Outline Brownian Process Continuity of Sample Paths Differentiability of Sample Paths Simulating Sample Paths Hitting times and Maximum

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1 Normal Distribution and Brownian Process Page 1 Outline Brownian Process Continuity of Sample Paths Differentiability of Sample Paths Simulating Sample Paths Hitting times and Maximum

2 Searching for a Continuous-time Process Page 2 Consider a process {B t : t 0} with standard deviation parameter σ and properties i) B(0) = 0, ii) B(t) has independent increments, iii) B t Normal(0, σ 2 t) for each t, i.e., 1 x2 f B t (x) = exp 2π σ 2 t 2σ 2 t Ex: Properties i), ii), iii) imply stationary increments. B t Normal(0, σ 2 t) and B s Normal(0, σ 2 t), and their moment generating functions respectively are m B t x = exp(σ 2 tx 2 /2) and m B s x = exp(σ 2 sx 2 /2). Since B t = B s + B t B s and B s B t B s for s t, m B t x = m B s x m B t B(s) x So m B t B(s) x = m B t x m B s x = exp(σ2 tx 2 /2) exp(σ 2 sx 2 /2) = exp σ2 t s x 2 2 B(t) This implies B t B(s) Normal 0, σ 2 t s, which depends only on t s but not separately on t or s. Use the process B(t) to represent particle or price movements over time. Set the current position of the process at origin: B(t = 0) = 0. After s time, the process arrives at B(s).» Assume: the future B(t) for t > s is independent of the past B(s), so B(t) has independent increments.» Do not particles/prices have momentum/trend? Location matters as well as the velocity/trend? Assume the process is as likely to go up as to come down, and it is likely to go further away from the origin in more time, so E(B(t)) = 0 and V(B(t))/t is constant. These are all reasonable but do not lead us to Normal distribution.

3 Linking to Normal Distribution via Random Walk: A Discrete-time Process Page 3 X n is called a random walk if X 0 = 0 and X n has increments X n+1 X n = σ with probability 1/2 σ with probability 1/2 Are the increments independent and stationary? Yes. E X n+1 X n = 0 and V X n+1 X n = σ 2 /2 + σ 2 /2 = σ 2. {Y i = X i X i 1 : i 1} is an iid sequence with mean E(Y i ) = 0 and variance V(Y i ) = σ 2. E X n = 0 and V X n = σ 2 n. n X n = i=1 X i X i 1 d Normal(0, σ 2 n) as n. For fixed t [nε, n + 1 ε), set of n = t ε, then X t ε x x as x. d Normal 0, σ 2 X t ε d Normal 0, σ 2 t ε as t. ε ε X t d Normal 0, σ 2 t as ε 0. ε Define continuous-time process B rw (t) as the limit of the random walk B rw t : = lim ε X t ε 0 ε B rw t Normal 0, σ 2 t. B rw (t) satisfies i), ii) and iii) listed on the previous page. t ε as t ε. If the process is viewed as sum of an iid sequence, central limit theorem can be invoked. This reasoning leads to Normal distribution.

4 Any Other Property - Continuity To Deduce or To Assume Page 4 Properties i), ii) and iii) do not ensure the continuity of the process. Does B(t + h) converge to B(t) almost surely? No. Example of making up a discontinuous process that satisfies i), ii) and iii). Let B(t) satisfy i), ii) and iii). Define a new process B (t) as B t except at s B s = 1 2 B s + 1 Normal 0, σ 2 s if Normal 0, σ 2 s is rational 2 0 otherwise Rational numbers are countable so P Normal 0, σ 2 s is rational = 0. B(t) and B (t) are the same almost surely. B (t) has properties i), ii) and iii) but is discontinuous at s. For a fixed sample point ω Ω, B(t; ω) is a function tracing the path of the process as time passes. B(t; ω) is called the sample path of the process B(t). Continuity of the sample path does nor follow from properties i), ii) and iii). Consider a process {B t : t 0} with standard deviation parameter σ and properties i), ii), iii) and iv) B(0; ω) = 0 and B(t; ω) is continuous for each ω Ω 0 such that P Ω 0 = 1. Existence of a process satisfying some properties is an issue. But it is already resolved for i), ii), iii), iv). There exists a process satisfying i), ii), iii), iv) and is called Brownian Process. Note this is the first reference to Brownian Process.

