Paper: arXiv 2510.18159

Authors: Andrey Itkin

Abstract

This paper introduces a semi-analytical method for pricing American options on assets (stocks, ETFs) that pay discrete and/or continuous dividends. The problem is notoriously complex because discrete dividends create abrupt price drops and affect the optimal exercise timing, making traditional continuous-dividend models unsuitable. Our approach utilizes the Generalized Integral Transform (GIT) method introduced by the author and his co-authors in a number of papers, which transforms the pricing problem from a complex partial differential equation with a free boundary into an integral Volterra equation of the second or first kind. In this paper we illustrate this approach by considering a popular GBM model that accounts for discrete cash and proportional dividends using Dirac delta functions. By reframing the problem as an integral equation, we can sequentially solve for the option price and the early exercise boundary, effectively handling the discontinuities caused by the dividends. Our methodology provides a powerful alternative to standard numerical techniques like binomial trees or finite difference methods, which can struggle with the jump conditions of discrete dividends by losing accuracy or performance. Several examples demonstrate that the GIT method is highly accurate and computationally efficient, bypassing the need for extensive computational grids or complex backward induction steps.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly mathematically dense, featuring advanced transformations like the Generalized Integral Transform and solving Volterra integral equations with singularities. However, it lacks empirical validation through backtesting, code implementations, or statistical performance metrics, focusing instead on theoretical derivation and numerical experiments.

Research Flowchart

  flowchart TD
  A["Research Goal: Price American Options with Hybrid Dividends"] --> B["Model: GBM with Discrete Cash & Proportional Dividends"]
  B --> C["Apply GIT Method"]
  C --> D["Transform PDE with Free Boundary"]
  D --> E["Solve Integral Volterra Equation"]
  E --> F{"Compute Price & Early Exercise Boundary"}
  F --> G["Outcomes: High Accuracy & Computational Efficiency"]
  G --> H["Comparison: Superior to Binomial/Finite Difference Methods"]
  style A fill:#e1f5fe
  style G fill:#e8f5e8
  style H fill:#e8f5e8

Cited References (307)

  1. Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein-Uhlenbeck process (2021)
  2. Closed form optimal exercise boundary of the American put option (2021)
  3. American options in time-dependent one-factor models: Semi-analytic pricing, numerical methods and ML support (2024)
  4. Semi-analytic pricing of American options in time-dependent jump-diffusion models with exponential jumps — → 2308.08760
  5. High-performance American option pricing (2016)
  6. Simple/Robust/Fast/Smooth Finite Difference Methods for American Options
  7. Fast American Option Pricing: The Double-Boundary Case
  8. Generalized Integral Transforms in Mathematical Finance (2021)
and 299 more
  1. Three-Month SOFR (2024) — link
  2. SOFR Derivative Pricing Using a Short Rate Model (2021) — link
  3. Trading SOFR Options (2019)
  4. SOFR Futures and Options (2022)
  5. Semi-closed form prices of barrier options in the Hull-White model (2020)
  6. Pricing Interest-Rate-Derivative Securities (1990)
  7. Interest Rate Modeling (2010)
  8. Analytic risk-free-rates option pricing with smile and skew (2023)
  9. Robust numerical methods for PDE models of Asian options (1997)
  10. A Simple Multi-Curve Model for Pricing SOFR Futures and Other Derivatives (2018) — link
  11. Finite difference scheme versus piecewise binomial lattice for interest rates under the skew CEV model (2023)
  12. The Value of an Asian Option. (1995)
  13. A P.D.E. approach to Asian options: analytical and numerical evidence (1997)
  14. Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach (2023)
  15. Discounted optimal stopping for maxima in diffusion models with finite horizon (2006)
  16. Pricing of Arithmetic Average Asian Option by Combining Variance Reduction and Quasi-Monte Carlo Method (2023)
  17. Negative Interest Rates for Dummies (2020) — [link](https://derivativelogic.com/dl-report-category/negative-interest-rates-for-dummies#:~:text=Could SOFR go negative?,read that right, 0.01%.)
