Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
133 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Multi-dimensional fractional Brownian motion in the G-setting (2312.12139v3)

Published 19 Dec 2023 in math.PR and q-fin.MF

Abstract: In this paper we introduce a definition of a multi-dimensional fractional Brownian motion of Hurst index $H \in (0, 1)$ under volatility uncertainty (in short G-fBm). We study the properties of such a process and provide first results about stochastic calculus with respect to a fractional G-Brownian motion for a Hurst index $H >\frac{1}{2}$ .

Definition Search Book Streamline Icon: https://streamlinehq.com
References (37)
  1. Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance, 2(2):73–88, 1995.
  2. Time-consistency of optimal investment under smooth ambiguity. European Journal of Operational Research, 293(2):643–657, 2021.
  3. Testing for a Change of the Long-Memory Parameter. Biometrika, 83(3):627–638, 1996.
  4. Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, 2008.
  5. A generalized stochastic process: Fractional G-Brownian motion. Methodology and Computing in Applied Probability, 25(22), 2023.
  6. Wei Chen. Fractional G-White Noise Theroy, Wavelet Decomposition for Fractional G-Brownian Motion, and Bid-Ask Pricing Application to Finance Under Uncertatiny. arXiv:1306.4070v1, 2013.
  7. Ambiguity, risk, and asset returns in continuous time. Econometrica, 70(4):1403–1443, 2002.
  8. Optimal stopping under ambiguity in continuous time. Mathematics and Financial Economics, 7:29–68, 2013.
  9. Sören Christensen. Optimal decision under ambiguity for diffusion processes. Mathematical Methods of Operations Research, 77:207–226, 2013.
  10. The microstructural foundations of leverage effect and rough volatility. Finance and Stochastics, 22(2):341–280, 2018.
  11. Perfect hedging in rough Heston models. The Annals of Applied Probability, 28(6):3823–3856, 2018.
  12. The characteristic functions of rough Heston models. Mathematical Finance, 29(1):3–38, 2019.
  13. Volatility is rough. Quantitative Finance, 18(6):933–949, 2018.
  14. G-Brownian motion as rough paths and differential equations driven by G-Brownian motion. In Séminaire de Probabilités XLVI, pages 125–193. Springer, 2014.
  15. The 1/H-variation of the divergence integral with respect to the fractional brownian motion for H>1/2 and fractional Bessel processes. Stochastic Processes and their Applications, 115(1):91–115, 2005.
  16. Independence under the G-expectation framework. Journal of Theroetical Probability, 27:1011–1020, 2014.
  17. G-Lévy processes under sublinear expectation. Probability, Uncertainty and Quantitative Risk, 6(1):1–22, 2021.
  18. Fractional white noise calculus and applications to finance. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 6(1):1–32, 2003.
  19. Convergences of random variables under sublinear expectations. Chinese Annals of Mathematics, Series B, 40(1):39–54, 2018.
  20. Spatial and temporal white noises under sublinear G-expectation. arXiv:1811.02901v1, 2018.
  21. Andrei N Kolmogorov. Wienersche spiralen und einige andere interessante kurven in hilbertscen raum, cr (doklady). Acad. Sci. URSS (NS), 26:115–118, 1940.
  22. Stochastic analysis of the fractional brownian motion. Potential Analysis, 10:177–214, 1997.
  23. Terry J Lyons. Uncertain volatility and the risk-free synthesis of derivatives. Applied mathematical finance, 2(2):117–133, 1995.
  24. A risk-neutral equilibrium leading to uncertain volatility pricing. Finance and Stochastics, 22:281–295, 2018.
  25. Ivan Nourdin. Selected Aspects of Fractional Brownian Motion. Bocconi & Springer Series, 2012.
  26. David Nualart. The Malliavin Calculus and Related Topics. Springer Verlag, 2006.
  27. Superhedging and dynamic risk measures under volatility uncertainty. SIAM Journal on Control and Optimization, 50(4):2065–2089, 2012.
  28. Shige Peng. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Processes an their Applications, 118:2223–2253, 2008.
  29. Shige Peng. G-Gaussian Processes under Sublinear Expectations and q-Brownian Motion in Quantum Mechanics. arXiv:1105.1055v1, 2011.
  30. Shige Peng. Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion. Springer Science & Business Media, 2019.
  31. Stochastic calculus with respect to G-Brownian motion viewed through rough paths. Science China Mathematics, 60(1):1–20, 2017.
  32. Differential equations driven by fractional Brownian motion. Collectanea Matematics, 53(1):55–81, 2002.
  33. L.C.G. Rogers. Arbitrage with fractional Brownian motion. Mathematical Finance, 7(1):95–105, 1997.
  34. Albert N Shiryaev. On arbitrage and replication for fractal models. Research Report, 20, 1998. MaPhySto, Department of Mathematical Sciences, University of Aarhus, Denmark.
  35. Sample path properties of G-Brownian motion. Journal of Mathematical Analysis and Applications, 467(1):421–431, 2018.
  36. Mishura Yuliya S. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer, 2008.
  37. Martina Zähle. Integration with respect to fractal functions and stochastic calculus. Probability Theory and Related Fields, 111:333–374, 1998.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now