Gost91
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Hi there!
I state that I'm an electronic engineer (undergraduate), then the my knowledges about the world of economics are almost null.
A colleague asked to me an help about one point of the proof of the theorem 1 in this paper.
The critical point is how to obtain [latex]\text{ d}Y_t [/latex] that at first glance it seemed to me enough trivial.
Unfortunately I get a wrong result, so I ask if is possible, without using advanced tools like stochastic calculus, get the solution, and in that case, how to get it.
I post my (absolutely not rigorous) calculations.[latex]
\begin{aligned}
\text{d}Y_t &= \lim_{\tau \to 0} \,\, [ Y_{t+\tau}-Y_{t} ] \\ &= \lim_{\tau \to 0} \left[- \int_{t+\tau}^s f(t+\tau,u) \text{ d}u -\left( - \int_t^s f(t,u) \text{ d}u\right) \right] \\ &=\lim_{\tau \to 0} \left[- \left( \int_t^s f(t+\tau,u) \text{ d}u - \int_t^{t+\tau} f(t+\tau,u) \text{ d}u \right) -\left( - \int_t^s f(t,u) \text{ d}u\right) \right] \\
&=\lim_{\tau \to 0} \left[ \int_t^{t+\tau} f(t+\tau,u) \text{ d}u -\left( \int_t^s f(t+\tau,u)-f(t,u) \text{ d}u\right) \right] \\
&=\lim_{\tau \to 0} \int_t^{t+\tau} f(t+\tau,u) \text{ d}u - \int_t^s \lim_{\tau \to 0} \, [f(t+\tau,u)-f(t,u)] \text{ d}u \\ &=f(t,s)\text{ dt}- \int_t^s \frac{\partial f}{\partial t} (t,u) \text{ d}t \text{ du} \end{aligned}
[/latex]
The question is: why it appear the total differential of [latex] f(t,u) [/latex] under the sign of integral?
Thank you in advance and sorry for my English.0
HJM model
in Applied Mathematics
Posted
Thanks for the reply mathematic, but the problem insist.
For my calculations, the result is
[latex] \text{d}Y_t=f(t,t) \text{ d}t- \int_t^s \frac{\partial f}{\partial t} (t,u) \text{ d}t \text{ d}u \tag{1} [/latex]
according to the Leibniz Integral rule. In fact, for that formula
[latex]
\begin{aligned}
\frac{ \text{d}Y_t}{\text{d}t} &= -\left ( f(t,s)\frac{\text{d} s}{\text{d} t}-f(t,t)\frac{\text{d} t}{\text{d} t}+\int_t^s \frac{\partial f}{\partial t} (t,u) \text{ d}u \right) \\
&=f(t,t)- \int_t^s \frac{\partial f}{\partial t} (t,u) \text{ d}u
\end{aligned} \tag{2}
[/latex]
where I have observed that [latex] s[/latex] is costant with respect to [latex] t[/latex].
By multiplying [latex] \text{d}t[/latex] each terms of [latex] (2) [/latex] we obtain [latex] (1) [/latex].
But in the paper the author have writed
[latex] \text{d}Y_t=f(t,t) \text{ d}t- \int_t^s \text{ d}f (t,u)\text{ d}u [/latex]
The previus expression is different respect [latex] (1) [/latex] because
[latex] \text{ d}f (t,u) =\frac{\partial f}{\partial t} (t,u) \text{ d}t +\frac{\partial f}{\partial u} (t,u) \text{ d}u [/latex]