  18. A penalty method for American options with jump diffusion processes (2004)
  19. Computational Methods for Solving System of Volterra Integral Equation (2011)
  20. Associated Weber–Orr Transform, Bio–Savart Law and Explicit Form of the Solution of 2D Stokes System in Exterior of the Disc (2019) — link
  21. Weber’s integral Theorem (1921)
  22. Diffusion Processes with Reflection (2009)
  23. Encyclopedia of Quantitative Finance (2010)
  24. Semi-closed form prices of barrier options in the time-dependent CEV and CIR models (2020)
  25. Tata Lectures on Theta (1983)
  26. American option (2022) — link
  27. American Call Options Under Jump‐Diffusion Processes – A Fourier Transform Approach (2009)
  28. Dynamic term structure models for SOFR futures (2021)
  29. Semi-analytical pricing of options written on SOFR futures — → 2409.04903
  30. Real Options and American Derivatives: The Double Continuation Region (2015)
  31. FX Open Forward (2024)
  32. Alternative Characterizations of American Put Options (1992)
  33. Pricing American options under negative rates (2021)
  34. Bitcoin: jumps, convenience yields, and option prices (2022)
  35. Fractional cointegration in Bitcoin spot and futures markets (2021)
  36. Mathematical Models of Financial Derivatives (2008)
  37. Controlled Diffusion Processes (1980)
  38. The stochastic behavior of commodity prices: Implications for valuation and hedging (1997)
  39. The Trap of Complacency in Predicting the Maximum (2007)
  40. A Change-of-Variable Time on Curves (2005)
  41. On American VIX options under the generalized 3/2 and 1/2 models (2017)
  42. Floating exercise boundaries for American options in time-inhomogeneous models — → 2502.00740
  43. A Fast and Accurate FFT-Based Method for Pricing Early-Exercise Options under Lévy Processes (2008)
  44. The H eston stochastic volatility model with piecewise constant parameters - efficient calibration and pricing of window barrier options (2018)
  45. A Change-of-Variable Formula with Local Time on Surfaces (2007)
  46. American option pricing under stochastic volatility: an efficient numerical approach (2008)
  47. Pricing American Options under Stochastic Volatility (2006)
  48. Change of Variables with Local Time on Surfaces for Jump Processes (2019) — link
  49. Predicting Functionals of Brownian Motion Through Local Time-Space Calculus (2009)
  50. Exercise boundary of the American Put near maturity in an exponential \LY model (2013)
  51. The evaluation of American option prices under stochastic volatility and jump-diffusion dynamics using the method of lines (2009)
  52. Pricing American options under stochastic volatility (2005)
  53. Limit Theorems for Stochastic Processes (1987)
  54. A survey of the integral representation of American option prices (2004)
  55. American and European options in multi-factor jump-diffusion models, near expiry (2008)
  56. Financial modelling with jump processes (2004)
  57. American and Bermudan Option Pricing by Integral Methods: the Case of Discrete Dividends. (2025)
  58. High-Performance Applications of the Non-Uniform Fast Fourier Transform to Option Pricing (2025) — link
  59. Options on realized variance by transform methods: a non-a ne stochastic volatility model (2012)
  60. Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model (2016)
  61. Pricing options on realized variance (2005)
  62. Non-Gaussian Merton-Black-Scholes Theory (2002)
  63. Estimating the parameters of 3/2 stochastic volatility model with jump (2023)
  64. Rational and Exponential Legendre Tau Method on Steady Flow of a Third Grade Fluid in a Porous Half Space
  65. On the Transition Density of the Time-Inhomogeneous 3/2 Model: A Unified Approach for Models Related to Squared Bessel Process (2025)
  66. Some Remarks on First Passage of \LY Processes, the American Put and Pasting Principles (2005)
  67. Advances in Finance and Stochastics (2002)
  68. Some applications of \LY processes in insurance and finance (2004)
  69. An efficient method for option pricing with discrete dividend payment (2008)
  70. A generalization of option pricing to price-limit markets (2020)
  71. The Valuation of American Options for a Class of Diffusion Processes (2002)
  72. The analytic valuation of American options (1990)
  73. An Alternative Approach to the Valuation of American Options and Applications (1996)
  74. An analysis of American options (1992)
