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For the joint dynamics of the best bid price, the best ask price, and the standing volume densities on both sides of the limit order book we derive a weak law of large numbers, which states that the limit order book model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the scaling limit the two volume densities each follow a nonlinear PDE coupled with two nonlinear ODEs that describe the best bid and ask price.<\/jats:p>","DOI":"10.1137\/15m1024226","type":"journal-article","created":{"date-parts":[[2017,4,25]],"date-time":"2017-04-25T16:47:19Z","timestamp":1493138839000},"page":"314-343","source":"Crossref","is-referenced-by-count":15,"title":["A Weak Law of Large Numbers for a Limit Order Book Model with Fully State Dependent Order Dynamics"],"prefix":"10.1137","volume":"8","author":[{"given":"Ulrich","family":"Horst","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D\u00f6rte","family":"Kreher","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,4,25]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/15M1011469"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1111\/j.1540-6261.1995.tb05192.x"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1002\/fut.20334"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jedc.2015.05.004"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/110856605"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1287\/opre.1090.0780"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/0304-405X(87)90029-8"},{"key":"atypb9","unstructured":"S. N. Elaydi,\n                      An Introduction to Difference Equations\n                      , 3rd ed., Springer, New York, 2005."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1080\/14697680400008627"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"X. Gao, J. G. Dai, A. B. Dieker, and S. J. Deng,\n                      Hydrodynamic Limit of Order Book Dynamics\n                      ,arXiv:1411.7502, 2014.","DOI":"10.2139\/ssrn.2530306"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/0304-405X(85)90044-3"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"X. Guo, Z. Ruan, and L. Zhu,\n                      Dynamics of Order Positions and Related Queues in a Limit Order Book\n                      ,arXiv:1505.04810v2, 2015.","DOI":"10.2139\/ssrn.2607702"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jedc.2011.09.012"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2014.982278"},{"key":"atypb17","unstructured":"W. Huang and M. Rosenbaum,\n                      Ergodicity and Diffusivity of Markovian Order Book Models: A General Framework\n                      ,arXiv:1505.04936v1, 2015."},{"key":"atypb18","unstructured":"P. Lakner, J. Reed, and F. Simatos,\n                      Scaling Limit of a Limit Order Book Via the Regenerative Characterization of Le\u0301vy Trees\n                      ,arXiv:1312.2340v2, 2014."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1142\/S2382626616500040"},{"key":"atypb20","first-page":"167","author":"Pisier G.","year":"1986","journal-title":"Heidelberg"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/hhp011"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1017\/S0021900200011566"},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"T.W. Yang and L. Zhu,\n                      A Reduced-Form Model for Level-1 Limit Order Books\n                      ,arXiv:1508.07891v3, 2015.","DOI":"10.2139\/ssrn.2653149"}],"container-title":["SIAM Journal on Financial Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/15M1024226","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:21:16Z","timestamp":1787318476000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/15M1024226"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/15M1024226"],"URL":"https:\/\/doi.org\/10.1137\/15m1024226","relation":{},"ISSN":["1945-497X"],"issn-type":[{"value":"1945-497X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}