## Arbitrage in a discrete time model of a financial market with taxation proportional to the portfolio size.(Ukrainian, English)Zbl 1224.91142

Teor. Jmovirn. Mat. Stat. 81, 155-163 (2009); translation in Theory Probab. Math. Stat. 81, 177-186 (2010).
The author introduces the notion of $$V^{\varepsilon}$$-arbitrage (in other words, an arbitrage under the taxation proportional to the portfolio size) for a multi-period discrete time model of a financial market. Since some of $$V^{\varepsilon}$$-arbitrage-free financial markets may have an arbitrage in the classical sense mathematical tools different from the classical ones are required to study these markets. The main conditions that are equivalent to the criteria of nonexistence of a bounded arbitrage in a one-period financial market are proposed by Yu. S. Mishura [Prykl. Stat., Aktuarna Finans. Mat. 2003, No. 1–2, 49–54 (2003; Zbl 1150.91381)]. Those results are generalized to one-period models with random initial data and to multiperiod models in the paper by Yu. S. Mishura, P. S. Shelyazhenko and G. M. Shevchenko [Teor. Jmovirn. Mat. Stat. 77, 122–131 (2007); translation in Theory Probab. Math. Stat. 77, 135–146 (2008; Zbl 1199.91277)]. The main result of this paper is similar to that of the classical arbitrage theory and gives a relationship between the existence of a bounded arbitrage and the existence of a measure with certain properties. In contrast to the classical case, where the measure is a martingale, the measure is an $$\varepsilon$$-martingale, that is $$| E_{P^*}\Delta X_t^i/{\mathcal F}_{t-1}| \leq\varepsilon$$ for all $$i$$ and $$t$$ (here $$\Delta X^i_t$$ is the discounted increment of the price of an $$i$$-th asset in the interval $$[t-1, t]$$). This condition becomes the classical arbitrage-free condition if $$\varepsilon=0$$.

### MSC:

 91G10 Portfolio theory 91G80 Financial applications of other theories 91B24 Microeconomic theory (price theory and economic markets)

### Citations:

Zbl 1150.91381; Zbl 1199.91277
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