Solve Textbook Exercises 

Compute the expected value of the product of two integrals and , where is the standard WienerProcess[].

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X pr = ItoProcess[{\[DifferentialD]x[t] == w[t] \[DifferentialD]t, \[DifferentialD]y[t] == t w[t] \[DifferentialD]t}, {x[t], y[t]}, {{x, y}, {0, 0}}, t, w \[Distributed] WienerProcess[]];

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X Expectation[ Subscript[\[ScriptCapitalI], 1][1] Subscript[\[ScriptCapitalI], 2][ 1], {Subscript[\[ScriptCapitalI], 1], Subscript[\[ScriptCapitalI], 2]} \[Distributed] pr]

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Compare with an alternative computation.

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X Integrate[ Expectation[t w[t] w[s], w \[Distributed] WienerProcess[]], {t, 0, 1}, {s, 0, 1}]

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Using the Ito formula, verify that the process is a martingale with respect to the filtration generated by the Wiener process .

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X x\[ScriptCapitalP] = ItoProcess[{0, 1, (w[t] + t) Exp[-t/2 - w[t]]}, {w, 0}, t]

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Apply the Ito formula by converting the process to its standard form. The diffusion coefficient of the standard Ito process must be zero for to be a martingale.

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X ItoProcess[x\[ScriptCapitalP]]

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Prove the martingale property directly.

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X Expectation[(w[t] + t) Exp[-w[t] - t/2] \[Conditioned] w[s] == ws, w \[Distributed] WienerProcess[], Assumptions -> 0 < s < t]

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