6
votes
Accepted
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 ...
- 54k
4
votes
Accepted
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 "...
- 185k
3
votes
Accepted
Stochastic integral with respect to a random field
The notations in the question are ambiguous (Bjørn Kjos-Hanssen showed that the other interpretation cannot be correct). I assume that the expression of interest is given by
$$
g(t) = \int_0^t \Sigma(...
- 9,958
3
votes
Accepted
Large deviation bound for O-U process
Fernique's theorem, valid for all continuous Gaussian processes, implies that there are constants $C,c$ (depending on $T$) such that $P(\max_{0 \le t \le T} |X_t| \ge z) \le C e^{-c z^2}$.
See also ...
- 29.6k
3
votes
Large deviation bound for O-U process
Yes: for white noise perturbations of the 1D dynamical system $\dot{x}=b(x)$, the action functional is $S(\phi)=\frac{1}{2} \int_0^T |b(\phi(s))-\dot{\phi}(s)|^2 ds$ for $\phi \in H^1$ and otherwise ...
- 325
3
votes
Compute inverse series for implicit equation $b=-\log(1-e^{-x})/x$
Taylor series are popular only because they are taught in calculus, but usually are quite bad for practical numerical approximation like you want here.
Look for Padé approximants, Chebyshev series, ...
- 19.8k
3
votes
Fuzzy Logic in Finance
Fuzzy Logic in Financial Management - InTechOpen
URL:http://cdn.intechopen.com/pdfs/32889.pdf
Industrial and commercial applications of fuzzy logic
URL:https://link.springer.com/chapter/10.1007/3-...
- 31
3
votes
Reference for elementary and "cool" statistics or financial math
I like the following example because a) it comes down to simple algebra, b) it's a window into a much deeper set of ideas, and c) the result is strikingly counterintuitive, which gets students' ...
2
votes
Are symplectic methods used in (classical) Economics?
Okay so here's the thing. Economists love math, but good economists recognize that they are doing a social science. They don't do math for the sake of doing math.
Math is a tool/technology for ...
- 121
2
votes
Accepted
Is the "hybrid" Black-Scholes Hull-White model arbitrage free?
The discounted stock price satisfies
$$
dX(t) = \big(\mu(t) - r(t)\big)X(t) dt + \sigma_S(t) X(t) dW_S^{\mathbb P}(t).
$$
The Girsanov density for $X$ is
$$
Z(T) = \exp\left\{\int_0^T \nu(t)dW^{\...
- 1,533
2
votes
Accepted
Construction of a probability measure from a sequence of probability measures
$\newcommand\R{\mathbb R}$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 ...
- 185k
1
vote
Accepted
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 ...
- 489
1
vote
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). ...
- 5,261
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