Clear["Global`*"];

T=1; (*Driving function defined on [0,T]*)
c=1; (*Scaling constant*)

Change file location to the location of driving function data. Note that the driving function should be for the downward LE. If it is for the upward LE, remove “Reverse” from next line of code.

data =Reverse[Flatten[Import["C:/Users/starnes/brownianmotion.mat"]]];
n=Length[data];
data=N[c*data];
dfdata=Transpose[{Table[T*i,{i,0,T,1/(n-1)}],data}];

ListPlot[dfdata]

flip[z_] := If [Im[z] ≥ 0, z, -z];
f[z0_, z_] := z0+flip[Sqrt[(z-z0)^2-4*T/(n-1)]];

zvals = {data[[1]]};

For[i=1, i≤n-1, i++, zvals=Prepend[Map[Function[z,f[data[[i]],z]],zvals],data[[i+1]] ]];

ListLinePlot[{Re[#],Im[#]} &/@ zvals]

ListPlot[{Re[#],Im[#]} &/@ zvals]