1 | initial version |

I don't know what you are doing in creating B1. Other than that, most of your time is being spent in converting between Sage numbers or arrays and numpy numbers or arrays. When writing your code, make sure to minimize these kinds of conversions.

Following is the code which is almost the same as your code, except everything is handled by numpy and there are hardly any conversions being done, except for a few floating values. Also, I removed the line containing B1 since it gives errors with zero variance.

```
import numpy
# General routine to sample a Brownian motion on a time grid
def rB(tvec):
dt = numpy.diff(tvec)
sdt = numpy.sqrt(dt)
dB = sdt * numpy.random.normal(0, 1, len(dt))
B = numpy.cumsum(dB)
return(B)
```

Here, we must ensure that the input `tvec`

is also a numpy array, otherwise there will be a lot of time wasted in converting it. After we make this change, the time taken by the code is 0.17s (as compared to over 4s with your code).

```
tvec = numpy.arange(0, 1e3, 1e-3) # srange(0,1e3,1e-3)
time B = rB(tvec)
Time: CPU 0.16 s, Wall: 0.17 s
```

Output of r code:

```
%r
tvec <- seq(0,1000,1e-3)
system.time(B <- c(0,cumsum(rnorm(length(tvec)-1,sd=sqrt(diff(tvec))))))
user system elapsed
0.440 0.003 0.451
```

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