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Introduction to Malliavin Calculus

Author

Listed:
  • Nualart,David
  • Nualart,Eulalia

Abstract

This textbook offers a compact introductory course on Malliavin calculus, an active and powerful area of research. It covers recent applications, including density formulas, regularity of probability laws, central and non-central limit theorems for Gaussian functionals, convergence of densities and non-central limit theorems for the local time of Brownian motion. The book also includes a self-contained presentation of Brownian motion and stochastic calculus, as well as Lévy processes and stochastic calculus for jump processes. Accessible to non-experts, the book can be used by graduate students and researchers to develop their mastery of the core techniques necessary for further study.

Suggested Citation

  • Nualart,David & Nualart,Eulalia, 2018. "Introduction to Malliavin Calculus," Cambridge Books, Cambridge University Press, number 9781107611986.
    • Handle: RePEc:cup:cbooks:9781107611986
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    Citations

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    Cited by:

    1. Nourdin, Ivan & Pu, Fei, 2022. "Gaussian fluctuation for Gaussian Wishart matrices of overall correlation," Statistics & Probability Letters, Elsevier, vol. 181(C).
    2. Nourdin, Ivan & Nualart, David & Peccati, Giovanni, 2021. "The Breuer–Major theorem in total variation: Improved rates under minimal regularity," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 1-20.
    3. Chen, Xingzhi & Xu, Xin & Tian, Baodan & Li, Dong & Yang, Dan, 2022. "Dynamics of a stochastic delayed chemostat model with nutrient storage and Lévy jumps," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    4. Masahiro Handa & Noriyoshi Sakuma & Ryoichi Suzuki, 2024. "A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$ L 1 -pure jump additive processes and its application to portfolio optimization," Annals of Finance, Springer, vol. 20(3), pages 329-352, September.
    5. Tsubasa Nishimura & Kenji Yasutomi & Tomooki Yuasa, 2022. "Higher-Order Error Estimates of the Discrete-Time Clark–Ocone Formula," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2518-2539, December.
    6. Jaehyun Kim & Hyungbin Park & Jonghwa Park, 2019. "Pricing and hedging short-maturity Asian options in local volatility models," Papers 1911.12944, arXiv.org, revised Apr 2024.
    7. Ehsan Azmoodeh & Yuliya Mishura & Farzad Sabzikar, 2022. "How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?," Journal of Theoretical Probability, Springer, vol. 35(1), pages 484-527, March.
    8. Hyungbin Park, 2021. "Influence of risk tolerance on long-term investments: A Malliavin calculus approach," Papers 2104.00911, arXiv.org.
    9. Čoupek, Petr & Duncan, Tyrone E. & Pasik-Duncan, Bozenna, 2022. "A stochastic calculus for Rosenblatt processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 853-885.
    10. Ernst, Philip A. & Huang, Dongzhou & Viens, Frederi G., 2023. "Yule’s “nonsense correlation” for Gaussian random walks," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 423-455.
    11. Fenge Chen & Bing Li & Xingchun Peng, 2022. "Portfolio Selection and Risk Control for an Insurer With Uncertain Time Horizon and Partial Information in an Anticipating Environment," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 635-659, June.
    12. Kohatsu-Higa, Arturo & Nualart, Eulalia & Tran, Ngoc Khue, 2022. "Density estimates for jump diffusion processes," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    13. Hiroaki Hata & Nien-Lin Liu & Kazuhiro Yasuda, 2022. "Expressions of forward starting option price in Hull–White stochastic volatility model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 101-135, June.
    14. Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2022. "On the implied volatility of Asian options under stochastic volatility models," Papers 2208.01353, arXiv.org, revised Mar 2024.
    15. Azmoodeh, Ehsan & Ljungdahl, Mathias Mørck & Thäle, Christoph, 2022. "Multi-dimensional normal approximation of heavy-tailed moving averages," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 308-334.
    16. Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2023. "On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model," Papers 2308.15341, arXiv.org, revised Sep 2024.
    17. Mauricio Elizalde & Carlos Escudero & Tomoyuki Ichiba, 2022. "Optimal investment with insider information using Skorokhod & Russo-Vallois integration," Papers 2211.07471, arXiv.org.
    18. Elisa Al`os & Eulalia Nualart & Makar Pravosud, 2023. "On the implied volatility of Inverse options under stochastic volatility models," Papers 2401.00539, arXiv.org, revised Sep 2024.
    19. Ji Huang, 2023. "A Probabilistic Solution to High-Dimensional Continuous-Time Macro and Finance Models," CESifo Working Paper Series 10600, CESifo.
    20. Jie Xiong & Zuo quan Xu & Jiayu Zheng, 2019. "Mean-variance portfolio selection under partial information with drift uncertainty," Papers 1901.03030, arXiv.org, revised Oct 2020.
    21. Ivan Nourdin & Giovanni Peccati & Xiaochuan Yang, 2022. "Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance," Journal of Theoretical Probability, Springer, vol. 35(3), pages 2020-2037, September.
    22. Levental, S. & Vellaisamy, P., 2023. "Formulas for the divergence operator in isonormal Gaussian space," Statistics & Probability Letters, Elsevier, vol. 194(C).
    23. Ayub Ahmadi & Mahdieh Tahmasebi, 2024. "Pricing and delta computation in jump-diffusion models with stochastic intensity by Malliavin calculus," Papers 2405.00473, arXiv.org.
    24. Ruzong Fan & Hong-Bin Fang, 2022. "Stochastic functional linear models and Malliavin calculus," Computational Statistics, Springer, vol. 37(2), pages 591-611, April.

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