Conditional Brownian Motion Between Observed Time Points

Summary

The document asks how to derive the conditional distribution of a Brownian motion value between two observed times. Given the values at the interval endpoints, it states that the intermediate value is normally distributed, with a mean formed by linear interpolation between those endpoints and a variance that depends on the distances from the intermediate time to each endpoint.

This is the Brownian bridge conditional law, relevant to modeling paths between discrete observations. The document supplies the formula but gives no derivation, examples, or application to trading or pricing. It also describes observations through sampled geometric Brownian asset prices, whose logarithms reveal the corresponding Brownian values when model parameters are known.

Key ideas

• Conditioning Brownian motion on its values at both ends of an interval produces a normal distribution at an interior time.
• The conditional mean interpolates linearly between the endpoint values.
• The conditional variance is the product of the two elapsed-time portions divided by the full interval length.
• The document asks for a proof but does not provide one.

Tags

# The conditionnal law of a brownian motion


# The conditionnal law of a brownian motion












Please, I have a question about the conditionnal law of a brownian motion.

Here is the statement:

We have $\mathcal{B}_{h}$ the $\sigma$-field generated by the $\left(S_{t_{k}}, k=0, \ldots, N\right)$ with $S_{t}=S_{T_{0}} \exp \left(\sigma W_{t}-\frac{\sigma^{2}}{2} t+r t\right)$

The law of $W_{u}$ with respect to $\mathcal{B}_{h}$ for $u \in\left[t_{k}, t_{k+1}\right]$, $h=t_{k+1}-t_{k}$, is given by

$\mathcal{L}\left(W_{u} \mid W_{t_{k}}=x, W_{t_{k+1}}=y\right)=\mathcal{N}\left(\frac{t_{k+1}-u}{h} x+\frac{u-t_{k}}{h} y, \frac{\left(t_{k+1}-u\right)\left(u-t_{k}\right)}{h}\right)$

I don't know how tu prove this équality.

Thank in advance

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