I am trying to read Karatzas/Shreve "Methods of Mathematical Finance". In ch. 1, Definition 5.5, a cumulative income process $\Gamma(t)=\Gamma^{\mathrm{fv}}(t)+\Gamma^\mathrm{lm}(t)$ (a semimartingale under the original measure $P$) is defined to be integrable if $$ E_0 \int_0^T \frac{d \hat\Gamma_0^{\mathrm{fv}}(u)}{S_0(u)} < \infty, E_0 \int_0^T\frac{d \langle \Gamma_0^{\mathrm{lm}} \rangle (u)}{S_0^2(u)} < \infty $$ where $E_0$ denotes expectation with respect to $d P_0 = Z_0\, dP, Z_0 =\exp \left[ - \int_0^t \theta'(s) dW(s) - \frac 1 2 \int_0^t \| \theta(s) \|^2 ds \right]$, and $\theta (\cdot )$ being the market price of risk process.

$d \hat\Gamma^{\mathrm{fv}}(u) = \vert d \Gamma^{\mathrm{fv}} (u)\vert $ denotes the absolute variation of the finite variation part of $\Gamma$. Moreover, the authors remind that, with respect to the new measure $P_0$, $$ d \Gamma_0^{\mathrm{fv}}(t) = d \Gamma^{\mathrm{fv}}(t) - \theta'(t) d \langle \Gamma^{\mathrm{lm}}, W \rangle (t) . $$

Now the authors claim in Remark 5.8 that with $H_0(t ) := Z_0 (t) / S_0 (t)$ (the state price density process), the integrability conditions for the cumulative income process can be rewritten as $$ E \int_0^T H_0(u) d \hat\Gamma^{\mathrm{fv}}(u) < \infty , $$ if $\Gamma^{\mathrm{lm}}(\cdot ) \equiv 0$.

I don't see how this is equivalent, and I don't know if it's supposed to be equivalent to both conditions or only the first one. I tried to rewrite the proposed condition. If I'm not mistaken,

$$ E \int_0^T H_0(u) d \hat\Gamma^{\mathrm{fv}}(u) = E_0 \left[ \int_0^T \frac{\exp \left( \int_{(u,T]} \theta'(s) dW(s) + \frac 1 2 \int_{(u,T]} \| \theta(s) \|^2 d s \right) }{S_0(u)} d \hat\Gamma_0^{\mathrm{fv}}(u) \right] $$

Now what do I do with the numerator?

| improve this question | | | | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.