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random - Simulating discrete time stochastic dynamic systems


What is the canonical way of simulating discrete time stochastic dynamical systems in Mathematica using the new functionality of Random processes?


To take a concrete example, lets consider the optimal gambling problem. A gambler comes to a casino with an initial fortune x1 and let Xn denote his fortune at time n. At each time he bets a fraction αxn. At each time he wins with probability p and loses with probability (1−p). Thus, the dynamics of the system can be written as:


Xn+1={(1+α)Xn,with probability p(1−α)Xn,with probability 1−p


I want to simulate this system for 20 time steps with x1=12, p=0.8, and then plot 50 sample paths and the mean value of X20.



Answer



I would proceed like the following. It will be natural to propose that the win-event occurs following BinomialDistribution with probability p=0.8 so that we can use the built-in BinomialProcess to simulate the win and losses in 20 time steps for 50 sample paths.


timstep = 20;
win = BinomialProcess[.8];

samplepaths=50;
process = RandomFunction[Evaluate[win], {0, timstep - 1}, samplepaths];
ListLinePlot[process, InterpolationOrder -> 0, Frame -> True,
PlotStyle -> {{Dashed, Opacity[.9]}}]

enter image description here


Then just use FoldList on the TemporalData to get your result simulation of 50 sample paths and the mean in a simple functional style!


x1 = 12;
alpha = .15; (* assume some value for alpha *)
WinEvents = process["States"];

ST[WinEvents_?ListQ, x1_, alpha_, SimTime_] := Module[{},
   FoldList[Max[If[#2 == 1, (1 + alpha) #1, (1 - alpha) #1], 0.] &,
            x1,
            Prepend[Differences@WinEvents, First@WinEvents]]
  ];
simdat = ST[#, x1, alpha, timstep] & /@ WinEvents;

For visualization


Show[ListLinePlot[Legended[Mean[simdat], "Mean"], PlotRange -> All, 
PlotStyle -> {{AbsoluteThickness[3.5], Opacity[.9], Red}}], 

ListLinePlot[simdat, PlotRange -> All, 
PlotStyle -> {{Dashed, Opacity[.7]}}], 
ListPlot[simdat, PlotRange -> All], Frame -> True]

enter image description here


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