[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120694-en":3,"doc-seo-120694-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},120694,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697",8,"Research & Report","Machine Learning architectures for price formation models with common noise","A machine learning framework is presented to numerically solve a mean-field game (MFG) price formation model under common noise. The commodity price is determined with a market-clearing condition driven by a stochastic supply process defined by an SDE with Brownian common noise. The method uses a dual recurrent neural network architecture to encode the noise dependence, combined with a particle approximation of the mean-field dynamics. A single adversarially trained loss enforces the constraint, and posterior convergence estimates are provided alongside numerical experiments.","Machine Learning architectures for price formation models with  \ncommon noise  \nDiogo Gomes 1 , Julian Gutierrez 1 , and Mathieu Laurire2  \narXiv :2305 . 17618v1 [math .OC] 28 May 2023  \nAbstract—We propose a machine learning method to solve a mean-field game price formation model with common noise. This involves determining the price of a commodity traded among rational agents subject to a market clearing condition imposed by random supply, which presents additional challenges compared to the deterministic counterpart. Our approach uses a dual recurrent neural network architecture encoding noise dependence and a particle approximation of the mean-field model with a single loss function optimized by adversarial training. We provide a posteriori estimates for convergence and illustrate our method through numerical experiments.  \nI. INTRODUCTION  \nIn this work, we extend the use of machine learning (ML) techniques for the numerical solution of the mean-field games (MFGs) price formation models, introduced in [8], to incorporate the common noise model from [9] (see also [10]) . The goal is to determine the price ϖ of a commodity with a noisy supply Q traded among rational agents within a finite time horizon T > 0, under a market-clearing condition. More precisely, we assume the supply function Q satisfies the following stochastic differential equation (SDE)  \n(dQQ(0(t))==qb0S (Q(t), t)dt + σS (Q(t), t)dW(t), (1) where q0 ∈ R and W is a one-dimensional Brownian motion acting as common noise. The coefficients bS and σ S satisfy the usual Lipschitz conditions for existence and uniqueness of solutions (see [7]) . Because of (1), our model explains the price formation for commodities with continuous and smooth fluctuations, such as stocks, bonds, currencies, and continuously produced or consumed goods such as oil or natural gas. Additional sources of noise can be considered. For instance, sudden and discontinuous fluctuations can be modeled by adding Poisson jumps to (1) .  \nLet (Ω , F, F, P) be a complete filtered probability space supporting W. Progressive measurability refers to the measurability with respect to this filtration, which we require for all stochastic processes. In this context, the MFG with common noise characterizing the price is the following.  \nProblem 1: Suppose that H : R2 → R is uniformly convex and differentiable in the second argument, m0 is a  \nKeywords: Mean Field Games; Price formation; Neural Networks  \n1 King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955- 6900. Saudi Arabia [diogo.gomes@kaust.edu.sa](diogo.gomes@kaust.edu.sa)[ ](diogo.gomes@kaust.edu.sa)[julian.gutierrezpineda@kaust.edu.sa](julian.gutierrezpineda@kaust.edu.sa)  \n2New York University Shanghai (NYU Shanghai), Shanghai 200122 . [China](China ml5197@nyu.edu)[ ml5197@nyu.edu](China ml5197@nyu.edu)  \nprobability measure on R, and uT : R → R is uniformly convex and differentiable. Find m : [0, T] × R → R, u, Z : [0, T] × R × Ω → R, and ϖ : [0, T] × Ω → R progressively measurable, satisfying m ⩾ 0 and  \n􀀾􀀸 −du + H(x,ϖ + ux )dt = Z (t, x)dW(t),  \n􀀾  \n􀀾  \n􀀾 u (T, x) = uT (x),  \n􀀼 mt − (Hp (x,ϖ + ux)m)x = 0 , (2)  \n􀀾 m(0, x) = m0 (x),  \n􀀾  \n􀀺􀀾 −RR Hp (x,ϖ + ux)mdx = Q (t) .  \nThe previous problem generalizes the one introduced in [11], which corresponds to the case σ S = 0 . The numerical solution of (2) presents additional challenges compared to the deterministic counterpart, as the state space becomes infinitedimensional. [10] showed that (2) is well-posed when bS and σ S are linear and H is quadratic, obtaining semi-explicit solutions. Section II presents the derivation of (2) .  \nIn the absence of common noise, several numerical schemes have been proposed: Fourier series [17], semiLagrangian schemes [4], fictitious play [12], and variational methods [3] . [6] proposes an ML-based approach to solve bilevel Stackelberg problems between a principal and a mean field of agents by reformulating the problem as a si","cbCaih9GxhRcDrYn","https://ap.wps.com/l/cbCaih9GxhRcDrYn","pdf",1027989,1,6,"English","en",105,"# I. Introduction\n## Problem setting and stochastic supply dynamics\n## Prior numerical and ML approaches\n## Proposed method and main result","[{\"question\":\"What problem does the proposed method solve?\",\"answer\":\"It solves a mean-field game price formation model where the supply is random and driven by common noise, including a market-clearing condition for the commodity price.\"},{\"question\":\"How is common noise incorporated into the model?\",\"answer\":\"Common noise enters through a stochastic differential equation for the supply function, with a one-dimensional Brownian motion acting as the shared noise term.\"},{\"question\":\"What machine learning architecture is used?\",\"answer\":\"The approach employs a dual recurrent neural network architecture to represent the noise-dependent price process, together with a particle approximation for the mean-field component and adversarial training with a single loss.\"}]","Machine Learning architectures for price formation models with common noise | PDF",1785731571,15,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"machine-learning-architectures-for-price-formation-models-with-common-noise","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/machine-learning-architectures-for-price-formation-models-with-common-noise/120694/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04","2026-08-03",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the proposed method solve?","Question",{"text":76,"@type":77},"It solves a mean-field game price formation model where the supply is random and driven by common noise, including a market-clearing condition for the commodity price.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is common noise incorporated into the model?",{"text":81,"@type":77},"Common noise enters through a stochastic differential equation for the supply function, with a one-dimensional Brownian motion acting as the shared noise term.",{"name":83,"@type":74,"acceptedAnswer":84},"What machine learning architecture is used?",{"text":85,"@type":77},"The approach employs a dual recurrent neural network architecture to represent the noise-dependent price process, together with a particle approximation for the mean-field component and adversarial training with a single loss.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]