GENERALIZED BROWNIAN MOTIONS WITH APPLICATION TO FINANCE

Title & Authors
GENERALIZED BROWNIAN MOTIONS WITH APPLICATION TO FINANCE
Chung, Dong-Myung; Lee, Jeong-Hyun;

Abstract
Let $\small{X\;=\;(X_t,\;t{\in}[0, T])}$ be a generalized Brownian motion(gBm) determined by mean function a(t) and variance function b(t). Let $\small{L^2({\mu})}$ denote the Hilbert space of square integrable functionals of $\small{X\;=\;(X_t - a(t),\; t {in} [0, T])}$. In this paper we consider a class of nonlinear functionals of X of the form F(. + a) with $\small{F{in}L^2({\mu})}$ and discuss their analysis. Firstly, it is shown that such functionals do not enjoy, in general, the square integrability and Malliavin differentiability. Secondly, we establish regularity conditions on F for which F(.+ a) is in $\small{L^2({\mu})}$ and has its Malliavin derivative. Finally we apply these results to compute the price and the hedging portfolio of a contingent claim in our financial market model based on a gBm X.ഀĀ᐀會Ā᐀ﶖ⨀㡚�瀀ꀏ會Ā䁇ﶖ⨀⎞ऀĀ᐀會Ā᐀顇ﶖ⨀烛�Ā㰀會Ā㰀ﶖ⨀ࣜ�Ā㈀會Ā㈀䡈ﶖ⨀ꃜ�Ā᐀會Ā᐀ꁈﶖ⨀㣝�Ā᐀會Ā᐀ﶖ⨀택�Ā저會Ā저偉ﶖ⨀颵⎞ऀĀ저會Ā저ꡉﶖ⨀ザ⎞ഀĀ저會Ā저Jﶖ⨀좶⎞Ā저會Ā저塊ﶖ⨀࡙�؀Ā؀會Ā؀끊ﶖ⨀ꁙ�䬀Ā切會Ā切ࡋﶖ⨀µ⎞Ā搀會Ā搀恋ﶖ⨀䠤璮Ā저會Ā저롋ﶖ⨀ႛ蒥Ā저會Ā저၌ﶖ⨀耢璮ࠀĀࠀ會Āࠀ桌ﶖ⨀ᠣ璮Ā저會Ā저쁌ﶖ⨀뀣璮ጀĀ저會Ā저ᡍﶖ⨀ꢛ蒥
Keywords
generalized Brownian motion;Malliavin derivative;Black-Scholes model;Hedging portfolio;
Language
English
Cited by
1.
Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral, Archiv der Mathematik, 2016, 106, 6, 591
References
1.
S. J. Chang, J. G. Choi, and D. Skoug, Integration by parts formulas involving generalized Fourier-Feynman transforms on function sapce, Trans. Amer. Math. Soc. 355 (2003), no. 7, 2925-2948

2.
S. J. Chang and D. M Chung, Conditional function space integrals with applica- tions. Rochy Mountain J. Math. 26 (1996), no. 1, 37-62

3.
S. J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a first variation on function space, Integral Transforms Spec. Funct. 14 (2003), no. 5, 375-393

4.
Z. -Y. Huang and J. A. Yan, Introduction to in finite dimensional stochastic anal- ysis, Kluwer Academic Publishers, Dordrecht; Science Press, Beijing, 2000

5.
H. -H. Kuo, White Noise Distribution Theory, CRC Press, Boca Raton, FL, 1996

6.
D. Lamberton and B. Lapeyre, Introduction to stochastic calculus applied to finance, Chapman & Hall, London, 1996

7.
J. H. Lee, The linear space of generalized Browian motions with applications, Proc. Amer. Math. Soc. 133 (2005), no. 7, 2147-2155

8.
R. C. Merton, Theory of rational option pricing, Bell J. Econom. and Management Sci. 4 (1973), 141-183

9.
D. Nualart, The Malliavin calculus and related topics, Springer-Verlag, 1995

10.
B. Oksendal, An introduction to Malliavin calculus with applications to econom- ics, Working paper, No 3/96, Norwegian School of Economics and Business Administration, 1996

11.
J. Potthoff and M. Timpel, On a dual pair of smooth and generalized random variables, Potential Anal. 4 (1995), 647-654

12.
P. Wilmott, Derivatives, John Wiley and Sons Ltd, 1998

13.
J. Yeh, Stochastic process and the Wiener integral, Marcel Dekker, Inc., New York, 1973