## No arbitrage under transaction costs, with fractional Brownian motion and beyond.(English)Zbl 1133.91421

Summary: We establish a simple no-arbitrage criterion that reduces the absence of arbitrage opportunities under proportional transaction costs to the condition that the asset price process may move arbitrarily little over arbitrarily large time intervals.
We show that this criterion is satisfied when the return process is either a strong Markov process with regular points, or a continuous process with full support on the space of continuous functions. In particular, we prove that proportional transaction costs of any positive size eliminate arbitrage opportunities from geometric fractional Brownian motion for $$H \in (0,1)$$ and with an arbitrary continuous deterministic drift.

### MSC:

 91G80 Financial applications of other theories 60G22 Fractional processes, including fractional Brownian motion 60J25 Continuous-time Markov processes on general state spaces
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### References:

 [1] DOI: 10.1109/18.21218 · Zbl 0664.94003 [2] DOI: 10.1007/s00780-004-0144-5 · Zbl 1092.91021 [3] Bouleau N., Dirichlet Forms and Analysis on Wiener Space, Vol. 14 of de Gruyter Studies in Mathematics (1991) · Zbl 0748.60046 [4] DOI: 10.1007/s007800300101 · Zbl 1035.60036 [5] Cutland N. J., Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), Vol. 36 of Progr. Probab. pp 327– (1995) [6] DOI: 10.1007/s002459911019 · Zbl 0960.91053 [7] Decreusefond L., Theory and Applications of Long-Range Dependence pp 203– (2003) [8] DOI: 10.1023/A:1008634027843 · Zbl 0924.60034 [9] DOI: 10.1007/BF01450498 · Zbl 0865.90014 [10] Delbaen F., Ann. Appl. Probab. 5 (4) pp 926– (1995) [11] Dellacherie C., Probabilities and Potential. B, Vol. 72 of North-Holland Mathematics Studies (1982) [12] Follmer H., Stochastic Finance, Vol. 27 of de Gruyter Studies in Mathematics (2002) [13] Grenander U., Abstract Inference, Wiley Series in Probability and Mathematical Statistics (1981) [14] DOI: 10.1214/aoap/1037125861 · Zbl 1016.60065 [15] DOI: 10.1007/s007800100062 · Zbl 1026.60051 [16] DOI: 10.1007/s007800200089 · Zbl 1064.60085 [17] DOI: 10.1016/S0304-4068(00)00064-1 · Zbl 0986.91012 [18] DOI: 10.1007/BF00538863 · Zbl 0194.49003 [19] Karatzas I., Brownian Motion and Stochastic Calculus, Vol. 113 of Graduate Texts in Mathematics, 2. ed. (1991) · Zbl 0734.60060 [20] Karatzas I., Methods of Mathematical Finance, Vol. 39 of Applications of Mathematics (New York) (1998) · Zbl 0941.91032 [21] Li W. V., Stochastic Processes: Theory and Methods, Vol. 19 of Handbook of Statist. pp 533– (2001) [22] Liptser R. S., Theory of Martingales, Vol. 49 of Mathematics and Its Applications (Soviet Series) (1989) · Zbl 0728.60048 [23] Maheswaran S., Models, Methods, and Applications of Econometrics: Essays in Honor of A. R. Bergstrom pp 301– (1993) [24] DOI: 10.2307/3318691 · Zbl 0955.60034 [25] DOI: 10.1007/s440-000-8016-7 · Zbl 04560196 [26] Revuz D., Continuous Martingales and Brownian Motion, Vol. 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2. ed. (1994) · Zbl 0804.60001 [27] DOI: 10.1111/1467-9965.00025 · Zbl 0884.90045 [28] Rogers L. C. G., Diffusions, Markov Processes, and Martingales, Vol. 2, 2. ed. (2000) · Zbl 0949.60003 [29] DOI: 10.1016/S0304-4149(98)00025-8 · Zbl 0934.91022 [30] Samko S. G., Fractional Integrals and Derivatives (1993) [31] DOI: 10.1111/j.0960-1627.2004.00180.x · Zbl 1119.91046 [32] A. N.Shiryaev(1998 ): On Arbitrage and Replication for Fractal Models , Technical report, MaPhySto. [33] DOI: 10.1007/PL00013536 · Zbl 0978.91037 [34] DOI: 10.1007/s007800050049 · Zbl 0924.90029
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