Question

(1 point) The Wiener process z- W satisfying is obtained by taking the limit as Δι 0, Δι > 0, of a random walk process such that For each t 2 0, e(t) is a random variable with mean 0 and standard deviation 1 This problem is an informal derivation of rules for simplification of expressions involving Δι and ΔΖ To approximate (1) we apply (2) but instead of taking the limit, we consider At as a small positive value Let Zo z(0), a constant (possibly zero) and let Z1 satisfy where 0-e(0) is a random variable with mean 0 and standard deviation 1 Then Z1 approximates z(At)- WAr Any random variable x may be written as a sum of two part:s where x and x have the properties EL] E[x], var[x] - Q The part x contains the "non-random" behavior of x. The part x contains the "truly random" behavior of x E)0 varvar[x] For the random variable ZI-Zo + €ονΔι from (3), we may write Z1 1 + 1 where (type Zo as z0) and ~ -( eps0 ) At (type є0 as epsO )

Answer 1