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we take expected values, the stochastic integral will also vanish. The initial value Xt = x and the boundary condition F (T, x) = Φ(x) will eventually leave us with the formula F (t, x) = Et,x [Φ (XT )] , where we have indexed the expectation operator in order to emphasize that the expected value is to be taken given the initial value Xt = x. Thus we have proved the following result, which is known as the Feynman–Kač stochastic representation formula.