Since the coin tosses are independent, we conclude that W(t) has independent increments. That is, for...

 

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Since the coin tosses are independent, we conclude that W(t) has independent increments. That is, for all 0 t t S S tn the random variable W(t)-W(t). W(t3)-W(t). W(t) - W(t-1) (P2) are independent. a) Let 0 t = 8n t = 8n. Fid E(W(t) -W(t)) and Var(W(t) -W(t)). b) Apply Central Limit Theorem to show that W(t)-W(t)-N(0.t-t) (P3) The stochastic process W(t) is called standard Brownian motion. A stochastic process W = {W(t),t 0) is said to be a standard Brownian motion if it satisfies the properties (P1), (P2), (P3) and with continuous function oft (P4). Let W = [W(t), t20) be a standard Brownian motion. Show that X = Brownian motion for any c> 0 and 20. = {cB, c20}is a Since the coin tosses are independent, we conclude that W(t) has independent increments. That is, for all 0 t t S S tn the random variable W(t)-W(t). W(t3)-W(t). W(t) - W(t-1) (P2) are independent. a) Let 0 t = 8n t = 8n. Fid E(W(t) -W(t)) and Var(W(t) -W(t)). b) Apply Central Limit Theorem to show that W(t)-W(t)-N(0.t-t) (P3) The stochastic process W(t) is called standard Brownian motion. A stochastic process W = {W(t),t 0) is said to be a standard Brownian motion if it satisfies the properties (P1), (P2), (P3) and with continuous function oft (P4). Let W = [W(t), t20) be a standard Brownian motion. Show that X = Brownian motion for any c> 0 and 20. = {cB, c20}is a

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a Since Wt is a standard Brownian motion we know that EWt Wt 0 and VarWt Wt t t b By the Central Lim... View the full answer