Kalman Filter see Filtering
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
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
When there are dividends Di accrued in the time interval, then the return is Ri(t1, t2) = (Pi(t2) + Di(t2 − t1))/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 . Then an investment or trading strategy at time t is the vector process
Given the investments wt1 X(t1) at time t1, the accumulated capital at time t2 is
The trajectory of returns between time t1 and time t2 depends ...