Applications of the Quadratic Covariation Differentiation Theory: Variants of the Clark-Ocone and Stroock's Formulas

References

• Allouba , H. 2006 . A differentiation theory for Itô's calculus . Stoch. Anal. Appl. 24 : 367 – 380 .
   Web of Science ®Google Scholar
• Allouba , H. 2011 . A universal p-covariation calculus. Preprint (In prep.).
   Google Scholar
• Allouba , H. A differentiation theory for Itô's calculus: Generalizations, applications, and ramifications. Preprint (In prep.).
   Google Scholar
• Allouba , H. , and Zheng , W. 2001 . Brownian-time processes: the PDE connection and the half-derivative generator . Ann. Probab. 29 : 1780 – 1795 .
   Web of Science ®Google Scholar
• Allouba , H. 2002. Brownian-time processes: the PDE connection II and the corresponding Feynman-Kac formula. Trans. Amer. Math. Soc. 354:4627–4637 (online).
   Web of Science ®Google Scholar
• Allouba , H. , and Fontes , R. Semi-SPDEs models in finance and their corresponding Black-Scholes-Merton system. (Submitted).
   Google Scholar
• Bhattacharya , R. , and Waymire , E. 1990 . Stochastic Processes with Applications . John Wiley & Sons , Inc .
   Google Scholar
• Clark , J. M. C. 1970 . The representation of functionals of Brownian motion as stochastic integrals. Ann. Math. Statist. 41:MR0270448. [Correction to the paper, Ann. Math. Statist. 42 (1971) 1778].
   Google Scholar
• Doleans-Dade , C. 1967 . Integrales stochastiques dependant d' un' paramètre . Publ. Inst. Stat. Univ. Paris 16 : 23 – 24 .
   Google Scholar
• Durrett , R. 1996 . Stochastic Calculus: A Practical Introduction . CRC Press , New York .
   Google Scholar
• Di Nunno , G. 2002 . Stochastic integral representations, stochastic derivatives, and minimal variance hedging . Stoch. Stoch. Rep. 73 : 181 – 198 .
   Google Scholar
• Di Nunno , G. , Øksendal , B. , and Proske , F. 2009 . Malliavin Calculus for Lévy Processes with Applications to Finance. Universitext. Springer-Verlag, Berlin.
   Google Scholar
• Fontes , R. 2010 . Applications of Allouba's differentiation theory and semi-SPDEs. Ph.D. Dissertation, Kent State University.
   Google Scholar
• Houdré , C. 1990 . Linear Fourier and stochastic analysis . Probab. Theory Related Fields 87 : 167 – 188 .
   Web of Science ®Google Scholar
• Itô , K. 1944 . Stochastic integral . Proc. Imperial. Acad. Tokyo 20 : 519 – 524 .
   Google Scholar
• Itô , K. 1951 . Multiple Wiener integral . J. Math. Soc. Japan 3 : 157 – 169 .
   Google Scholar
• Jacod , J. 1979 . Calcul Stochastique et Problémes de Martingales. Lecture Notes in Math. 714 . Springer-Verlag , Berlin .
   Google Scholar
• Karatzas , I. , Ocone , D. L. , and Li , J. 1991 . An extension of Clark's formula . Stoch. Stoch. Rep. 37 : 127 – 131 .
   Google Scholar
• Karatzas , I. , and Shreve , S. E. 1991 . Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics, 113. Springer-Verlag, New York.
   Google Scholar
• Malliavin , P. 1978 . Stochastic Calculus of Variation and Hypoelliptic Operators. Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), pp. 195–263, Wiley, New York.
   Google Scholar
• Ocone , D. 1984 . Malliavin's calculus and stochastic integral representations of functionals of diffusion processes . Stochastics 12 : 161 – 185 .
   Google Scholar
• Ocone , D. L. , and Karatzas , I. 1991 . A generalized Clark representation formula, with application to optimal portfolios . Stoch. Stoch. Rep. 34 : 187 – 220 .
   Google Scholar
• Øksendal , B. 2003 . Stochastic Differential Equations. An Introduction with Applications, 6th ed. Universitext. Springer-Verlag, Berlin.
   Google Scholar
• Royden , H. 1988 . Real Analysis, , 3rd ed. Macmillan , New York .
   Google Scholar
• Shreve , S. E. 2004 . Stochastic Calculus for Finance. II. Continuous-Time Models. Springer Finance, New York.
   Google Scholar
• Stroock , D. W. 1987 . Homogeneous Chaos Revisited. Séminaire de Probabilités, XXI, 1–7, Lecture Notes in Math., 1247 . Springer , Berlin .
   Google Scholar
• van Neerven , J. , and Veraar , M. C. 2006 . On the Stochastic Fubini Theorem in Infinite Dimensions. Stochastic Partial Differential Equations and Applications|VII, 323–336, Lect. Notes Pure Appl. Math., 245 . Chapman & Hall/CRC , Boca Raton , FL .
   Google Scholar