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### Compute inverse series for implicit equation $b=-\log(1-e^{-x})/x$

Writing $e^{-x} = t$, your equation is $t + t^b = 1$. I'll assume $0 < b < 1$ (for the case $b > 1$, write $s = t^b$ and the equation becomes $s + s^{1/b} = 1$). Now the slightly more ...
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### Elliptic PDEs in Finance

For elliptic PDE applications to options these would need be independent of time, they need to be perpetual (i.e. never expire), which is not a typical scenario. If your definition of "...
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$\newcommand\R{\mathbb R}$This is impossible to do in such generality. E.g., suppose that $\Omega=\R$, $\mathcal{F}$ is the Borel $\sigma$-algebra over $\R$, $P$ is the standard normal distribution, $$... • 122k 2 votes ### Beginning books on stochastic calculus and finance Shreve, Stochastic Calculus for Finance, volumes 1 and 2. • 24.8k 1 vote Accepted ### Are there any known results on the probability distributions of perpetuities with power law discount rates? This is more of an extended comment trying to study this integral. In the similar spirit as here we study the Laplace transform. As explained here we have the Lévy-Khintchine formula for compound ... • 5,221 1 vote ### Integral over a Markov process Regarding question (1): There are many ways to phrase the Markov property. One of the more convenient ones is as follows: if \Phi is a non-negative function measurable with respect to \mathcal F_{t,... • 16.4k 1 vote ### Reference request in optimal stopping It seems related to optimal stopping. First you have to decide when to make the first purchase. Then you have a new problem of when to make the next purchase. Although it's more complicated than that ... • 24.8k 1 vote ### Inverting the cumulative probability function to find roots of stochastic function Since you are considering a numerical approach: If you have access to Mathematica, you can use the "Reduce" functionality to find all roots in a given interval. ... • 185k 1 vote ### Is it possible to solve P = Cny^{-1}(1-1/(1+y/n)^{nT}) + M/(1+y/n)^{nT} for y? Let z=1+y/n, then your equation is equivalent to$$ Pz^{nT}=C\sum_{k=0}^{nT-1}z^k + M.  So you're basically asking, is a polynomial equation $\sum_{k=0}^N c_kx^k=0$ solvable if you know that $... • 24.8k 1 vote ### Compute inverse series for implicit equation$b=-\log(1-e^{-x})/x\$

here is the series up to order 15, extension to higher order is no big deal using Mathematica or Maple or the like (which will also give you the numerical coefficients symbolically, I omit these ...
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### Taylor Series expansion for an implicitely defined family of functions

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 ...
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### Discrete version of Ito's lemma

Proposition 1.13.1 of this book. The result is based on the Clarke-Ocone formula and is a discrete-time (as opposed to a simple process in continuous time approach, as posted by most of the others). ...
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