{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:17:08Z","timestamp":1787321828214,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 1517664"],"award-info":[{"award-number":["DMS 1517664"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Finan. Math."],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:p>We study an optimal investment and consumption problem on infinite horizon, under the assumption that one of the investment opportunities is a fund charging high-watermark fees. The fund and the additional risky assets follow a multidimensional geometric L\u00e9vy structure. The interest rate is constant and the utility function has constant relative risk aversion. Identifying the wealth of the investor together with the distance to paying fees as the appropriate states, we obtain a two-dimensional stochastic control problem with both jumps and reflection. We derive the Hamilton--Jacobi--Bellman integro-differential equation, reduce it to one dimension, and then show it has a smooth solution. Using verification arguments the optimal strategies are obtained in feedback form. Some numerical results display the impact of the fees on the investor.<\/jats:p>","DOI":"10.1137\/18m1205066","type":"journal-article","created":{"date-parts":[[2020,7,28]],"date-time":"2020-07-28T15:34:28Z","timestamp":1595950468000},"page":"750-787","source":"Crossref","is-referenced-by-count":4,"title":["Optimal Investment with High-Watermark Fee in a Multidimensional Jump Diffusion Model"],"prefix":"10.1137","volume":"11","author":[{"given":"Karel","family":"Jane\u010dek","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4652-8789","authenticated-orcid":true,"given":"Zheng","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mihai","family":"S\u00eerbu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,7,28]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1540-6261.2009.01499.x"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/s0294-1449(16)30106-8"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.57262\/die\/1356060597"},{"key":"atypb4","unstructured":"D. Applebaum,\n                      L\u00e9vy Processes and Stochastic Calculus, 2nd ed.\n                      , Cambridge Stud. Adv. Math. 116, Cambridge University Press, Cambridge, UK, 2009,https:\/\/cds.cern.ch\/record\/1253366."},{"key":"atypb5","unstructured":"G. O. Aragon and J. Qian,\n                      The Role of High-Water Marks in Hedge Fund Compensation\n                      , preprint, 2007."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2007.02.007"},{"key":"atypb7","unstructured":"J. Bertoin,\n                      L\u00e9vy Processes\n                      , Cambridge Tracts in Math. 121, Cambridge University Press, Cambridge, UK, 1996."},{"key":"atypb8","unstructured":"M. Braun and H. Roche,\n                      Hedge Fund Fee Structure and Risk Exposure: Theory and Empirical Evidence\n                      , preprint, 2018,https:\/\/mipp.cl\/wp-content\/uploads\/2018\/10\/HFund17A.pdf."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1992-00266-5"},{"key":"atypb10","first-page":"35","volume":"65","author":"Cvitanic J.","year":"1995","journal-title":"constraints, IMA"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1287\/moor.15.4.676"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1287\/moor.13.4.588"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-008-0066-8"},{"key":"atypb14","unstructured":"L. Evans,\n                      Partial Differential Equations\n                      , Grad. Stud. Math. 19, AMS, Providence, RI, 1998,https:\/\/books.google.com\/books?id=kUiuCVQtK7EC."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4068(00)00067-7"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1111\/1540-6261.00581"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9965.1993.tb00044.x"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1111\/mafi.12057"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s00780-015-0266-y"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/100790884"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1287\/moor.11.2.261"},{"key":"atypb22","unstructured":"N. Katzourakis,\n                      An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L-infinity\n                      , Springer Briefs in Math., Springer, New York, 2014,https:\/\/books.google.com\/books?id=rfObBQAAQBAJ."},{"key":"atypb23","unstructured":"T. Konstantopoulos,\n                      The Skorokhod Reflection Problem for Functions with Discontinuities (Contractive Case)\n                      , Technical report, ECE Department, University of Texas at Austin, 1999."},{"key":"atypb24","unstructured":"A. Kontaxis,\n                      Asymptotics for Optimal Investment with High-Water Mark Fee\n                      , Ph.D. thesis, University of Texas at Austin, 2015."},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.2307\/1926560"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0531(71)90038-X"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9965.1995.tb00071.x"},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"B. \u00d8ksendal and A. Sulem,\n                      Applied Stochastic Control of Jump Diffusions\n                      , Universitext, Springer, New York, 2007,https:\/\/books.google.com\/books?id=ALHJxprw4ksC.","DOI":"10.1007\/978-3-540-69826-5"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1111\/j.1540-6261.2008.01427.x"},{"key":"atypb30","first-page":"1","volume":"8","author":"Pham H.","year":"1998","journal-title":"J. Math. Systems Estimation Control"},{"key":"atypb31","unstructured":"H. Roche,\n                      Optimal Consumption and Investment Strategies Under Wealth Ratcheting\n                      , preprint, 2006."},{"key":"atypb32","unstructured":"M. Schroder,\n                      Optimal Portfolio Selection with Fixed Transaction Costs: Numerical Solutions\n                      , preprint, 1995."},{"key":"atypb33","unstructured":"S. P. Sethi,\n                      Optimal Consumption and Investment with Bankruptcy\n                      , Springer, New York, 2012."},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177004966"},{"key":"atypb35","unstructured":"P. Tankov,\n                      Financial Modelling with Jump Processes\n                      , Chapman & Hall\/CRC Financ. Math. Ser., CRC Press, Boca Raton, FL, 2003,https:\/\/books.google.com\/books?id=bVlieV8GBrIC."},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1155\/2015\/413072"}],"container-title":["SIAM Journal on Financial Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1205066","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:15:25Z","timestamp":1787318125000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1205066"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":36,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/18M1205066"],"URL":"https:\/\/doi.org\/10.1137\/18m1205066","relation":{},"ISSN":["1945-497X"],"issn-type":[{"value":"1945-497X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}