Comments to the reviewer's feedback
We thank the reviewer for your quick feedback. We have the following response.
**Application of Machine learning in finance:**
We respectfully disagree with the comment that machine learning methods have "not proved much relevance in finance." In fact, the use of artificial neural networks for financial SDE modeling and optimization dates back to the early 1990s. Survey paper [9] summarizes **more than 150 papers** that use neural network-parameterized SDEs to model prices in option pricing and hedging. Additionally, recent papers surveyed in [9] actively explore the use of deep network architectures. This underscores the relevance of our proposed methodology in cutting-edge finance applications. Furthermore, based on the private consulting activities of some of the authors, neural networks and other machine learning techniques are very much alive and of significant interest in financial applications.
Given these academic interests and industry experiences, we believe it is hard to argue otherwise. *We respectfully ask why you think that ML has not proven to be relevant in finance.*
**Affine processes:**
We do not intend to use our method to optimize simple affine processes. Our methodology is designed for estimating sensitivity or optimizing values driven by high-dimensionally parameterized jump-diffusions. Our method excels in modern ML settings where deep neural networks are used in the SDE parameters.
**Application in sciences and engineering:**
Other than financial applications, our methodology is well-motivated and an important contribution considering its application in science and engineering. For example, neural SDEs aim to learn the behavior of natural (physical) random processes with parameterized SDEs. Highly cited papers in this field include [7] and [5] (with and without jump). Neural SDEs typically optimize $\min_\theta E[g(X_\theta^x(T))]$ subject to an SDE
$$ dX_\theta^x(t) = \mu_\theta(t, X_\theta^x(t)) dt + \sigma_\theta(t, X_\theta^x(t)) dB(t) + \chi_\theta(t,X_\theta^x(t))dN(t)$$
where $\mu_\theta,\sigma_\theta,\chi_\theta$ are neural network parameterized and $N(t)$ is a Poisson process.
Gradient methods for approximating stochastic optimal control with a parameterized policy class [3] are also important applications.
We remark that many neural network-parameterized SDEs are not widely used primarily because of the lack of scalable gradient estimators. Our contribution directly addresses this issue.
Given these numerous applications across multiple disciplines, we find it **hard to believe** that NeurIPS, where practitioners and theorists, scientists and engineers exchange ideas and advance machine learning theory and applications, is not a suitable venue to publish this work.
**Rigorous validation of limit interchange:**
We respectfully disagree with the claim that the interchange of limits is what "everyone in the ML community would believe anyways." Wrong intuition regarding the interchange of limits can lead to serious errors. A researcher might explore the following gradient estimation idea: From semigroup theory, write $E_x[g(X_{\theta}(t))] = (e^{tL_{\theta}}g)(x)$. Formal differentiation (taking the limit quotient) yields: $$\partial_\theta (e^{tL_{\theta}}g)(x) \stackrel{?}{=} t e^{tL_{\theta}} \partial_\theta L_{\theta} g(x) = t E_x[\partial_{\theta}L_{\theta} g(X(t))].$$ Although this can work for Markov jump processes, it is incorrect, unfortunately, for diffusions. The correct expression is precisely the representation in our Theorem 1. The reason readers find the ideas in Section 2 natural is that we have invested research effort into identifying a simple and intuitive justification, all backed by rigorous proof.
Nevertheless, given this helpful discussion with the reviewer, we think including this incorrect reasoning in the paper could help to illustrate the need for caution when working with generators of SDEs.
**Assumption 2:**
We have considered including sufficient conditions on the model primitives to imply Assumption 2 but decided against it for the following reason: The primary focus of this paper is to rigorously justify the **validity** of the generator gradient estimator. Proving Assumption 2 using sufficient conditions could detract from this goal.
The differentiability and moment bounds for the value are well-established in the literature. Also, justifying Assumption 2 from model primitives would introduce another set of assumptions typical in jump-diffusion analysis but potentially confusing for users, as we have explained in the previous response.
If you still have concerns about establishing Assumption 2, we can provide a proof in the Appendix with smooth and bounded rewards (so that there will be no interference with the variance's growth power).
[9] Ruf, Johannes, and Weiguan Wang. Neural networks for option pricing and hedging: a literature review. Journal of Computational Finance.