{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:25:35Z","timestamp":1772252735675,"version":"3.50.1"},"reference-count":59,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T00:00:00Z","timestamp":1648771200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministry of Economy, Industry and Competitiveness","doi-asserted-by":"publisher","award":["PID2019-106811GB-C33 (AEI\/10.13039\/501100011033)"],"award-info":[{"award-number":["PID2019-106811GB-C33 (AEI\/10.13039\/501100011033)"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002809","name":"Government of Catalonia","doi-asserted-by":"publisher","award":["2017 SGR 608"],"award-info":[{"award-number":["2017 SGR 608"]}],"id":[{"id":"10.13039\/501100002809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We address the process of discounting in random environments, which allows valuation of the future in economic terms. We review several approaches to the problem regarding different well-established stochastic market dynamics in the continuous-time context and include the Feynman\u2013Kac approach. We also review the relation between bond-pricing theory and discounting and introduce both the market price of risk and the risk neutral measure from an intuitive point of view devoid of excessive formalism. We provide the discount for each economic model and discuss their key results. We finally present a summary of our previous empirical studies for several countries on the long-run discount problem.<\/jats:p>","DOI":"10.3390\/e24040496","type":"journal-article","created":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T21:22:39Z","timestamp":1648848159000},"page":"496","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Valuing the Future and Discounting in Random Environments: A Review"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5810-879X","authenticated-orcid":false,"given":"Jaume","family":"Masoliver","sequence":"first","affiliation":[{"name":"Departament de F\u00edsica de la Mat\u00e8ria Condensada, Universitat de Barcelona, 08028 Barcelona, Spain"},{"name":"Universitat de Barcelona Institute of Complex Systems (UBICS), 08028 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3221-1211","authenticated-orcid":false,"given":"Miquel","family":"Montero","sequence":"additional","affiliation":[{"name":"Departament de F\u00edsica de la Mat\u00e8ria Condensada, Universitat de Barcelona, 08028 Barcelona, Spain"},{"name":"Universitat de Barcelona Institute of Complex Systems (UBICS), 08028 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8533-6539","authenticated-orcid":false,"given":"Josep","family":"Perell\u00f3","sequence":"additional","affiliation":[{"name":"Departament de F\u00edsica de la Mat\u00e8ria Condensada, Universitat de Barcelona, 08028 Barcelona, Spain"},{"name":"Universitat de Barcelona Institute of Complex Systems (UBICS), 08028 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7871-073X","authenticated-orcid":false,"given":"J. Doyne","family":"Farmer","sequence":"additional","affiliation":[{"name":"Institute for New Economic Thinking at the Oxford Martin School, Oxford OX1 3UQ, UK"},{"name":"Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK"},{"name":"Santa Fe Institute, Santa Fe, NM 87501, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John","family":"Geanakoplos","sequence":"additional","affiliation":[{"name":"Santa Fe Institute, Santa Fe, NM 87501, USA"},{"name":"Department of Economics, Yale University, New Heaven, CT 06511, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mantegna, R.N., and Stanley, H.E. (1999). Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press.","DOI":"10.1017\/CBO9780511755767"},{"key":"ref_2","unstructured":"Bouchaud, J.-P., and Potters, M. (2011). Theory of Financial Risk and Derivative Pricing: From Statistical Mechanics to Risk Management, Cambridge University Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1051\/epn\/2019103","article-title":"Econophysics: Still fringe after 30 years?","volume":"50","author":"Bouchaud","year":"2019","journal-title":"Europhys. News"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"21","DOI":"10.24033\/asens.476","article-title":"Th\u00e9orie de la sp\u00e9culation","volume":"17","author":"Bachelier","year":"1900","journal-title":"Ann. Sci. \u00c9cole Norm. Sup."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1287\/opre.7.2.145","article-title":"Brownian motion in stock markets","volume":"7","author":"Osborne","year":"1959","journal-title":"Oper. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1080\/713665670","article-title":"Empirical properties of asset returns: Stylized facts and statistical issues","volume":"1","author":"Cont","year":"2001","journal-title":"Quant. Financ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1086\/294632","article-title":"The variation of certain speculative prices","volume":"35","author":"Mandelbrot","year":"1963","journal-title":"J. Bus."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1086\/294633","article-title":"Mandelbrot and the stable Paretian hypothesis","volume":"35","author":"Fama","year":"1963","journal-title":"J. Bus."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1093\/rfs\/4.4.727","article-title":"Stock price distributions with stochastic volatility: An analytic approach","volume":"4","author":"Stein","year":"1991","journal-title":"Rev. Financ. Stud."},{"key":"ref_10","unstructured":"Fouque, J.