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Assume I know the SDE of a stochastic process as:

$$ d\theta(t) = \frac{\sigma^2 cot(\theta(t))}{2 }dt + \sigma dB(t). $$ What are the steps to generate random angles (random walks) corresponding to the SDE? (assume I can generate uniformly distributed random numbers within a specified range, such as [-1,1])

K252
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  • @KurtG. I don't think the answer you mentionned answers the question because according to it, we need to find a function $f(\theta, t)$ such that $df(\theta, t) = a(t) dt + b(t) dB(t)$ (in the answer, $f = \ln (x)$). However, finding $f$ is not easy. – NN2 Dec 31 '23 at 17:59
  • @NN2 . In OP's case we only need the essence of Lutz Lehmann's answer which is the Euler scheme $X_{t+\Delta t}=X_t+(α-\tfrac12σ_t^2)\Delta t+σ_tz^1_t\sqrt{\Delta t},.$ – Kurt G. Dec 31 '23 at 18:03
  • @K252 OP: Can you figure out in the answer to the duplicate question what $z^1_t$ is and adapt that scheme to your SDE? The only potential problem I see is that your drift $\cot\theta(t)$ has a singularity. – Kurt G. Dec 31 '23 at 18:06
  • @KurtG. As I said, the Lutz LEhmann's answer required a transfomation $X_t = \color{red}{\ln}( S_t) $ ($f(x) = \ln(x)$ as I wrote in the second comment). It's not a duplicate question. If you solve it, you will see. – NN2 Dec 31 '23 at 18:12
  • @NN2 You are overthinking it. OP's Euler scheme is plainly $$ \theta(t+\Delta t)=\theta(t)+\frac{\sigma^2\cot\theta(t)}{2}\Delta t+\sigma,z_t^1\sqrt{\Delta t} $$ OP's question was essentially if the random numbers $z^1_t$ are uniform or not. The answer to that is in the duplicate. We can forget about the function $f,.$ – Kurt G. Dec 31 '23 at 18:16
  • @KurtG. No, the scheme (in Lutz' answer) is only applied to $X_t$ (and not $S_t$) after the transformation $X_t = f(S_t)$. All the difficulty is to find the transformation $X_t = f(S_t)$. You can answer this question (you can even use the Lutz' method if necessary), you will see. – NN2 Dec 31 '23 at 18:18
  • @NN2 Let's agree that we disagree. Happy new year before it is getting too late. :) Last hint: the Euler scheme can be applied to any SDE. Lutz Lehmann solved a more general problem in which the function $f$ was a convenient transformation. Over here I see the singularity in the drift as the real problem. – Kurt G. Dec 31 '23 at 18:21
  • @KurtG. Why don't you answer the question if it is not difficult? And yes, happy new year to you! :) – NN2 Dec 31 '23 at 18:22
  • @NN2 I may answer it next year but prefer OP to do it themselves. Better learning experience than getting it on a silver plate. – Kurt G. Dec 31 '23 at 18:24
  • Thank you both. I will check if the solution given in the mentioned link can solve my problem. – K252 Dec 31 '23 at 19:20

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