All Questions
Tagged with mathematical-finance pr.probability
36 questions
0
votes
1
answer
51
views
Reconstruction of law of diffusion process from call option values
Let $X_{\cdot}$ be a $1$-dimensional diffusion process. If I know the value of the
$$\big\{\mathbb{E}[\max\{X_t,c\}\big| X_0 =x\big]:\, c\in \mathbb{R} \text{ and } \,\, t\in (0,1] \big\}.$$
Then, ...
- 5,405
0
votes
1
answer
81
views
Stochastic Geometric Progression [closed]
Let $\mu_1, \mu_2, \ldots, \mu_n, \ldots \in \mathbb{R}$, let $\sigma_1, \sigma_2,
\ldots \in [0, \infty)$ be sequences of numbers.
Let $z_1, z_2, \ldots, z_n, \ldots$ be independent random variables ...
- 13
5
votes
1
answer
437
views
Elliptic PDEs in Finance
In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
- 5,405
2
votes
0
answers
59
views
How to determine speed (rate) in large deviation principle for geometric Brownian motion
By reading Asymptotics for volatility derivatives in multi-factor rough
volatility models by Lacombe, Muguruza and Stone, I am not familiar with the way they deduce the speed (or rate) when showing ...
- 21
3
votes
1
answer
159
views
Are there any known results on the probability distributions of perpetuities with power law discount rates?
Currently I am working on studying stochastic integrals of the form: $$Z_\infty = \int_0^\infty e^{-f(t)}\mathop{d}S_t$$
where $S_t$ is a Compound-Poisson process with Exponentially-distributed ...
- 33
8
votes
0
answers
304
views
"Meritocratic" pyramid schemes
There have been a couple of times in my life when people from multi-level marketing organizations attempted to recruit me. I listened to what they had to say, and both times I did not get involved ...
- 2,075
1
vote
0
answers
328
views
Preservation of variance for log-normal variables under change of measure
Aim: to show that changing a probability measure via the application of a Radon-Nikodym derivative preserves variance of a log-normally distributed random variable (for the case when variance is non-...
- 141
0
votes
1
answer
151
views
Construction of a probability measure from a sequence of probability measures
Summary
I would like to pass from a sequence of probability measures whose "limit" satisfies a desired property to a new probability measure that satisfies this property.
Details
We work on ...
user475819
0
votes
0
answers
340
views
Why are financial markets modeled by càdlàg processes?
When opening a book or reading an article on mathematical finance, financial markets (e.g. stock prices) are always modeled by càdlàg semimartingales. I was wondering why it is that these processes ...
- 309
2
votes
1
answer
495
views
Stochastic integral with respect to a random field
I came across a generalized Black-Scholes equation formulation in this paper.
Let me highlight the basic idea below. Consider a random field $W(t,T)$ where for a fixed $T$, $W$ is a Brownian motion ...
- 77
1
vote
0
answers
68
views
Recovering a Log-Correlated Gaussian Field from a limit-lognormal singular measure
In a paper I (didn't write, but) co-authored, Forecasting Volatility with the Multifractal Random Walk Model, we use explicit formulas that give the law of $(X(t),t>0)$ conditional on $(X(t),t<0)...
- 3,085
3
votes
1
answer
214
views
Inverting the cumulative probability function to find roots of stochastic function
Given a function:
$$f[x]=a\, \Phi \left[-x+\sigma \sqrt{\tau}\right]-\left(b+c\, e^{-d \tau}\right)\Phi \left[-x\right]$$
where $\Phi$ is the cumulative density function of the standard normal ...
3
votes
2
answers
380
views
Large deviation bound for O-U process
Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X_t = -\alpha X_t dt + \sigma dB_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max_{t\le T} |X_t| \ge z
\...
- 719
1
vote
0
answers
114
views
Extending risk neutral measure to insurance/mortality filtration
In insurance mathematics, one often models the underlying of an insurance policy with a Black Scholes model on a filtered probability space $(\Omega,\mathbb{Q},\mathcal{F},\mathbb{F}=(\mathcal{F}_{t}))...
- 203
3
votes
2
answers
1k
views
Is the "hybrid" Black-Scholes Hull-White model arbitrage free?
Given a "hybrid" Black-Scholes Hull White (BSHW) model. That is, the stock price is modelled by a Black Scholes SDE:
\begin{equation} dS(t) = \mu(t)S(t)dt + \sigma_{S}(t)S(t)dW^{\mathbb{P}}_{S}(t)
\...
- 203
2
votes
1
answer
461
views
Is it safe to work on a Cadlag modification of a Feller process?
Let $f$ be a continuous bounded function.
$X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write
$$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...
- 1,399
1
vote
2
answers
134
views
Is zero a regular point for a drifted $\alpha$-stable process?
We consider 1-d process of the form $Y_{t} = bt + M_{t}^{\alpha}$,
where $M_{t}^{\alpha}$ is $\alpha$-stable process for some $\alpha
\in (0,2)$ with its levy symbol $\eta(u) = - |u|^{\alpha}.$,
and $...
