Theory - 2023.2 English

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2023.2 English

Suppose \(W\) is an 1-dimensional Brownian motion, \(a,b \in \mathbb{R}, 0 < s < t < u\), then with known \((W_{u}, W_{s})\), the conditional probability distribution is as follows:

\[W_{t} | (W_{u}=b, W_{s}=a) \sim \mathcal N (\dfrac{(u-t)a+(t-s)b}{u-s},\dfrac{(u-t)(t-s)}{u-s})\]

The backward simulation method for Brownian bridge is to generate a sequence between \(a\) and \(b\). Suppose we have constructed \(k\) points \(W_{0}, W_{t_{1}}, ..., W_{t_{k-2}}, W_{T}\), we want to generate another point at time \(s\), \(t_{i} < s < t_{i+1}\). According to Markov and independent increments property of Brownian motion, we have

\[W_{s} | (W_{0}=a,...,W_{t_{i}}=x,W_{t_{i+1}}=y,...,W_{T}=b) \sim W_{s} | (W_{t_i} = x, W_{t_{i+1}}=y)\]


\[W_{s} | (W_{t_i} = x, W_{t_{i+1}}=y) \sim \mathcal N (\dfrac{(t_{i+1}-s)x+(s-t_{i})y}{t_{i+1}-t_{i}},\dfrac{(t_{i+1}-s)(t_{i+}-t_{i})}{t_{i+1}-s})\]