  75. American options valuation in time-dependent jump-diffusion models via integral equations and characteristic functions — → 2506.18210
  76. American Options: Symmetry Properties (2010)
  77. Pricing and Hedging American Options: A Recursive Integration Method (1996)
  78. Handbook of stochastic methods for physics, chemistry and the natural sciences (2004)
  79. Handbook of Stochastic Methods (1996)
  80. A New Version of the Adaptive Fast Gauss Transform for Discrete and Continuous Sources (2024)
  81. A Tree-Based Method to Price American Options in the Presence of Discrete Dividends (2006)
  82. Calibration to American options: numerical investigation of the de-Americanization method (2018)
  83. Local Volatility from American Options (2017)
  84. Fast Trees for Options with Discrete Dividends (2013)
  85. An integral equation for American put options on assets with general dividend processes (2011)
  86. On the early exercise boundary of the American put option (2002)
  87. The Implied Strike (2015) — link
  88. A new integral equation formulation for American put options (2018)
  89. American options on assets with dividends near expiry (2002)
  90. An Analytic Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends (1977) — link
  91. A Note on an Analytical Valuation Formula for Unprotected American Call Options on Stocks with Known Dividends (1979) — link
  92. On the Valuation of American Call Options on Stocks with Known Dividends (1981) — link
  93. Finessing Fixed Dividends (2002)
  94. Computational Methods in Finance (2013)
  95. Option Pricing: Mathematical Models and Computation (1993)
  96. Valuing American Options by Simulation: A Simple Least-Squares Approach (2001) — link
  97. Fast Option Pricing with Discrete Dividends (2024)
  98. The Numerical Solution of the American Option Pricing Problem: Finite Difference and Transform Approaches (2006)
  99. American Option Pricing by the Finite Difference Method (2025)
  100. Tools for computational finance (2017)
  101. Advanced mathematical methods for scientists and engineers (1978)
  102. Finite-difference solution ansatz approach in least-squares M onte C arlo (2025) — link
  103. Introduction to stochastic calculus with applications (2005)
  104. Analytical Methods for Solution of Non-Stationary Heat Conductance Boundary Problems in Domains with Moving Boundaries (1999)
  105. Handbook of Integral Equations: Second Edition (2008)
  106. The Cauchy method of residues: Theory and Applications (1984)
  107. A Closed-Form Solution for Optimal Mean-Reverting Trading Strategies (2020)
  108. Local time and the pricing of time-dependent barrier options (2010)
  109. Differential relations of theta functions (1995)
  110. Mpmath Library (2007) — link
  111. A new Tikhonov regularization mathod (2001)
  112. Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein-Uhlenbeck process (2020)
  113. An Elementary Introduction to Stochastic Interest Rate Modeling (2012)
  114. Boundary conditions for the single-factor term structure equation (2011)
  115. Implementing Models in Quantitative Finance: Methods and Cases (2008)
  116. Pricing European options on zero-coupon bonds with a fitted finite volume method (2017)
  117. Partial Differential Equations of Parabolic Type (1964)
  118. Equations of mathematical physics (1963)
  119. On the First Hitting Time Density of an Ornstein-Uhlenbeck Process (2018) — link
  120. The Fast Generalized Gauss Transform (2010)
  121. Analytic Swaption Pricing in the Black-Karasinski Model (2020) — link
  122. Options, Futures, and Other Derivatives (2011)
  123. Handbook of linear partial differential equations for engineers and scientists (2002)
  124. General Short-Rate Analytics (2011)
  125. An Effective Approximation for Zero-Coupon Bonds and Arrow-Debreu Prices in the Black-Karasinski Model (2014)
  126. Analytic Option Prices for the Black-Karasinski Short Rate Model (2017) — link
  127. The One-Dimensional Heat Equation (1984)
  128. Linear and Nonlinear Integral Equations (2011)
  129. Semi-closed form prices of barrier options in the Hull-White model (2020)
  130. A jump to default extended CEV model: an application of B essel processes (2006)
  131. A Theory of the Term Structure of Interest Rates (1985)
  132. Notes on the B essel Process (2018) — link
  133. Further Results on the Constant Elasticity of Variance Call Option Pricing Model (1982)
