𝒫=DiscreteMarkovProcess[1,(({{0, 1/2, 1/2}, {1/2, 0, 1/2}, {1/2, 1/2, 0}}))];

data=RandomFunction[𝒫,{0,50}];

ListPlot[data,Filling->Axis,Ticks->{Automatic,{1,2,3}},PlotRange->All]

𝒫=ContinuousMarkovProcess[1,(({{0, 1/2, 1/2}, {1/2, 0, 1/2}, {1/2, 1/2, 0}})),{2,7,3}];

data=RandomFunction[𝒫,{0,10}];

ListLinePlot[data,Filling->Axis,Ticks->{Automatic,{1,2,3}},PlotRange->All,InterpolationOrder->0]