A continuous non-Brownian motion martingale with Brownian motion marginal distributions. (English) Zbl 1137.60325
Summary: We construct a continuous martingale that has the same univariate marginal distributions as Brownian motion, but that is not Brownian motion.
MSC:
| 60G44 | Martingales with continuous parameter |
| 60J65 | Brownian motion |
| 33-XX | Special functions |
| 35K57 | Reaction-diffusion equations |
| 60E99 | Distribution theory |
| 60G18 | Self-similar stochastic processes |
| 60J60 | Diffusion processes |
References:
| [1] | Albin, J. M.P., Extremes of diffusions over fixed intervals, Stochast. Proc. Appl., 48, 211-235 (1993) · Zbl 0793.60088 |
| [2] | Bibby, B. M.; Skovgaard, I. M.; Sørensen, M., Diffusion-type models with given marginal distribution and autocorrelation function, Bernoulli, 11, 191-220 (2005) · Zbl 1066.60071 |
| [3] | Bondesson, L., Factorization theory for probability distributions, Scand. Actuar. J., 1, 44-53 (1995) · Zbl 0833.60016 |
| [4] | Campi, L., Arbitrage and completeness in financial markets with given \(N\)-dimensional distributions, Decis. Econ. Finance, 27, 57-80 (2004) · Zbl 1091.91032 |
| [5] | Carr, P.; Madan, D. B., A note on sufficient conditions for no arbitrage, Finance Res. Lett., 2, 125-130 (2005) |
| [6] | Cox, J. C.; Ingersoll, J. E.; Ross, S. A., A theory of the term structure of interest rates, Econometrica, 53, 385-406 (1985) · Zbl 1274.91447 |
| [7] | Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions, vol. I (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0051.30303 |
| [8] | Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions, vol. II (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0052.29502 |
| [9] | Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Tables of Integral Transforms, vol. I (1954), McGraw-Hill: McGraw-Hill New York · Zbl 0055.36401 |
| [10] | Feller, W., Two singular diffusion problems, Ann. Math., 54, 173-182 (1951) · Zbl 0045.04901 |
| [11] | Hamza, K.; Klebaner, F. C., On nonexistence of non-constant volatility in the Black-Scholes formula, Discrete Contin. Dyn. Syst. Ser. B, 6, 829-834 (2006) · Zbl 1133.60337 |
| [12] | Hamza, K., Klebaner, F.C., 2006b. A family of non-Gaussian martingales with Gaussian marginals. arXiv:math.PR 0604127.; Hamza, K., Klebaner, F.C., 2006b. A family of non-Gaussian martingales with Gaussian marginals. arXiv:math.PR 0604127. · Zbl 1152.60039 |
| [13] | Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Springer: Springer New York · Zbl 0734.60060 |
| [14] | Kellerer, H. G., Markov-Komposition und eine Anwendung auf Martingale, Math. Ann., 198, 99-122 (1972) · Zbl 0229.60049 |
| [15] | Madan, D. B.; Yor, M., Making Markov martingales meet marginals: with explicit constructions, Bernoulli, 8, 509-536 (2002) · Zbl 1009.60037 |
| [16] | Schachermayer, W., Teichmann, J., 2006. How close are the option pricing formulas of Bachelier and Black-Merton-Scholes?. Math. Finance, to appear.; Schachermayer, W., Teichmann, J., 2006. How close are the option pricing formulas of Bachelier and Black-Merton-Scholes?. Math. Finance, to appear. · Zbl 1138.91479 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