  134. Numerical analysis of an extended structural default model with mutual liabilities and jump risk (2018)
  135. Representations of the First Hitting Time Density of an Ornstein-Uhlenbeck Process (2005)
  136. Modeling Volatility and Valuing Derivatives Under Anchoring (2014)
  137. Boundary-Crossing Identities for Diffusions Having the Time-Inversion Property. (2010)
  138. On Three Important Problems in Mathematical Finance (2020)
  139. Kinetic Theory of Phase Transformations (1978)
  140. First Passage Time Problem: A Fokker-Planck Approach (2004)
  141. On the First Hitting Time Density of an Ornstein-Uhlenbeck Process (2018) — link
  142. A guide to first-passage processes (2001)
  143. Model-Free Backward and Forward Nonlinear PDEs for Implied Volatility (2020)
  144. Two-step collocation methods for two-dimensional Volterra integral equations of the second kind (2019)
  145. Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM (2013)
  146. The Numerical Computation of the Product of Conjugate Imaginary Gamma Functions (1940)
  147. Uniform Asymptotic Expansions of ConfluentHypergeometric Functions (1978)
  148. A Review of the Fed’s Unconventional Monetary Policy (2018)
  149. A short history of mathematical population dynamics (2011)
  150. The Logistic SDE (2015)
  151. Notice sur la loi que la population suit dans son accroisseement (1838)
  152. Decisions modeling the dynamics of commodity prices for investement decisions under uncertainty (2010)
  153. A new logistic-type model for pricing European options (2015)
  154. Market Self-Learning of Signals, Impact and Optimal Trading:Invisible Hand Inference with Free Energy (2018)
  155. Second order equations with non-negative characteristic form (1973)
  156. A Heat Transfer Textbook (2019)
  157. Fast Integral Equation Methods for the Modified Helmholtz Equation (2011)
  158. Boundary Integral Operators for the Heat Equation (1990)
  159. Jump conditions for single and double layer potentials (2009) — link
  160. Integrals depending on parameters (2016)
  161. Potential Theory and Its Applications to Basic Problems of MathematicalPhysics (1967)
  162. Works on the theory of potential (1949)
  163. Semi-analytical solution of a McKean-Vlasov equation with feedback through hitting boundary (2019)
  164. Negative oil prices put spotlight on investors (2020)
  165. Options Traders Adapt to Electronic Markets in Pandemic (2020) — link
  166. Valuing scenarios with real option pricing (2020)
  167. Podcast: Kaminski and Ronn on negative oil and options pricing (2020)
  168. Multilayer heat equations: application to finance (2022)
  169. Operational calculus based on the two- sided Laplace integral (1950)
  170. From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed’s policy (2020) — link
  171. NIST Digital Library of Mathematical Functions (2020) — link
  172. Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit (2022)
  173. Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit (2020) — link
  174. Drift without flux: Brownian walker with a space-dependent diffusion coefficient (2001)
  175. On the constructions of the skew Brownian motion (2006)
  176. Fokas method for a multi-domain linear reaction-diffusion equation with discontinuous diffusivity (2014)
  177. Iontophoretic transdermal drug delivery: A multi-layered approach (2016)
  178. Multilayer diffusion growth insilicon-molybdenum interactions (2002)
  179. Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions (2018)
  180. Option Pricing Model: Comparing Louis Bachelier with Black-Scholes Merton (2016) — link
  181. Switch to Bachelier Options Pricing Model (2020)
  182. Pricing Financial Instruments. The Finite-Difference method. (2000)
  183. Pricing Barrier Bond options with one-factor interest rate models (2003)
  184. Methods of Mathematical Physics (1962)
  185. Fractional Integrals and Derivatives. Theory and Applications (1993)
  186. Numerical simulation of random walks on regular fractal sets (2014)
  187. Theory of ordinary differential equations (1955)
  188. Numerical Solution for First Kind Fredholm Integral Equations by Using Sinc Collocation Method (2016)
  189. Collocation Methods for Volterra Integral and Related Functional Differential Equations (2004)
  190. A new approach to the numerical solution of weakly singular Volterra integral equations (2004)