-P., Papanicolau, G., and Sircar, K.R. (2000). Derivatives in Financial Markets with Stochastic Volatility, Cambdrige University Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1080\/14697680600727547","article-title":"Multiple time scales and the exponential Ornstein\u2013Uhlenbeck stochastic volatility model","volume":"6","author":"Masoliver","year":"2006","journal-title":"Quant. Financ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"155","DOI":"10.2307\/2967612","article-title":"A note on measurement of utility","volume":"4","author":"Samuelson","year":"1937","journal-title":"Rev. Econ. Stud."},{"key":"ref_13","unstructured":"Brigo, D., and Mercurio, F. (2006). Interest Rate Models\u2014Theory and Practice, Springer."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1126\/science.1235665","article-title":"Determining benefits and costs for future generations","volume":"341","author":"Arrow","year":"2013","journal-title":"Science"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Stern, N. (2006). The Economics of Climate Change: The Stern Review, Cambridge University Press.","DOI":"10.1017\/CBO9780511817434"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1257\/jel.45.3.686","article-title":"The Stern Review on the economics of climate change","volume":"45","author":"Nordhaus","year":"2007","journal-title":"J. Econ. Lit."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1126\/science.1137316","article-title":"Critical assumptions in the Stern Review on Climate Change","volume":"317","author":"Nordhaus","year":"2007","journal-title":"Science"},{"key":"ref_18","unstructured":"Dasgupta, P. (2006). Comments on the Stern Review\u2019s Economics of Climate Change, Cambridge University Press."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1257\/jel.45.3.703","article-title":"A review on the Stern review on the economics of climate change","volume":"45","author":"Weitzman","year":"2007","journal-title":"J. Econ. Lit."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Nordhaus, W.D. (2008). A Question of Balance, Yale University Press.","DOI":"10.2307\/j.ctt1npzkh"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1017\/S0266267114000297","article-title":"Ethics, equity and the economics of climate change. Paper 1","volume":"30","author":"Stern","year":"2014","journal-title":"Sci. Philos. Econ. Philos."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1017\/S0266267114000303","article-title":"Ethics, equity and the economics of climate change. Paper 2","volume":"30","author":"Stern","year":"2014","journal-title":"Sci. Philos. Econ. Philos."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1257\/pol.20160240","article-title":"Discounting disentangled","volume":"10","author":"Drupp","year":"2018","journal-title":"Am. Econ. J. Econ. Policy"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3695","DOI":"10.1073\/pnas.1315987111","article-title":"Agreeing to disagree on climate policy","volume":"111","author":"Heal","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"543","DOI":"10.2307\/2224098","article-title":"A mathematical theory of saving","volume":"38","author":"Ramsey","year":"1928","journal-title":"Econ. J."},{"key":"ref_26","unstructured":"Farmer, J.D., Geanakoplos, J., Masoliver, J., Montero, M., and Perell\u00f3, J. (2022, March 24). Discounting the Distant Future. University of Yale, Cowles Foundation Discussion Paper No. 1951. Available online: http:\/\/ssrn.com\/Abstract=1448811."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"052816","DOI":"10.1103\/PhysRevE.91.052816","article-title":"Value of the future: Discounting in random environments","volume":"91","author":"Farmer","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_28","unstructured":"Masoliver, J. (2018, January 17). The value of the distant future: Discounting in random environments. Proceedings of the 3er Congr\u00e9s d\u2019Economia i Empresa de Catalunya, Col\u00b7legi d\u2019Economistes de Catalunya, Barcelona, Spain."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Perell\u00f3, J., Montero, M., Masoliver, J., Farmer, J.D., and Geanakoplos, J. (2020). Statistical analysis and stochastic interest rate modeling for valuing the future with implications in climate change mitigation. J. Stat. Mech., 0432110.","DOI":"10.1088\/1742-5468\/ab7a1e"},{"key":"ref_30","unstructured":"Farmer, J.D., Geanakoplos, J., Masoliver, J., Montero, M., Perell\u00f3, J., and Richiardi, M.G. (2021). Discounting the distant future: What do historical bond prices imply about the long term discount rate?. J. Math. Econ., to appear."},{"key":"ref_31","unstructured":"Andersen, L.B.G., and Piterbarg, V.V. (2010). Interest Rate Modeling, Atlantic Financial Press."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2751","DOI":"10.1016\/j.jbankfin.2005.02.006","article-title":"Credit risk modeling with affine processes","volume":"29","author":"Duffie","year":"2005","journal-title":"Bank. Financ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/0304-405X(77)90016-2","article-title":"An equilibrium characterization of the terms structure","volume":"5","author":"Vasicek","year":"1977","journal-title":"J. Financ. Econ."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Masoliver, J. (2018). Random Processes, First-Passage and Escape, World Scientific.","DOI":"10.1142\/10578"},{"key":"ref_35","unstructured":"Masoliver, J., Montero, M., and Perell\u00f3, J. (2021). Valuing the future under random structural conditions: Non-stationary models for discounting, in preparation."},{"key":"ref_36","first-page":"769","article-title":"A re-examination of the traditional hypothesis about the term structure of interest rates","volume":"35","author":"Cox","year":"1981","journal-title":"J. Financ."