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Kalman Filter see Filtering

Kelly Problem

Consider a financial market with K assets whose prices Pi(t), i = 1, …, K are stochastic, dynamic processes, and a risk-free asset whose price is P0(t). The vector of prices at time t is

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If the prices are given at points in time t1 and t2, with t1 < t2, then the rate of return over that time on a unit of capital invested in asset i is

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When there are dividends Di accrued in the time interval, then the return is Ri(t1, t2) = (Pi(t2) + Di(t2t1))/Pi(t1).

Suppose an investor has wt units of capital at time t, and that capital is fully invested in the assets, with the proportions invested in each asset given by xi(t), i = 0, … K, where images. Then an investment or trading strategy at time t is the vector process

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Given the investments wt1 X(t1) at time t1, the accumulated capital at time t2 is

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The trajectory of returns between time t1 and time t2 depends ...

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