5 Any Other Property - Differentiability To Deduce or To Assume or To Refute Ex: For a given c > 0, B c (t) = B c2 t c is a derived process starting from Brownian process B(t). Does B c (t) satisfy i), ii), iii), iv)? i) B c (0) = B(0)/c = 0. ii) For r < s < t, B c (t) B c (s) = B(c 2 t)/c B(c 2 s)/c which is independent of B(c 2 r)/c = B c (r), i.e., B c (t) B c (s) B c (r). iii) B c (t) has the same distribution as B(c 2 t)/c Normal(0, c 2 tσ 2 )/c = Normal 0, tσ 2 B(t). iv) B(c 2 t; ω) is continuous almost surely and then B c (t; ω) = B(c 2 t; ω)/c is also continuous. Composition of continuous functions is continuous. Hence, B c (t) is a Brownian process with standard deviation parameter σ. B(t) and B c (t) are two versions of the same Brownian process; They are identical probabilistically. For a given c > 0, B nc t : t 0 : n = 1,2, is a sequence of identical Brownian processes B 2c t = B c c 2 t = B c4 t c c 2, B 3c t = B 2c c 2 t = B c6 t c c 3,, B nc t = B c2n t c n B nc is obtained by contracting time by a factor of c 2n and magnitude by c n. Ex: For c = n = 2, we have B 4 t = B(16t)/4 Slope of the derived process is 4 times the slope of the original process Slope of the derived = B 4 t B 16t B 16t = = 4 = 4 Slope of the original t 4t 16t Slope increases in the derived process by 4 = 2 2 = c n. Page 5

6 Differentiability Refuted Page 6 B 8c B 4c B 2c B(t) 0 time c = 2 B nc t has a slope that is c n times of the slope of B t. The slope of B nc around 0 is increasing without a bound as n increases B nc (t) is not differentiable around 0; B t is not differentiable around 0. Because of stationary increments, B t is not differentiable anywhere. Continuous but non-differentiable sample paths to model Particle movements» Stock prices

7 Simulation over [0, T] Page 7 Sampling iid sequence {Z i } of standard normal random variables Generate random walk X n X 0 = 0 and X n = n i=1 Z i for 1 n N σ X n N has Normal 0, σ2 n N σ X n N and B n N distribution has the same distribution Fill in the gaps between n n+1 and by N N interpolation, justified on the next page Hence, B(t) is defined for t n N. n+1 N. To simulate over [0,2], simulate [0,1] twice and concatenate sample paths

8 Interpolation Page 8 B(t)=v B(t)-B(s)=v-x v-u B(s)=x B(s)-B(r)=x-u B(r)=u (v-u)/(t-r) E(B(s) B(r)=u, B(t)=v) u+(s-r)(v-u)/(t-r) (s-r)(v-u)/(t-r) u r s r s t s t? v u t r s r t r E B s B r = u, B t = v = Brownian paths are self-similar s r t s v u v + u = u + (s r) t r t r t r

9 Hitting Times For a given constant a, the first time Brownian process reaches a is called hitting time τ a. Page 9 τ a B t Let us search for the cdf P(τ a t) of hitting time for a > 0 First note from conditioning that P(B t a) = P(B t a τ a t) P(τ a t) + P(B t a τ a > t)p(τ a > t) = P B t a τ a t P τ a t Because P B t a τ a > t = 0 Let us focus on the first term P B t a τ a t = P(B t a B(τ a ) = a, τ a t) = P(B t τ a a a = 0 B(τ a ) = a, τ a t) = 1 2» The first equality is from definition of τ a, the second is from stationary and independent increments, the third is from symmetry of Brownian process. These yield P τ a t = 2 P(B t a). Since B t Normal(0, σ 2 t), P τ a t = 2 P Normal 0, σ 2 t a = 2 P Normal 0,1 a σ t Because of the symmetry, P τ a t = P(τ a t). Hence, for a < 0 P τ a t = P τ a t = 2 P Normal 0, σ 2 t a In general, regardless of positive or negative a, P τ a t = 2 P Normal 0, σ 2 t a = 2 2π a/(σ t) exp x2 2 dx

10 Maximum of the Process Page 10 max B s s 0,t B t The maximum of the process over [0,t] is max s 0,t P max s 0,t B s a = P τ a t = 2 2π a/(σ t) B(s) and is nonnegative. Its tail probability is exp x2 2 dx. max s 0,t implies τ a t τ a t implies max B s a s 0,t Two events above are equivalent

11 Summary Page 11 Brownian Process Continuity of Sample Paths Differentiability of Sample Paths Simulating Sample Paths Hitting times and Maximum

12 Counting, Measurability, Independence Random Vectors Conditioning Inequalities & Limits Classifying & Ordering utdallas Summary of the Course Page 12 Probability and Stochastic Processes Uncountable Range Continuous Random Variables Brownian Process Continuous State Space Countable Range Discrete Random Variables Probability Defining Manipulating Markov Chains Discrete time Poisson Process Continuous time Stochastic Processes Indexing Discrete State Space