},{"key":"ref_37","first-page":"975","article-title":"A note on the local expectation hypothesis","volume":"41","author":"Gilles","year":"1986","journal-title":"J. Financ."},{"key":"ref_38","unstructured":"Sahala, Y.A., and Hansen, M.P. (2009). Affine term structure models. The Handbook of Financial Econometrics, Elsevier."},{"key":"ref_39","unstructured":"Gardiner, C.W. (1986). Handbook of Stochastic Methods, Springer."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1086\/294846","article-title":"Mutual fund performance","volume":"39","author":"Sharpe","year":"1966","journal-title":"J. Business"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1016\/S0378-4371(03)00619-8","article-title":"Option pricing and perfect hedging on correlated stocks","volume":"330","author":"Masoliver","year":"2003","journal-title":"Physica A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"385","DOI":"10.2307\/1911242","article-title":"A theory of the term structure of interest rate","volume":"53","author":"Cox","year":"1985","journal-title":"Econometrica"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"041116","DOI":"10.1103\/PhysRevE.86.041116","article-title":"First-passage and escape problems in the Feller process","volume":"86","author":"Masoliver","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/0304-405X(78)90020-X","article-title":"On the term structure of interest rates","volume":"6","author":"Dothan","year":"1978","journal-title":"J. Financ. Econ."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"173","DOI":"10.2307\/1969318","article-title":"Two singular diffusion processes","volume":"54","author":"Feller","year":"1951","journal-title":"Ann. Math."},{"key":"ref_46","unstructured":"Pitt, H.R. (1958). Tauberian Theorems, Oxford University Press."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1137\/0505045","article-title":"Asymptotic expansion of Laplace convolutions for large argument and tail densities for certain sums of random variables","volume":"5","author":"Handelsman","year":"1974","journal-title":"SIAM J. Math. Anal."},{"key":"ref_48","unstructured":"Farmer, J.D., and Geanakoplos, J. (2022, March 24). Hyperbolic Discounting Is Rational: Valuing the Far Future with Uncertain Discount Rates. Cowles Foundation Discussion Paper No. 1719. Available online: http:\/\/ssrn.com\/abstract=1448811."},{"key":"ref_49","first-page":"150","article-title":"Asymptotic behavior of stochastic discount rates","volume":"76 A","author":"Geanakoplos","year":"2014","journal-title":"Ind. J. Stat."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/S0095-0696(02)00031-1","article-title":"Discounting the distant future: How much do uncertain rates increase valuations?","volume":"46","author":"Newell","year":"2003","journal-title":"J. Environ. Econ. Manag."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1111\/j.1468-0327.2008.00211.x","article-title":"Declining discount rates: Economic justifications and implications for long-run policy","volume":"23","author":"Gollier","year":"2008","journal-title":"Econ. Policy"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/qje\/qju036","article-title":"Very long-run discount rates","volume":"130","author":"Giglio","year":"2015","journal-title":"Q. J. Econ."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1047","DOI":"10.3982\/ECTA13447","article-title":"No-bubble conditions: Model-free test in housing markets","volume":"84","author":"Giglio","year":"2016","journal-title":"Econometrica"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"3527","DOI":"10.1093\/rfs\/hhab032","article-title":"Climate change and long-run discount rates: Evidence from real estate","volume":"34","author":"Giglio","year":"2021","journal-title":"Rev. Financ. Stud."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1006\/jeem.1998.1052","article-title":"Why the far-distant future should be discounted at its lowest possible rate","volume":"36","author":"Weitzman","year":"1998","journal-title":"J. Environ. Econ. Manag."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1002\/jae.937","article-title":"Discounting distant future: How much selection affect the certainty equivalent rate","volume":"22","author":"Groom","year":"2007","journal-title":"J. Appl. Econom."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.jeem.2008.04.004","article-title":"Social discounting under uncertainty: A cross-country comparison","volume":"57","author":"Hepburn","year":"2007","journal-title":"J. Environ. Econ. Manag."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.jeem.2015.06.003","article-title":"Declining discount rates and the Fisher Effect: Inflated past, discounted future?","volume":"73","author":"Freeman","year":"2015","journal-title":"J. Environ. Econ. Manag."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.jeconom.2008.11.001","article-title":"Parameter estimation and bias correction for diffusion processes","volume":"149","author":"Tang","year":"2009","journal-title":"J. Econom."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/496\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:48:11Z","timestamp":1760136491000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/4\/496"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,1]]},"references-count":59,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2022,4]]}},"alternative-id":["e24040496"],"URL":"https:\/\/doi.org\/10.3390\/e24040496","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints202111.0291.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,1]]}}}