{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:31:07Z","timestamp":1787319067819,"version":"build-2736575974"},"reference-count":22,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>We study properties of solutions to fully nonlinear versions of the standard Black\u2013Scholes partial differential equation. These equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility. We show that typical nonlinear Black\u2013Scholes equations can be viewed as dynamic programming equation of an associated control problem. We establish existence and comparison results and show that the equation induces a convex risk measure on the set of all continuous terminal value claims. Moreover, we study the asymptotic behavior of solutions as market frictions get \u201clarge.\u201d Finally, the pricing of individual contracts relative to a book of derivatives is discussed.<\/jats:p>","DOI":"10.1137\/090773647","type":"journal-article","created":{"date-parts":[[2011,2,1]],"date-time":"2011-02-01T18:07:53Z","timestamp":1296583673000},"page":"185-204","source":"Crossref","is-referenced-by-count":15,"title":["Nonlinear Black\u2013Scholes Equations in Finance: Associated Control Problems and Properties of Solutions"],"prefix":"10.1137","volume":"49","author":[{"given":"R\u00fcdiger","family":"Frey","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ulrike","family":"Polte","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,2,1]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.00068"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1080\/13504869500000005"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1111\/j.0960-1627.2004.00179.x"},{"key":"R4","unstructured":"G. Barles,\n                      Solutions de Viscosit\u00e9 des Equations de Hamilton\u2013Jacobi\n                      , Springer, Paris, 1994."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/hhj014"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-004-0123-x"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-009-0116-x"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1992-00266-5"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1032374469"},{"key":"R10","unstructured":"W. H. Fleming and H. M. Soner,\n                      Controlled Markov Processes and Viscosity Solutions\n                      , 2nd ed., Springer, New York, 2006."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/s007800200072"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/s007800050035"},{"key":"R13","unstructured":"R. Frey,\n                      Market illiquidity as a source of model risk in dynamic hedging\n                      , in Model Risk, R. Gibson, ed., Risk Publications, London, 2000, pp. 125\u2013136."},{"key":"R14","doi-asserted-by":"crossref","unstructured":"R. Frey and P. Patie,\n                      Risk management for derivatives in illiquid markets: A simulation study\n                      , in Advances in Finance and Stochastics, K. Sandmann and P. J. Sch\u00f6nbucher, eds., Springer-Verlag, Berlin, 2002, pp. 137\u2013159.","DOI":"10.1007\/978-3-662-04790-3_8"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.00036"},{"key":"R16","unstructured":"A. Friedman,\n                      Partial Differential Equations of Parabolic Type\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1964."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.2307\/2331224"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9965.2005.00227.x"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"N. Krylov,\n                      Nonlinear Elliptic and Parabolic Equations of the Second Order\n                      , D. Reidel, Dordrecht, The Netherlands, 1987.","DOI":"10.1007\/978-94-010-9557-0"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jedc.2004.11.004"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9965.00045"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139996308534"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090773647","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:44:52Z","timestamp":1787316292000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090773647"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/090773647"],"URL":"https:\/\/doi.org\/10.1137\/090773647","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}