  191. Approximation Theory and Approximation Practice (2013)
  192. Analytical Solution of the Single-Phase Stefan Problem (2009)
  193. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (2006)
  194. Commodities and Commodity Derivatives: Modeling and Pricing for Agriculturals, Metals and Energy (2009)
  195. Integral Transforms and Their Applications (2014)
  196. Linear integral equations: Theory and technique (1997)
  197. The Numerical Solution of Volterra Equations (1986)
  198. Solving Frontier Problems of Physics: The Decomposition Method (1994)
  199. Taylor-series expansion of density functionals (1994)
  200. Linear algebra and its applications (2006)
  201. Fredholm theorem for integral equations (2001)
  202. Introduction to nonlinear differential and integral equations (1962)
  203. Rank-defficient and discrete ill-posed problems (1998)
  204. The Theory of T ikhonov Regularization for F redholm Equations of the First Kind. (1984)
  205. Color Constancy and Image Segmentation Techniques for Applications to Mobile Robotics (2005)
  206. Stochastic Volatility and Local Volatility (2002) — link
  207. The Reduction Method for Valuing Derivative Securities (2002) — link
  208. A reduction algorithm for an ordinary differential equations admitting a solvable Lie group (1990)
  209. Similarity Methods for Differential Equations (1974)
  210. S tochastic P rocesses in P hysics and C hemistry (1992)
  211. Stochastic processes in physics and chemistry (2007)
  212. The Fokker-Planck Equation: Methods of Solution and Applications Second Edition (1989)
  213. The Reduction Method for Valuing Derivative Securities (2018)
  214. On the Analytical and Numerical Properties of the Truncated L aplace Transform (2014)
  215. Weber-Type Integral Transform Connected with Robin-Type Boundary Conditions (2020)
  216. A class of integral transforms (1964)
  217. Index Transforms (1996)
  218. Eigenfunction Expansions Associated with Second-Order Differential Equations (1962)
  219. On integral transform pairs arising from second order differential equations (1962)
  220. On a method of solution of some problems in diffraction theory (1939)
  221. Volterra Integral Equations. An Introduction to Theory and Applications (2017)
  222. Elements of the Theory of Functions and Functional Analysis (1999)
  223. Stochastic Processes (2017)
  224. Semigroups,Boundary Value Problems and Markov Processes (2014)
  225. An Expanded Local Variance Gamma model (2020) — link
  226. An Expanded Local Variance Gamma model (2018)
  227. Financial Modelling: Theory, Implementation and Practice with MATLAB Source (2012)
  228. Fitting Local Volatility: Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models
  229. Geometric Local Variance Gamma model (2019)
  230. The Volatility Smile (2016)
  231. Stochastic implied trees: Arbitrage pricing with stochastic term and strike structure of volatility (1998)
  232. A method of recursive images to solve transient heat diffusionin multilayer materials (2015)
  233. Advanced Equity Derivatives: Volatility and Correlation (2014)
  234. Tables of Fourier Transforms and Fourier Transforms of Distributions (1990)
  235. Vorlesungen " u ber Fouriersche Integrale (1932)
  236. Scattered Data Approximation (2005)
  237. Calculating Prices and Sensitivities for Path-Independent Derivatives Securities in Multifactor Models (1997)
  238. Universal barriers (2002)
  239. An analytical approximation method for pricing barrier options under the double H eston model (2019)
  240. Singular Perturbations for Boundary Value Problems Arising from Exotic Options (2004)
  241. An asymptotic expansion formula for Up-and-Out barrier option price under stochastic volatility model (2013)
  242. Approximate pricing of E uropean and barrier claims in a local-stochastic volatility setting (2017)
  243. Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results (2014)
  244. Semi-analytical prices for lookback and barrier options under the H eston model (2019)
  245. Modern SABR Analytics (2019)
  246. The Time-Dependent FX-SABR Model: Efficient Calibration based on Effective Parameters (2015)
  247. Arbitrage-Free SABR (2014)
  248. Pricing Continuously Monitored Barrier Options under the SABR Model: A Closed form approximation (2017)
  249. A Spectral Approach to Pricing of Arbitrage-Free SABR Discrete Barrier Options (2019)
