On the stochastic integral representation of Brownian functionals
-
Ekaterine Namgalauri
Abstract
In this paper, we study the question of representing one class of Brownian functionals as a stochastic Itô integral with an explicit form of the integrand. The considered class of functionals also includes functionals that are not smooth in the sense of Malliavin, to which both the well-known Clark–Ocone formula (1984) and its generalization, the Glonti–Purtukhia formula (2017), are inapplicable.
Funding statement: The work was partially supported by the grant STEM-22-226.
References
[1] J. M. C. Clark, The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Statist. 41 (1970), 1282–1295. 10.1214/aoms/1177696903Search in Google Scholar
[2] C. Dellacherie and P.-A. Meyer, Probabilités et potentiel. Chapitres V à VIII, Act. Sci. Indust. 1385, Hermann, Paris, 1980. Search in Google Scholar
[3] O. Glonti and O. Purtukhia, Hedging of European option of integral type, Bull. Georgian Natl. Acad. Sci. (N. S.) 8 (2014), no. 3, 4–13. Search in Google Scholar
[4] O. Glonti and O. Purtukhia, Hedging of one european option of integral type in Black–Scholes model, Int. J. Eng. Innov. Technol. 4 (2014), no. 5, 51–61. Search in Google Scholar
[5] O. A. Glonti and O. G. Purtukhiya, On an integral representation of a Brownian functional (in Russian), Teor. Veroyatn. Primen. 61 (2016), no. 1, 158–164; translation in Theory Probab. Appl. 61 (2017), no. 1, 133–139. Search in Google Scholar
[6] O. A. Glonti and O. G. Purtukhiya, European option hedging with a nonsmooth payment function (in Russian), Ukraïn. Mat. Zh. 70 (2018), no. 6, 773–787; translation in Ukrainian Math. J. 70 (2018), no. 6, 890–905. Search in Google Scholar
[7] S. E. Graversen, A. N. Shiryaev and M. Ĭor, On stochastic integral representations of functionals of Brownian motion. II (in Russian), Teor. Veroyatn. Primen. 51 (2006), no. 1, 64–77; translation in Ukrainian Math. J. 51 (2007), no. 1, 65–77. Search in Google Scholar
[8] J. M. Harrison and S. R. Pliska, Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process. Appl. 11 (1981), no. 3, 215–260. 10.1016/0304-4149(81)90026-0Search in Google Scholar
[9] U. G. Haussmann, On the integral representation of functionals of Itô processes, Stochastics 3 (1979), no. 1, 17–27. 10.1080/17442507908833134Search in Google Scholar
[10] I. Karatzas, D. L. Ocone and J. Li, An extension of Clark’s formula, Stochastics Stochastics Rep. 37 (1991), no. 3, 127–131. 10.1080/17442509108833731Search in Google Scholar
[11] H. Livinska and O. Purtukhia, Stochastic integral representation of one stochastically non-smooth Wiener functional, Bull. TICMI 20 (2016), no. 2, 24–36. Search in Google Scholar
[12] D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Probab. Appl. (New York), Springer, Berlin, 2006. Search in Google Scholar
[13] D. Nualart and E. Pardoux, Stochastic calculus with anticipating integrands, Probab. Theory Related Fields 78 (1988), no. 4, 535–581. 10.1007/BF00353876Search in Google Scholar
[14] D. Ocone, Malliavin’s calculus and stochastic integral representations of functionals of diffusion processes, Stochastics 12 (1984), no. 3–4, 161–185. 10.1080/17442508408833299Search in Google Scholar
[15] D. L. Ocone and I. Karatzas, A generalized Clark representation formula, with application to optimal portfolios, Stochastics Stochastics Rep. 34 (1991), no. 3–4, 187–220. 10.1080/17442509108833682Search in Google Scholar
[16] J.-F. Renaud and B. Rémillard, Explicit martingale representations for Brownian functionals and applications to option hedging, Stoch. Anal. Appl. 25 (2007), no. 4, 801–820. 10.1080/07362990701420092Search in Google Scholar
[17] A. N. Shiryaev and M. Ĭor, On stochastic integral representations of functionals of Brownian motion. I (in Russian), Teor. Veroyatnost. i Primenen. 48 (2003), no. 2, 375–385; translation in Theory Probab. Appl. 48 (2004), no. 2, 304–313. Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1
- On completely non-Baire unions in category bases
- Square-free values of n 2 + n + 1
- The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition
- Sign-changing solutions for parametric Neumann problems with broken symmetry and arbitrary growth
- The triharmonic equation on the Heisenberg group
- On Busemann–Feller extensions of translation invariant density differentiation bases
- Multiplicative bi-skew Jordan triple derivations on prime ∗-algebra
- Nonmeasurable products of absolutely negligible sets in uncountable solvable groups
- The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
- On the stochastic integral representation of Brownian functionals
- Generalized Bessel matrix functions
- Degenerate time-fractional diffusion equation with initial and initial-boundary conditions
- Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution
- 𝐿𝑝(⋅) − 𝐿𝑞(⋅) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension
- Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras
Articles in the same Issue
- Frontmatter
- The second cohomology spaces of 𝒦(1) with coefficients in the superspace of weighted densities and deformations of the superspace of symbols on S 1|1
- On completely non-Baire unions in category bases
- Square-free values of n 2 + n + 1
- The stability and convergence analysis for singularly perturbed Sobolev problems with Robin type boundary condition
- Sign-changing solutions for parametric Neumann problems with broken symmetry and arbitrary growth
- The triharmonic equation on the Heisenberg group
- On Busemann–Feller extensions of translation invariant density differentiation bases
- Multiplicative bi-skew Jordan triple derivations on prime ∗-algebra
- Nonmeasurable products of absolutely negligible sets in uncountable solvable groups
- The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation
- On the stochastic integral representation of Brownian functionals
- Generalized Bessel matrix functions
- Degenerate time-fractional diffusion equation with initial and initial-boundary conditions
- Some new characterizations of EP elements, partial isometries and strongly EP elements in rings with involution
- 𝐿𝑝(⋅) − 𝐿𝑞(⋅) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension
- Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras
