Optional Switching. Suppose that Xn and Yn are two martingales with respect to Fn that represent values of an investment in a fair market that evolve under two different investment strategies. Suppose...

Optional Switching. Suppose that Xn and Yn are two martingales with respect to Fn that represent values of an investment in a fair market that evolve under two different investment strategies. Suppose an investor begins with the X-strategy and then switches to the Y -strategy at an a.s. finite stopping time τ of Fn, and that Xτ = Yτ . Then the investment value would be