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Can we find a Taylor Series expansion for $y(x)$ implicitly defined by: $$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ?$$

In financial mathematics, the two-additive-factors Model G2++ is commonly used for interest rates forecasting. G2++ Model parameters are calibrated on market Caps and Swaptions prices. On the basis of these calibrated parameters complex financial product prices are computed using Monte-Carlo methods.

For further details on G2++ models refer for example to Brigo & Mercurio

Numerical computational challenges arise due to the necessity to integrate an implicitly defined function of the form:

$$\sum _{i=1}^nA_ie^{a_ix+b_iy} = 1 ,$$

which implicitly defines $y(x)$. All other parameters $A_i$, $a_i$ and $b_i$ are constant positive real numbers.

Now comes my question: What is the Taylor of $y(x)$ around $x=0$?

I started with a highly simplified version of the equation:

$$e^{x+y}+e^{x+by} = 1,$$

And found explicit closed form solutions for $y(x)$ when $b \in \{0,1,2,3,4\}$. This corresponds to the cases when the equation becomes Galois solvable. But what if $b$ is irrational, or arbitrarily large?

At least for this simplified single parametric version, wouldn't there be a strategy to obtain Taylor Series expansions?

Thanks for your answers or suggestions! They are greatly appreciated.

Looking at the case $b=2$, it seems that $y(x)$ is really close to its linear asymptotes $x+y=0$ and $x+by=0$. maybe a clue? see here.

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The coefficient of the Taylor expansion of y at x=0 can be found recursively. The convergence of the resulting series can be analyzed by mayorizing the coefficients and verifyig that the majorant series converges in a neighborhood of x=0. In your case, bounding the tail of the Taylor series may be important to decide how many terms to keep.

You can find the recursion in any proof of the Implicit Functions Theorem for analytic functions in the complex plain.

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  • $\begingroup$ Thanks for your answer. It makes me confident that such a Taylor expansion can be derived. Despite searching litterature on Implicit Functions Theorem I could not find anything on recursively computing the coefficients of the Taylor expansion? Would you happen to know specific reference? $\endgroup$
    – Aobara
    Commented Aug 22, 2016 at 8:14
  • $\begingroup$ I can't give you a reference. May be somone else will have a suggestion. What I had in mind is more or less in "Elementary Theory of Analytic Functions of One or Several Complex Variables" by H. Cartan. In Chapter 1, section 9 he goes over the inverse function theorem. The recurssion in (9.1) involves the polynomial $P_n$ that has coefficients in $\mathbb N_0$, so $|P_n(a_1,..b_{n-1})| \le P_n( |a_1|,...,|b_{n-1}|)$, which leads to the majorization in (9.2), etc. $\endgroup$
    – VictorZurkowski
    Commented Aug 22, 2016 at 8:41
  • $\begingroup$ Thanks a lot that reference really helped! Even if we are not dealing with inverse functions here it is totally applicable to the situation. It is sufficient to know $a_0$ (by numerically solving $e^{a_0}+e^{b a_0}=1$), to recursively deduct the following coefficients. $$a_1 = -1/{(e^{a_0}*(1-b)+b)},$$ $$a2 = -((1-e^{a_0} (a_1 b+1)^2+e^{a_0}*(a_1+1)^2)/(2*(b*(1-e^{a_0})+e^{a_0})), etc... $$ However, the convergence appears to be rather slow and there is no benefit gained from the fact that $y(x)+x$ tends towards $0$ at $+\infty$. Still a great step forward. Thanks a lot!! $\endgroup$
    – Aobara
    Commented Aug 22, 2016 at 12:21

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