- 1,399
3
votes
0
answers
171
views
compactness of a probability set
I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of ...
- 1,835
16
votes
2
answers
2k
views
On mathematical aspects of the most recent Nobel Prize in economics winners' work
Can somebody briefly introduce the mathematical aspects, in particular, those related to mathematical finance, of the three economists who were just awarded this year's Nobel Memorial Prize in ...
- 622
0
votes
1
answer
161
views
Ratios of random variables with weak moment condition
Let $X_n$ be a sequence of iid positive random variables. Assume that $X_n$ has finite $\alpha$th moment for some value $\alpha \in (0,1)$, but infinite first moment. Assume also that the reciprocal $...
- 8,502
2
votes
0
answers
263
views
A strange Weakly Compactness in $L^1 ( \Omega, \mathcal{F}, \mathbb{P})$
Hi to everyone,
The ingredients of my problem are the following:
I have a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, a set (continuum cardinality) $\mathcal{Q}$ of probability measures on $...
- 21
5
votes
3
answers
1k
views
One can earn nothing on the Brownian motion, true ?
Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$).
Consider some "trading strategy" ...
- 24.9k
0
votes
1
answer
502
views
Mathematical properties of financial prices
Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes.
What is known about their mathematical properties ?
I know ...
- 24.9k
12
votes
0
answers
1k
views
American put option pricing by "binomial trees"
I'm teaching a financial mathematics course and have found a fascinating (to me) numerical phenomenon and wonder if anyone has studied it, or knows anything similar.
I'll try and give a description ...
- 23.2k
12
votes
3
answers
2k
views
Compactness of the set of densities of equivalent martingale measures
Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal P^\...
- 243
11
votes
10
answers
1k
views
Is there any straightforward way to substitute for Gaussian/Brownian assumptions in financial mathematics?
A huge amount of financial mathematics assumes Gaussian distributions of risks and Brownian movement of prices. What efforts have there been to replace these with heavy-tailed distributions? For ...
- 2,383
3
votes
0
answers
518
views
Laplace transform of a stopping time for stochastic volatility models
Let $V_t$ be a solution of the SDE
$$dV_t=V_t(rdt+\sigma_t dW_t) $$
where $\sigma_t$ satisfies some other SDE
$$d\sigma_t=\alpha(t,\sigma_t)dt+\beta(t,\sigma_t)dW^{\\ \prime}_t $$
and $W_t$ and $...
20
votes
10
answers
4k
views
Expected value as decision criterion in the context of rare events
I have often seen discussions of what actions to take in the context of rare events in terms of expected value. For example, if a lottery has a 1 in 100 million chance of winning, and delivers a ...
- 3,475
5
votes
1
answer
478
views
Stieltjes integrals of predictable processes
I am looking for a direct proof of the fact that, roughly speaking, if $S=S_0+A+M$ is an $L^2$ semimartingale, and $M$ (the martingale part) has the martingale representation property, then for any ...
- 71
3
votes
2
answers
919
views
Characteristic operator
Let $X_t\in\mathbb{R}$ be an Ito diffusion process given by $$ dX_t=a(b-X_t)dt+\sigma dW_t$$, then the characteristic operator of $X_t$ is given by $$L=a(b-x)\frac{\partial}{\partial x}+\frac{\sigma^...
- 29
1
vote
1
answer
22k
views
Covariance and standard deviation relationship
I would like to know if an increase in the covariance between two variables would imply that the standard deviation for one of the variables has increased?
This is assuming that the standard ...
- 21
8
votes
1
answer
6k
views
Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory
I am confused and don't get the big picture concerning the connection between
Ito integral
Stratonovich integral
Standard results in probability theory concerning skewed distributions.
Example: Take ...
- 5,935
2
votes
1
answer
254
views
Brownian Bridge under observational error
Suppose that $Z_t$ follows a simple discrete random walk $Z_t=Z_{t-1}+e_t$ , where $e_t$ are a bunch of uncorrelated normal variables with arbitrary variance sigma^2, and that there are observations ...
- 457
3
votes
3
answers
316
views
Finding a distribution family that is preserved under mixture.
Consider the following
$f_{t+1}(z)=p_{12} f_{t}(z/A)+ p_{21} f_{t}(z/B)+p_{22} f_{t}(z/(A+B))$, where $A$, $B$, and the $p$'s are constants and $f_t$ is a probability distribution. Are there any nice ...
- 457
6
votes
3
answers
4k
views
Rigorous definition, detection and test for trending vs. mean-reverting behaviour of stochastic processes
This is a question that has haunted me for some time. In the domain of time series you always talk about trends and mean reversion. But at least to me these concepts are either defined axiomaticly ...
- 5,935
4
votes
2
answers
543
views
maximizing function (stochastic calculus)
S is a price process which follows Geometric Brownian motion with no drift:
dS=S*vol*dW, vol=const., W is a Wiener process.
Define the following ratio: R=E[Max(f(S)-S(T),0)]/E[f(S)], where S(T) is ...
- 41