# Brownian Motion with stopping threshold

I just want to generate one stochastic process, whereas I want to be able to manipulate the start value and the volatility. Furthermore, the process should stop when it reaches 5

However, as soon as I use the SeedRandom[seed] function some error occurs. Hope you can help me. I also want the function to start at 0 and not at 1 Definition function of: (* x[t+1]=x[t]+μ (drift) + shock[t]; which stops at threshold*)

My code:

σ = 0.2
SeedRandom; shock1 =
RandomVariate[NormalDistribution[0, σ],(*time interval/Length of paths*) 10]

μ = 0.1

step[x_, μ_, shock1_] :=
If[x == 5, 5, If[x + μ + shock1 < 5, 5, x + μ + shock1]]

p1 = FoldList[step, 10, shock1]

ListLogPlot[p1, Joined -> True]


This Works perfectly, however, as soon as I enter the following i get some Problems:

Manipulate[SeedRandom[seed];
shock1 = RandomVariate[NormalDistribution[0, σ1], 10]; μ =
0.1; step[x_, μ_, shock1_] :=
If[x == 5, 5, If[x + μ + shock1 < 5, 5, x + μ + shock1]];
ListLogPlot[FoldList[step, S, shock1], Joined -> True], {S, 10,
50}, {σ1, 0.1, 1}]


I get the following error-message: "Argument seed in SeedRandom[seed] should be an integer or a string. "

But when I enter:

Manipulate[SeedRandom;
shock1 = RandomVariate[NormalDistribution[0, σ1], 10]; μ =
0.1; step[x_, μ_, shock1_] :=
If[x == 5, 5, If[x + μ + shock1 < 5, 5, x + μ + shock1]];
ListLogPlot[FoldList[step, S, shock1], Joined -> True], {S, 10,
50}, {σ1, 0.1, 1}]


and try to manipulate the volatility just nothing happens.

do you have any solutions for my problem?

• When you say that the first part works perfectly. It doesn't for me when I copy and paste that first section of code. If you quite the kernel, does the first part still work correctly? – Jonathan Shock Jun 5 '13 at 14:34
• The Initialization option might be useful here... – J. M. will be back soon Jun 5 '13 at 14:34
• You might want to use Clip in place of the two nested If. – b.gates.you.know.what Jun 5 '13 at 14:40
• @Jonathan Shock I alos have your problem now. I do not know why. Although I enter Remove["Global*"] before I enter everything. just before it worked, no other notebooks are open. – Milan Ivica Jun 5 '13 at 14:45
• @MilanIvica It would be easier to simply generate a matrix of processes and then choose those which satisfy your condition, IMO. – Rod Jun 6 '13 at 15:22

There are a couple of details that make your function misbehave. Here is a version that takes in {currentx, mu} and a term of the shock list and outputs {newx, mu}. I changed your Ifs to a single Which. The manipulate now seems to work fine. I also moved the function definition outside the Manipulate, which is more efficient (there's no need to define the function at each pass through the Manipulate).

step[{x_, mu_}, s_] := {Which[x == 5, 5, x + mu + s > 5, 5, True, x + mu + s], mu};

Manipulate[sig = 0.2; mu = 0.05; len = 500; SeedRandom[seed]; start=1;
shock = RandomVariate[NormalDistribution[0, sig], len];
p1 = FoldList[step, {start, mu}, shock];
ListLogPlot[p1[[All, 1]], Joined -> True], {seed, 1, 85, 1}] There are a couple of subtleties in this problem. For instance, one of the comments to the OP suggested replacing the logic of the Which (or If in the OP) with Clip`. This might not be a good idea because the goal is to freeze the process once it reaches 5. When clipping, the process might go above 5 (and be clipped) but later drop below 5 again. It is also more tricky than it seems to generate a lot of paths and then choose the ones you want: again, the paths would need to be processed to get the proper clipping.

• thank you for your help an remarks – Milan Ivica Jun 7 '13 at 6:49