t + Wt  e (ts) Ws ds.
0 If X0 = x0 , then X is also a Gaussian process.
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 35 / 42 Stochastic Calculus Stochastic Differential Equations The EulerMaruyama Scheme The EulerMaruyama method is one of the most popular schemes for numerically solving an SDE. Consider the following problem: dXt = b(Xt )dt + a(Xt )dWt . To solve this using the EulerMaruyama scheme, we first solve the stochastic difference equation, ~ ~ ~ Xn+1 = Xn + b(Xn )t + ~ a(Xn )t Zn , where Z1 , Z2 , is a sequence of IID standard normal RVs, and then set ~ Xnt = Xn .
Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 36 / 42 Stochastic Calculus Multidimensional Diffusions Multidimensional Brownian Motion Definition: An Rd valued process B = (Bt ; t 0) is said to be a ddimensional Brownian motion if B has independent increments: if t0 < < tn , then Bt0 , Bt1  Bt0 , , Btn  Btn1 are independent. B has stationary Gaussian increments: Bt+s  Bt N(0, sId ), where Id is the d d identity matrix. B has continuous sample paths. In particular, if B 1 , , B d are d independent onedimensional Brownian motions, then B = (B 1 , , B d ) is a ddimensional Brownian motion. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 37 / 42 Stochastic Calculus Multidimensional Diffusions Multidimensional Diffusion Processes Definition: A Rd valued continuoustime Markov process X = (Xt ; t 0) is said to be a diffusion process if The map t Xt is continuous with probability one. The infinitesimal generator of X is a secondorder partial differential operator Gf (x) =
d d 1 aij (x)xi xj f (x) + bi (x)xi f (x), 2 i,j=1 i=1 where for each x, a(x) = (aij (x)) is a positive semidefinite matrix. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 38 / 42 Stochastic Calculus Multidimensional Diffusions Characterization of Multidimensional Diffusions The infinitesimal drift and covariance coefficients can be calculated from 1 bi (x) = lim Ex Xti  x i t0 t 1 aij (x) = lim Ex Xti  x i Xtj  x j . t0 t The transition density p(t; x, y ) is the fundamental solution to the following parabolic PDE: t p(t; x, y ) = p(0; x, y ) = x (y ).
Jay Taylor (ASU) d d 1 yi yj aij (x)p(t; x, y )  yi bi (y )p(t; x, y ) 2 i,j=1 i=1 APM 530  Lecture 9 Fall 2010 39 / 42 Stochastic Calculus Multidimensional Diffusions We can also obtain the diffusion X as the limit (in distribution) as t 0 of the solutions to a sequence of stochastic difference equations X0 = x0 Xn+1 = Xn + b(Xn )t + (Xn )(t)1/2 Zn . Here (x) is any d d matrix such that (x)(x)T = A(x) and Zn is any sequence of independent Rd valued random variables with mean 0 and covariance matrix Id . We get the EulerMaruyama scheme by taking Zn N(0, Id ). Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 40 / 42 Stochastic Calculus Multidimensional Diffusions Multivariate SDEs The diffusion process X can also be expressed as the solution to a multivariate stochastic differential equation dXt = b(Xt )dt + (Xt )dWt or, equivalently, t Xt = X0 + b(Xs )ds + 0 t (Xs )dWs 0 where W = (Wt ; t 0) is a ddimensional Brownian motion. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 41 / 42 Stochastic Calculus References References Fristedt, B. and Gray, L. (1997) A Modern Approach to Probability. Birkhauser. Karatzas, I. and Shreve, S. E. (1991) Brownian Motion and Stochastic Calculus. Springer. Revusz, D. and Yor, M. (1999) Continuous Martingales and Brownian Motion. Springer. Jay Taylor (ASU) APM 530  Lecture 9 Fall 2010 42 / 42 ...
View
Full
Document
 Fall '10
 Staff

Click to edit the document details