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### About Ito integral of power of brownian motion

Using Ito's lemma, one can get the following expression for Ito integral of monomials:
$\int_0^TW(t)^ndW(t) = \frac{1}{n+1}W(t)^{n+1} - \frac{n}{2}\int_0^TW(t)^{n-1}dt.$
What can we say about the ...

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95 views

### European call option pricing under mean reverting stock return

Consider the stock price process satisfies the following SDE:
$dS_t=\mu_t S_tdt + \sigma S_t dW_t , S_0=s $
and the mean return $\mu_t$ satisfies the following SDE:
$d\mu_t=(a-\mu_t)dt +dB_t, \...

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**1**answer

144 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 ...

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**2**answers

97 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 $...

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**1**answer

84 views

### About the boundary conditions of the Black-Scholes-Merton PDE [closed]

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve.
Let $c(t,x)$ be the value of the ...

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105 views

### Asymptotics of Variable Drift Ornstein–Uhlenbeck Process

The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:
$dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t$
where $\theta>0$, $\mu$ and $\sigma>0$ are ...

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58 views

### The Stratonovich formulation of the Double Mean Reverting Model

I am writing my Bachelor's Thesis on the fast Ninomiya-Victoir calibration of the Double Mean Reverting model and have a question to its Stratonovich formulation. I am new to mathoverflow and a novice ...

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**1**answer

699 views

### Algebraic Number Theory in Financial Mathematics

I am currently doing my masters studies in financial mathematics. However, I have had a good background in number theory and I don't feel like leaving it just like that. I am thus inquiring on any ...

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**1**answer

687 views

### The Stock Market Polytope: Explanation?

Ovidiu Racorean.
"Crossing Stocks and the Positive Grassmannian I: The Geometry behind Stock
Market."
(arXiv Abstract link)
Anyone care to offer a summary of what's going on here?
(The ...

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131 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 ...

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1k 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 math finance, of the three economists who were just awarded this year's Nobel Memorial Prize in Economic ...

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**1**answer

114 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 $...

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231 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 $...

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**3**answers

833 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" ...

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**3**answers

885 views

### How much one can earn on a white noise ?

Consider the simplfied math. model for asset price (it is nevertheless quite practical for specific situations see "PS" part below) assume price "p(n)" at moment "n" is equal to N(0,1) - i.i.d - ...

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**1**answer

364 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 ...

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**0**answers

802 views

### American put option pricing by “binomial trees”

Dear MO World,
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 ...

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**1**answer

310 views

### Trajectorial version of Doob's $L^2$ inequality

In the paper http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf
you can find a trajectorial version of Doob's inequality. It is given by:
$$\bar{s}^2_T+4\sum_{k=0}^{T-1}\bar{...

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**1**answer

385 views

### Solving an Ornstein-Uhlenbeck-like SDE $y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t]$

I have asked a similar question involving some finance background some time ago here math.stackexchange, however no really good answer came up. I was able to find a solution at least for a special ...

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**2**answers

191 views

### market completion in stochastic volatility model

Hi all,
Consider a stochastic volatility model. As there are two sources of risk and one asset only, this is an imcomplete market. One can complete the market by considering a derivative V1 used to ...

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**0**answers

116 views

### stochastic volatility valuation equation

I'm trying to derive the valuation equation under a general stochastic volatility model. What one can read in the litterature is the following reasonning:
One consider a replicating self-financing ...

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**1**answer

244 views

### Arbitrage free price of a derivative when the price is collected over the lifetime of the derivative [closed]

Let $X_t$ be an american style financial derivative with random exercise time $T$
where $t$ and $T$ belongs to some finite set $A$.
Buying this derivative requires the buyer to pay $p_t$ up to time $T$...

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**3**answers

900 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^\...

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**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 ...

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**0**answers

354 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 $...

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**10**answers

3k 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 ...

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**1**answer

347 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 ...

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**2**answers

465 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^...

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**1**answer

751 views

### Responses from mathematicians concerning Flash trading [closed]

Have there been any responses from the mathematics community regarding flash trading, for example from a game theory or system dynamics point of view? Please answer with personal comments or ...

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**1**answer

15k 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 ...

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**1**answer

3k 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: ...

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**1**answer

237 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 ...

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**3**answers

259 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 ...

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2k views

### Convergence and non-convergence of left-point and mid-point Riemann sums

In standard calculus it is a well known fact that left-point and mid-point Riemann sums do become equal in the limit. When it comes to stochastic integration this is no longer the case. Taking the ...

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**5**answers

3k views

### Discrete version of Ito's lemma

Could anyone give me some references where I could find
(a) discrete version(s) of Ito's lemma
(b) a proof how it converges to the continuous form in the limit
(c) its usage within stochastic ...

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**2**answers

2k views

### Fuzzy Logic in Finance

Has fuzzy logic been commercially applied in finance fields and has it been successful ?
I have got knowledge that it has been applied in Algorithmic trading and operational risk, but I want to know ...

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**7**answers

832 views

### Reference for elementary and “cool” statistics or financial math

I signed up for a Math Mentorship Program (for high school students) this term, but one of the students assigned to me is more interested in Statistics and Finance - something that would help him to ...

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**3**answers

3k 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 ...

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**2**answers

2k views

### Ito's lemma in differential form

Basically you'll find two versions of ito's lemma in the literature: an integral and a differential form. The integral form is based on an Riemann-Stieltjes-integral approach, the differential form is ...

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993 views

### Matching Dynamic Trading Strategies with Derivatives

The famous Black-Scholes Framework is usally derived using a hedging approach where a self-financing portfolio is constructed and the resulting stochastic differential equation is being solved under ...

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**2**answers

488 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 ...

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**1**answer

3k views

### Transformation of the Black-Scholes PDE into the diffusion equation - shift of coordinate system

The aim of transforming the Black-Scholes PDE is of course to find a form where an relatively easy solution exists. Most of the steps seem to be straightforward - please use this reference:
http://...