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byouness
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This type of integral has appeared so many times and in so many places; for example, here, here and here. Basically, for each sample $$\omega$$, we can treat $$\int_0^t W_s ds$$ as a Riemann integral. Moreover, note that \begin{align*} d(tW_t) = W_t dt + tdW_t. \end{align*} Therefore, \begin{align*} \int_0^t W_s ds &= tW_t -\int_0^t sdW_s \tag{1}\\ &= \int_0^t (t-s)dW_s, \end{align*} which can also be treated as a (parametrized) Ito integral. Then, it is easy to see that \begin{align*} E\left(\int_0^t W_s ds\right) = 0, \end{align*} and that \begin{align*} Var\left(\int_0^t W_s ds\right) &= \int_0^t(t-s)^2 ds\\ &=\frac{1}{3}t^3. \end{align*}

Regarding the martingality, note that, from $$(1)$$, \begin{align*} \int_0^{t_2} W_s ds -\int_0^{t_1} W_s ds &=t_2W_{t_2}-t_1W_{t_1} + \int_{t_1}^{t_2}sdW_s\\ &=t_2(W_{t_2}-W_{t_1}) + (t_2-t_1) W_{t_1} + \int_{t_1}^{t_2}sdW_s\\ &=(t_2-t_1) W_{t_1} + \int_{t_1}^{t_2}(t_2+s)dW_s, \end{align*} for $$t_2>t_1\ge 0$$. Therefore, \begin{align*} E\left(\int_0^{t_2} W_s ds\mid \mathscr{F}_{t_1} \right) &= \int_0^{t_1} W_s ds + (t_2-t_1) W_{t_1}. \end{align*} It is not a martingale. Another way to see this is based the equation \begin{align*} d\left(\int_0^t W_s ds\right) = W_t dt, \end{align*} which is not driftless.

EDIT:

One other approach for the martingality can proceed as follows. For $$t_2>t_1 >0$$, \begin{align*} E\left(\int_0^{t_2} W_s ds \mid \mathscr{F}_{t_1}\right) &= \int_0^{t_1} W_s ds + E\left(\int_{t_1}^{t_2} W_s ds \mid \mathscr{F}_{t_1}\right)\\ &= \int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s \mid \mathscr{F}_{t_1}\right) ds\\ &=\int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s \mid \mathscr{F}_{t_1}\right) ds\\ &=\int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s-W_{t_1}+ W_{t_1}\mid \mathscr{F}_{t_1}\right) ds\\ &= \int_0^{t_1} W_s ds + (t_2-t_1)W_{t_1}. \end{align*}\begin{align*} E\left(\int_0^{t_2} W_s ds \mid \mathscr{F}_{t_1}\right) &= \int_0^{t_1} W_s ds + E\left(\int_{t_1}^{t_2} W_s ds \mid \mathscr{F}_{t_1}\right)\\ &=\int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s \mid \mathscr{F}_{t_1}\right) ds\\ &=\int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s-W_{t_1}+ W_{t_1}\mid \mathscr{F}_{t_1}\right) ds\\ &= \int_0^{t_1} W_s ds + (t_2-t_1)W_{t_1}. \end{align*}

replaced http://quant.stackexchange.com/ with https://quant.stackexchange.com/

This type of integral has appeared so many times and in so many places; for example, herehere, here and here. Basically, for each sample $$\omega$$, we can treat $$\int_0^t W_s ds$$ as a Riemann integral. Moreover, note that \begin{align*} d(tW_t) = W_t dt + tdW_t. \end{align*} Therefore, \begin{align*} \int_0^t W_s ds &= tW_t -\int_0^t sdW_s \tag{1}\\ &= \int_0^t (t-s)dW_s, \end{align*} which can also be treated as a (parametrized) Ito integral. Then, it is easy to see that \begin{align*} E\left(\int_0^t W_s ds\right) = 0, \end{align*} and that \begin{align*} Var\left(\int_0^t W_s ds\right) &= \int_0^t(t-s)^2 ds\\ &=\frac{1}{3}t^3. \end{align*}

