Simulating jump Markov process

Question

I am trying to simulate a jump Markov process as follows. I define the process $N_j = N_{j-1} + 1$ with probability 1/2 or $N_j = N_{j-1} - 1$ with probability $1/2,$ with $N_j = 0$ as an absorbing state. I tried

p = DiscreteMarkovProcess[{0, 1}, {{1/2, 1/2}, {1/2, 1/2}}];
data = RandomFunction[p, {0, 50}]
ListPlot[data, Filling -> Axis, Ticks -> {Automatic, {0, 1, 2}}]

but I am not getting any results that are in the absorbing state. Could someone help clarify this? I feel I am not understanding this...

Answer 1

Make the matrix:

nStates = 6;
mat = DiagonalMatrix[ConstantArray[1/2, nStates - 1], 1] + 
    DiagonalMatrix[ConstantArray[1/2, nStates - 1], -1];
mat = #/Total[#] & /@ mat; (*Sum normalize per row*)
 mat[[-1]] = ConstantArray[0, nStates]; mat[[-1, -1]] = 1;(*Make the last state an absorbing state*)
 MatrixForm[mat]

Simulate:

SeedRandom[13];
p = DiscreteMarkovProcess[mat[[1]], mat];
data = RandomFunction[p, {0, 50}]
ListPlot[data, Filling -> Axis, Ticks -> {Automatic, {0, 1, 2}}]

Interactive interface to visualize the simulation:

path = data["Values"];
Manipulate[
  Graph[p, GraphHighlight -> path[[time + 1]], GraphHighlightStyle -> "VertexConcaveDiamond", PlotLabel -> "Time \[LongEqual] " <> ToString[time], ImageSize -> Medium], {time, 0, Length[path] - 1, 1}, SaveDefinitions -> True]