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JohnLeM
Topic Author
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### The Bachelier problem

Following @bearish advise, I open a dedicated thread to discuss a numerical study of the Bachelier problem, that can be found here at SSRN.
Indeed, I did not complete still the conclusion section of this paper, this thread could help here. My first idea was to include a basic argumentation to sustain the numerical analysis, like
1) Why are the first two methods non convergent ?
2) Why the two other methods are convergent ?

The paper abstract is: We present a general approach to compute conditional expectations, materialized in this report by a python function $p(z | x) = \Pi^k(x,z)$, where $x,z$ are any iid samples of a martingale process at two different times, and $k$ is a kernel. This function is embedded in a framework, CodPy, and we benchmark it against two other methods. The first one is a Neural Network (NN) based method, using an open-source implementation. The second one is a kernel-based methods, that is quite similar to the neural network one. We will show numerically that these two last methods provides non convergent methods, since the function $\Pi^k$ provides a convergent method at statistical rate  $\frac{1}{\sqrt{N}} \%$, where $N := card(x)$ is the number of samples used. We illustrate also how to boost convergence rates using sharp discrepancy sequences.

JohnLeM
Topic Author
Posts: 515
Joined: September 16th, 2008, 7:15 pm

### Re: The Bachelier problem

Here is the good link to this technical analysis, the previous link seems erroneous.