Maximization of the portfolio growth rate under fixed and proportional transaction costs

Palczewski, Jan; Stettner, Łukasz

doi:10.4310/cis.2007.v7.n1.a3

Cited by 4 publications

(8 citation statements)

References 19 publications

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“…The results of this paper can be applied to an incomplete information case and extend [25]. Let us first sketch some motivation for this development.…”

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confidence: 84%

“…Here (z 1 , ρ) ∈ E 1 × P(Z 2 ) denotes the initial distribution of Z 1 (t), Z 2 (t) and P(Z 2 ) stands for the space of probability measures on (Z 2 , E 2 ). Now, we can follow a similar reasoning as in [25] to apply Theorem 5.1 and prove existence of an optimal portfolio. Here, however, we improve several aspects of the result; firstly, the transaction costs structure covers important examples (4) and (5).…”

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confidence: 96%

“…Here, due to a different method of proof of Theorem 5.1 we allow E 2 to be a general complete separable metric space (in this case, P(E 2 ) is also a complete separable metric space). For further details see [25]. ′ (n) ≤ − ln e.…”

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confidence: 99%

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Growth-optimal portfolios under transaction costs

Palczewski

Stettner

2008

Appl. Math. (Warsaw)

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This paper studies a portfolio optimization problem in a discrete-time Markovian model of a financial market, in which asset price dynamics depend on an external process of economic factors. There are transaction costs with a structure that covers, in particular, the case of fixed plus proportional costs. We prove that there exists a self-financing trading strategy maximizing the average growth rate of the portfolio wealth. We show that this strategy has a Markovian form. Our result is obtained by large deviations estimates on empirical measures of the price process and by a generalization of the vanishing discount method to discontinuous transition operators.

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“…The results of this paper can be applied to an incomplete information case and extend [25]. Let us first sketch some motivation for this development.…”

mentioning

confidence: 84%

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confidence: 96%

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Growth-optimal portfolios under transaction costs

Palczewski

Stettner

2008

Appl. Math. (Warsaw)

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“…From stochastic control point of view to study risk neutral GOP we have to solve additive Bellman equation, while in the case of risk sensitive GOP we have to solve multiplicative Bellman equations. If we additionally assume fixed costs our strategies are of impulse form (see [33]). …”

Section: Long Time Portfolio Functionalsmentioning

confidence: 99%

Problems of Mathematical Finance by Stochastic Control Methods

Stettner

2009

IFIP Advances in Information and Communication Technology

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Abstract. The purpose of this paper is to present main ideas of mathematics of finance using the stochastic control methods. There is an interplay between stochastic control and mathematics of finance. On the one hand stochastic control is a powerful tool to study financial problems. On the other hand financial applications have stimulated development in several research subareas of stochastic control in the last two decades. We start with pricing of financial derivatives and modeling of asset prices, studying the conditions for the absence of arbitrage. Then we consider pricing of defaultable contingent claims. Investments in bonds lead us to the term structure modeling problems. Special attention is devoted to historical static portfolio analysis called Markowitz theory. We also briefly sketch dynamic portfolio problems using viscosity solutions to HamiltonJacobi-Bellman equation, martingale-convex analysis method or stochastic maximum principle together with backward stochastic differential equation. Finally, long time portfolio analysis for both risk neutral and risk sensitive functionals is introduced. Pricing of Financial DerivativesOne of the fundamental problems of mathematics of finance is pricing of the derivative securities (shortly derivatives) i.e. securities the value of which depends on the basic securities such as stocks or bonds. In this section we restrict ourselves to stocks, although similar problems (unfortunately much harder) concern also derivatives of bonds. Modeling of Asset PricesWe start with modeling of asset prices (stocks). We assume that we are given d assets on the market and denote the price of the i-th asset at time t by S i (t). We shall consider in parallel way two approaches: in discrete and in continuous time. We assume a given probability space (Ω , F , (F t ), P). In the case of discrete time the asset prices satisfy the relation S i (t + 1) S i (t) = ζ i (t, z(t + 1), ξ (t + 1)),Research supported by MNiSzW grant no. 1 P03A 013 28.

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“…[Akien et al (2001)], [Davis and Norman (1990)], [Eastham and Hastings (1988)], [Korn (1998)], [Morton and Pliska (1995)], [Palczewski and Stettner (2006)], [Pliska and Suzuki (2004)], [Shreve et al (1991)], [Shreve and Soner (1994)], [Taksar et al (1988)…”

Section: Introductionmentioning

confidence: 99%

Log-Optimal Portfolio-Selection Strategies with Proportional Transaction Costs

Györfi¹,

Walk²

2012

Advances in Computer Science and Engineering: Texts

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Discrete time growth optimal investment in stock markets with proportional transactions costs is considered. The market process is a sequence of daily relative prices (called returns), and it is modelled by a first order Markov process. Assuming that the distribution of the market process is known, we show sequential investment strategies such that, in the long run, the growth rate on trajectories achieves the maximum with probability 1. Investment with consumption and with fixed transaction cost where the cost depends on the number of the shares involved in the transaction is also analyzed. IntroductionThe purpose of this chapter is to investigate sequential investment strategies for financial markets such that the strategies are allowed to use information collected from the past of the market and determine, at the beginning of a trading period, a portfolio, that is, a way to distribute their current capital among the available assets. The goal of the investor is to maximize his wealth on the long run. If there is no transaction cost then the only assumption used in the mathematical analysis is that the daily price relatives form a stationary and ergodic process. Under this assumption the best strategy 117

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Maximization of the portfolio growth rate under fixed and proportional transaction costs

Cited by 4 publications

References 19 publications

Growth-optimal portfolios under transaction costs

Growth-optimal portfolios under transaction costs

Problems of Mathematical Finance by Stochastic Control Methods

Log-Optimal Portfolio-Selection Strategies with Proportional Transaction Costs

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