{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:04:53Z","timestamp":1760241893391,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T00:00:00Z","timestamp":1539648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The present paper attains a Harnack inequality for the option pricing (or Kolmogorov) equation with gradient estimate arguments. We then perform a no-arbitrage analysis of a financial market.<\/jats:p>","DOI":"10.3390\/sym10100517","type":"journal-article","created":{"date-parts":[[2018,10,16]],"date-time":"2018-10-16T11:07:51Z","timestamp":1539688071000},"page":"517","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Harnack Inequality and No-Arbitrage Analysis"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1343-4994","authenticated-orcid":false,"given":"Wanxiao","family":"Tang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fanchao","family":"Zhou","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0262-5544","authenticated-orcid":false,"given":"Peibiao","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s11118-004-6456-4","article-title":"The Parabolic Harnack Inequality for the Time Dependent Ginzburg-Landau Type SPDE and its Application","volume":"22","author":"Kawabi","year":"2005","journal-title":"Potential Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.bulsci.2005.10.001","article-title":"Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below","volume":"130","author":"Arnaudon","year":"2006","journal-title":"Bull. Sci. Math."},{"key":"ref_3","first-page":"15","article-title":"Harnack inequality and no-arbitrage bounds for self-financing portfolios","volume":"49","author":"Carciola","year":"2009","journal-title":"Bol. Soc. Esp. Mat. Apl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1214\/009117906000001204","article-title":"Harnack Inequality and Applications for Stochastic Generalized Porous Media Equations","volume":"35","author":"Wang","year":"2007","journal-title":"Ann. Probab."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1023\/A:1016378210944","article-title":"Harnack inequalities for jump processes","volume":"17","author":"Bass","year":"2002","journal-title":"Potential Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0304-4149(81)90026-0","article-title":"Martingales and stochastic integrals in the theory of continuous trading","volume":"11","author":"Harrison","year":"1981","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0304-405X(76)90023-4","article-title":"The valuation of options for alternative stochastic processes","volume":"3","author":"Cox","year":"1976","journal-title":"J. Financ. Econ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s10479-004-5037-7","article-title":"Necessary and Sufficient Conditions for Weak No-Arbitrage in Securities Markets with Frictions","volume":"133","author":"Deng","year":"2005","journal-title":"Ann. Oper. Res."},{"key":"ref_9","unstructured":"Sandhu, R., Georgiou, T., and Tannenbaum, A. (arXiv, 2015). Market Fragility, Systemic Risk, and Ricci Curvature, arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1098\/rspa.2000.0722","article-title":"Interest Rates and Information Geometry","volume":"457","author":"Brody","year":"2011","journal-title":"R. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"862","DOI":"10.1119\/1.19139","article-title":"Foreign exchange market as a lattice gauge theory","volume":"67","author":"Young","year":"1999","journal-title":"Am. J. Phys."},{"key":"ref_12","unstructured":"Ilinski, K. (2001). Physics of Finance: Gauge Modelling in Non-Equilibrium Pricing, Wiley."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"431","DOI":"10.3934\/jgm.2015.7.431","article-title":"Geometric Arbitrage Theory and Market Dynamics","volume":"7","author":"Farinelli","year":"2015","journal-title":"J. Geom. Mech."},{"key":"ref_14","unstructured":"Choi, Y.H. (2007). Curvature Arbitrage. [Ph.D. Thesis, University of Iowa]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1002\/cpa.3160170106","article-title":"A Harnack inequality for parabolic differential equations","volume":"17","author":"Moser","year":"1964","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s10455-012-9342-0","article-title":"Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds","volume":"43","author":"Huang","year":"2013","journal-title":"Ann. Glob. Anal. Geom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4537","DOI":"10.1090\/S0002-9939-2015-12576-2","article-title":"Harnack estimate for the endangered species equation","volume":"143","author":"Cao","year":"2015","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1090\/S0002-9947-1990-0998126-5","article-title":"Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type","volume":"321","author":"Garofalo","year":"1990","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Barucci, E., and Fontana, C. (2017). Financial Markets Theory, Springer.","DOI":"10.1007\/978-1-4471-7322-9"},{"key":"ref_20","unstructured":"Delbaen, F., and Schachermayer, W. (2006). The Mathematics of Arbitrage, Springer."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/10\/517\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:26:06Z","timestamp":1760196366000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/10\/517"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,16]]},"references-count":20,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["sym10100517"],"URL":"https:\/\/doi.org\/10.3390\/sym10100517","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,10,16]]}}}