2
$\begingroup$

My reference is here : https://arxiv.org/pdf/1504.05309.pdf

My question is related to the example 2.6.1 : page 21-22; 2.6 Girsanov Theorem

It said in equation (2.8) $Z_t = exp(-\frac{1}{2}∫^t_0θ_s^2ds+∫^t_0θ_sdW_s)$.

And there is outcome for the exponential martingale, $Z_t = exp(-\frac{t}{2}(\frac{(r-μ)}{σ})+\frac{(r-μ)}{σ}W_t)$.

But I don't understand the logic behind this derivation. I want to the procedure how $θ_s$ term changed to $Z_t = exp(-\frac{t}{2}(\frac{(r-μ)}{σ})+\frac{(r-μ)}{σ}W_t)$.

Improve this question
$\endgroup$
2
  • $\begingroup$ Papanicolaou simply sets $\theta_s=\frac{\mu-r}{\sigma}$. This fraction is known as ''market price of risk'' or ''Sharpe ratio''. $\endgroup$
    – Kevin
    Commented Apr 6, 2021 at 11:22
  • $\begingroup$ Then Is it right to solve this equation like this? $df = f_tdt+ f_wdW_t+\frac{1}{2}f_{ww}dt$ and if $f = W_t then, ∫^t_0df = 0 + ∫^t_01dW_t + 0$ So.. because $W_t = ∫^t_01dW_t = f$, in exponential term, the constant term $θ_s$ multiplied $f$. As a outcome, in the exponential term $∫^t_0θ_sdW_t$ goes to $θ_s*f = θ_sW_t$. $\endgroup$
    – user13232877
    Commented Apr 7, 2021 at 0:09

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.