  250. Approximate pricing of European and Barrier claims in a local-stochastic volatility setting (2017)
  251. Implied Volatilities for Mean Reverting SABR Models (2020)
  252. A Non-Linear Stochastic Volatility Model (1997)
  253. A General Asymptotic Implied Volatility for Stochastic Volatility Models (2005) — link
  254. Black’s Model of Interest Rates as Options, Eigenfunction Expansions and J apanese Interest Rates (2004)
  255. Computer graphics and a new Gibbs phenomenon for Fourier-Bessel series (1992)
  256. Divergent Series (1991)
  257. Solving the Linear Integral Equations Based on Radial Basis Function Interpolation (2014)
  258. Stable Evaluation of Gaussian Radial Basis Function Interpolants (2012)
  259. Solution of Sparse Indefinite Systems of Linear Equations (1975)
  260. Option pricing formulae using F ourier transform: Theory and application. (2010) — [link](\url{https://pfadintegral.com/docs/Schmelzle2010 Fourier Pricing.pdf})
  261. A stochastic approach to the valuation of barrier options in H eston’s stochastic volatility model (2013) — link
  262. Semi-analytical pricing of barrier options in the time-dependent $\lambda$-SABR model (2021) — link
  263. Chebfun guide (2014)
  264. Implementing Clenshaw-Curtis quadrature I and II (1972)
  265. An adaptive Filon quadrature for stochastic volatility models
  266. Semi-Analytical Pricing of Barrier Options in the Time-Dependent $\lambda$- SABR Model: Uncorrelated Case (2022)
  267. Semi-analytical pricing of barrier options in the time-dependent Heston model (2022)
  268. American Options: A Comparison of Numerical Methods (1997) — link
  269. A Review on Regression-based Monte Carlo Methods for Pricing American Options (2010)
  270. A Hedged Monte Carlo Approach to Real Option Pricing (2015)
  271. Operator Splitting Methods for American Option Pricing (2004)
  272. The Analytic Valuation of American Options (1990)
  273. Nonlinear Equations: Numerical Methods for Solving (2018)
  274. Methods for Solving Systems of Nonlinear Equations (1998)
  275. Rapidly-convergent methods for evaluating elliptic integrals and theta elliptic functions (1984)
  276. Elfun18 - A collection of MATLAB functions for the computation of elliptic integrals and Jacobian elliptic functions of real arguments (2019)
  277. A Mathematical Analysis of the Optimal Exercise Boundary for American Put Options (2007)
  278. Real Option: Definition, Valuation Methods, Example (2021) — link
  279. The Promise and Peril of Real Options (2008) — link
  280. A Brownian Model of Financial Markets (1998)
  281. A product integration method for the approximation of the early exercise boundary in the American option pricing problem (2019)
  282. Approximate solutions to severalclasses of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine (2023)
  283. Deep learning calibration of option pricing models: some pitfalls and solutions (2020)
  284. On the numerical solution of Fredholm integral equations utilizing the local radial basis function method (2019)
  285. Lie Symmetry Analysis of Differential Equations in Finance (1998)
  286. Applications of Lie Groups to Differential Equations (1993)
  287. American options in time-dependent one-factor models: Semi-analytic pricing, numerical methods and ML support — link
  288. Multilayer heat equations and their solutions via oscillating integral transforms
  289. Exploring Time-Varying Jump Intensities: Evidence from S&P500 Returns and Options
  290. Jump-diffusion stock-return model with weighted fitting of time-dependent parameters (2003)
  291. Analysis of a mixed space-time diffusion equation
  292. Development of a Numerical Phase Optimized Upwinding Combined Compact Difference Scheme for Solving the Camassa–Holm Equation with Different Initial Solitary Wave
  293. Analysis of a pseudo-parabolic equation by potential wells
  294. Partial differential equations (2010)
  295. Notes on Partial Differential Equations — link
  296. Existence and probabilistic representation of the solutions of semilinear parabolic PDEs with fractional Laplacians
  297. A spectral collocation method with piecewise trigonometric basis functions for nonlinear Volterra–Fredholm integral equations
  298. Optimal exercise boundary via intermediate functionwith jump risk (2017)
  299. A PDE method for estimation of implied volatility (2020)

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