Random WALK, BROWNIAN MOTION and SDEs Continuation…

Properties of Continuous Brownian Motion For each 𝑡, 𝐵 𝑡 is normally distributed with mean 0 and variance 𝑡. For each 𝑡 1 < 𝑡 2 , the normal random variable 𝐵 𝑡 2 − 𝐵 𝑡 1 is independent of the random variable 𝐵 𝑡 1 , and in fact independent of all 𝐵 𝑠 , 0≤𝑠≤𝑡 1 . Brownian motion 𝐵 𝑡 can be represented by continuous paths.

Using normal Random Number Starting from 𝐵 0 =0, 𝐵 𝑡 1 can be made by choosing from the normal distribution 𝑁 0, 𝑡 1 = 𝑡 1 − 𝑡 0 𝑁(0,1). In general, 𝑩 𝒕 𝒌+𝟏 = 𝑩 𝒕 𝒌 + 𝒕 𝒌+𝟏 − 𝒕 𝒌 𝑵(𝟎,𝟏).* Note that 𝐵 𝑡 𝑘+1 − 𝐵 𝑡 𝑘 = 𝑡 𝑘+1 − 𝑡 𝑘 𝑁(0,1). *We can use this to discretize the continuous Brownian motion.