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In How were these SDE derived? I don't understand one part of Gordon's answer, specifically:

$$\ln S_t=\ln F_{0,t}-\frac{\sigma^2}{4\lambda}(1-e^{-2\lambda t})+\sigma e^{-\lambda t}\int_0^t e^{\lambda s}dB_s$$ $$d\ln S_t=\bigg(\frac{\partial \ln F_{0,t}}{\partial t}-\frac{\sigma^2 }{2}e^{-2\lambda t}-\lambda \sigma e^{-\lambda t}\int_0^t e^{\lambda s}dB_s\bigg)dt+\sigma dB_t$$

My question is, how to get from the first line to the second. Evidently, Ito's Lemma was applied, however it has been used in a way in which I have not encountered. It seems that the integral with the Brownian motion term was treated as a constant, but I don't see how that is allowed when there is $t$ in the upper limit.

Any help is appreciated.

Edit: I have come up with a solution, as long as the following is valid. Can anyone please confirm? The first line uses the product rule. \begin{align} d\bigg(\sigma e^{-\lambda t}\int_0^t e^{-\lambda s}dB_s\bigg)&=d(\sigma e^{-\lambda t})\int_0^t e^{-\lambda s}dB_s+\sigma e^{-\lambda t}d\bigg(\int_0^t e^{\lambda s}dB_s\bigg)\\ &=-\lambda\sigma e^{-\lambda t}dt \int_0^t e^{-\lambda s}dB_s+\sigma e^{-\lambda t}e^{\lambda t}dB_t\\ &=-\lambda\sigma e^{-\lambda t}dt \int_0^t e^{-\lambda s}dB_s+\sigma dB_t \end{align}

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    $\begingroup$ If we differentiate $\sigma e^{-\lambda t}\int_0^t e^{\lambda s} dB_s$ we just use the multiplication rule and the fact that $\frac{d}{dx} \int_0^t x(s) dB_s = x(t) dB_t $ if we assume some reasonable restrictions on $x$. Your edit gets the exponential wrong in the second part, it is $e^{\lambda t}$ without the '-'. $\endgroup$ – Forgottenscience Oct 18 '16 at 7:04
  • $\begingroup$ Very clear, thanks. Fixed the exponential as well. $\endgroup$ – none Oct 18 '16 at 9:08
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You are right in your use of product rule.

Since $\sigma e^{-\lambda t}$ is a finite variation process, you do not have crochet term and this is the standard product rule.

In your second line, there is an extra $t$ (which is a typo I presume)

$$\ln S_t=\ln F_{0,t}-\frac{\sigma^2}{4\lambda}(1-e^{-2\lambda t})+\sigma e^{-\lambda t}\int_0^t e^{\lambda s}dB_s$$ $$d\ln S_t=\bigg(\frac{\partial \ln F_{0,t}}{\partial t}-\frac{\sigma^2 }{2}e^{-2\lambda t}-\lambda \color{red}{t}\sigma e^{-\lambda t}\int_0^t e^{\lambda s}dB_s\bigg)dt+\sigma dB_t$$ the correct equation is: $$d\ln S_t=\bigg(\frac{\partial \ln F_{0,t}}{\partial t}-\frac{\sigma^2 }{2}e^{-2\lambda t}-\lambda \sigma e^{-\lambda t}\int_0^t e^{\lambda s}dB_s\bigg)dt+\sigma dB_t$$

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  • $\begingroup$ Excellent, fixed the typo. $\endgroup$ – none Oct 18 '16 at 9:18

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