Questions tagged [mathematical-finance]
For questions about mathematical problems arising from the study of financial markets.
71 questions
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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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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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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 $...
CommunityBot
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3
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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" ...
- 3,069
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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 - ...
CommunityBot
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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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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{s_k}(...
CommunityBot
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1
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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 ...
CommunityBot
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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 ...
CommunityBot
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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 ...
CommunityBot
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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$...
CommunityBot
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3
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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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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 $$
and $W_t$ and $...
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Characteristic operator
Let $X_t\in\mathbb{R}$ be an Ito diffusion process given by $$ 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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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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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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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 ...
- 3,069
2
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1
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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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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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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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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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