Regarding the martingality, note that, from $$(1)$$, \begin{align*} \int_0^{t_2} W_s ds -\int_0^{t_1} W_s ds &=t_2W_{t_2}-t_1W_{t_1} + \int_{t_1}^{t_2}sdW_s\\ &=t_2(W_{t_2}-W_{t_1}) + (t_2-t_1) W_{t_1} + \int_{t_1}^{t_2}sdW_s\\ &=(t_2-t_1) W_{t_1} + \int_{t_1}^{t_2}(t_2+s)dW_s, \end{align*} for $$t_2>t_1\ge 0$$. Therefore, \begin{align*} E\left(\int_0^{t_2} W_s ds\mid \mathscr{F}_{t_1} \right) &= \int_0^{t_1} W_s ds + (t_2-t_1) W_{t_1}. \end{align*} It is not a martingale. Another way to see this is based the equation \begin{align*} d\left(\int_0^t W_s ds\right) = W_t dt, \end{align*} which is not driftless.

EDIT:

One other approach for the martingality can proceed as follows. For $$t_2>t_1 >0$$, \begin{align*} E\left(\int_0^{t_2} W_s ds \mid \mathscr{F}_{t_1}\right) &= \int_0^{t_1} W_s ds + E\left(\int_{t_1}^{t_2} W_s ds \mid \mathscr{F}_{t_1}\right)\\ &= \int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s \mid \mathscr{F}_{t_1}\right) ds\\ &=\int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s \mid \mathscr{F}_{t_1}\right) ds\\ &=\int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s-W_{t_1}+ W_{t_1}\mid \mathscr{F}_{t_1}\right) ds\\ &= \int_0^{t_1} W_s ds + (t_2-t_1)W_{t_1}. \end{align*}

replaced http://math.stackexchange.com/ with https://math.stackexchange.com/

This type of integral has appeared so many times and in so many places; for example, here, herehere and herehere. Basically, for each sample $$\omega$$, we can treat $$\int_0^t W_s ds$$ as a Riemann integral. Moreover, note that \begin{align*} d(tW_t) = W_t dt + tdW_t. \end{align*} Therefore, \begin{align*} \int_0^t W_s ds &= tW_t -\int_0^t sdW_s \tag{1}\\ &= \int_0^t (t-s)dW_s, \end{align*} which can also be treated as a (parametrized) Ito integral. Then, it is easy to see that \begin{align*} E\left(\int_0^t W_s ds\right) = 0, \end{align*} and that \begin{align*} Var\left(\int_0^t W_s ds\right) &= \int_0^t(t-s)^2 ds\\ &=\frac{1}{3}t^3. \end{align*}

Regarding the martingality, note that, from $$(1)$$, \begin{align*} \int_0^{t_2} W_s ds -\int_0^{t_1} W_s ds &=t_2W_{t_2}-t_1W_{t_1} + \int_{t_1}^{t_2}sdW_s\\ &=t_2(W_{t_2}-W_{t_1}) + (t_2-t_1) W_{t_1} + \int_{t_1}^{t_2}sdW_s\\ &=(t_2-t_1) W_{t_1} + \int_{t_1}^{t_2}(t_2+s)dW_s, \end{align*} for $$t_2>t_1\ge 0$$. Therefore, \begin{align*} E\left(\int_0^{t_2} W_s ds\mid \mathscr{F}_{t_1} \right) &= \int_0^{t_1} W_s ds + (t_2-t_1) W_{t_1}. \end{align*} It is not a martingale. Another way to see this is based the equation \begin{align*} d\left(\int_0^t W_s ds\right) = W_t dt, \end{align*} which is not driftless.

EDIT:

One other approach for the martingality can proceed as follows. For $$t_2>t_1 >0$$, \begin{align*} E\left(\int_0^{t_2} W_s ds \mid \mathscr{F}_{t_1}\right) &= \int_0^{t_1} W_s ds + E\left(\int_{t_1}^{t_2} W_s ds \mid \mathscr{F}_{t_1}\right)\\ &= \int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s \mid \mathscr{F}_{t_1}\right) ds\\ &=\int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s \mid \mathscr{F}_{t_1}\right) ds\\ &=\int_0^{t_1} W_s ds + \int_{t_1}^{t_2} E\left(W_s-W_{t_1}+ W_{t_1}\mid \mathscr{F}_{t_1}\right) ds\\ &= \int_0^{t_1} W_s ds + (t_2-t_1)W_{t_1}. \end{align*}

Gordon
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edited body
Gordon
• 20k
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• 32
• 78
Gordon
• 20k
• 1
• 32
• 78