Paths with drift θ are reweighted by the density dQ/dP = exp(−θX_T + ½θ²T). The weighted law is driftless Brownian motion, at the price of fewer effective samples.
Measure change
0.80
03
1.00
0.255
Random seed: 7
Mean weight
1.032
Effective sample size
2074 / 4000
ESS / N
51.8%
Weighted mean of X_T
-0.048
The weights average to one. As θ grows a few paths carry nearly all the weight, which is why importance sampling with a large shift is noisy.