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Optimal Position Size with Transaction Costs given Forecast Mean and StDev

I have rather a challenging question. I'm hoping that someone can share their experience. I will build up the problem in steps. Let's start our thinking with the idea of a buy and hold strategy of ...
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Log returns vs Relativizing to Portfolio size of $1

In a current empirical research project, I am tracking a non-parametric measure of a transaction cost. To this extent, I track this cost in two ways Cost in terms of log returns Cost in terms of ...