dt = 1/1000

tmax = 100

n = Floor[tmax/dt]

dw = Table[{i, Random[NormalDistribution[0, Sqrt[dt]]]}, {i, 1, n}];

stepw[{i_, wi_}] := {i + 1, wi + dw[[i + 1, 2]]}

w = NestList[stepw, {0, 0}, n];

ListLinePlot[w]