American Institute of Mathematical Sciences

Advanced
August  2010, 27(3): 907-917. doi: 10.3934/dcds.2010.27.907

A nonlinear partial integro-differential equation from mathematical finance

Frederic Abergel 1, and  Remi Tachet 1,

1. 

Grande Voie des Vignes, Chatenay-Malabry, 92295, France, France

Received  July 2009 Revised  January 2010 Published  March 2010

Consistently fitting vanilla option surfaces is an important issue when it comes to modeling in finance. As far as local and stochastic volatility models are concerned, this problem boils down to the resolution of a nonlinear integro-differential pde. The non-locality of this equation stems from the quotient of two integral terms and is not defined for all bounded continuous functions. In this paper, we use a fixed point argument and suitable a priori estimates to prove short-time existence of solutions for this equation.
Keywords: Analysis of pde's, local and stochastic volatility models, nonlinear parabolic partial integro-differential equation, fixed point methods, short-time existence, calibration of vanillas..
Mathematics Subject Classification: Primary: 35A01, 35K55, 35R09, 47G20; Secondary: 60H1.
Citation: Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907
[1]

Yong-Kui Chang, Xiaojing Liu. Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability. Evolution Equations and Control Theory, 2020, 9 (3) : 845-863. doi: 10.3934/eect.2020036
[2]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217
[3]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[4]

Changling Xu, Tianliang Hou. Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations. Electronic Research Archive, 2020, 28 (2) : 897-910. doi: 10.3934/era.2020047
[5]

Wenxiu Gong, Zuoliang Xu. An alternative tree method for calibration of the local volatility. Journal of Industrial and Management Optimization, 2022, 18 (1) : 137-156. doi: 10.3934/jimo.2020146
[6]

Elena Bonetti, Elisabetta Rocca. Global existence and long-time behaviour for a singular integro-differential phase-field system. Communications on Pure and Applied Analysis, 2007, 6 (2) : 367-387. doi: 10.3934/cpaa.2007.6.367
[7]

Laura Cremaschi, Carlo Mantegazza. Short-time existence of the second order renormalization group flow in dimension three. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5787-5798. doi: 10.3934/dcds.2015.35.5787
[8]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885
[9]

Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021021
[10]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977
[11]

Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044
[12]

Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059
[13]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053
[14]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[15]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537
[16]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129
[17]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57
[18]

Martino Bardi, Annalisa Cesaroni, Daria Ghilli. Large deviations for some fast stochastic volatility models by viscosity methods. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 3965-3988. doi: 10.3934/dcds.2015.35.3965
[19]

Hyung Ju Hwang, Thomas P. Witelski. Short-time pattern formation in thin film equations. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 867-885. doi: 10.3934/dcds.2009.23.867
[20]

Julio Guerrero, Giuseppe Orlando. Stochastic local volatility models and the Wei-Norman factorization method. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022026

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (93)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy