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2
votes
0answers
291 views

Finding jump probabilities from mean-occupancy values for positions on a one-dimensional random walk

Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as ...
2
votes
2answers
880 views

Examples of deterministic processes of quadratic variation which are of unbounded variation

In [Föllmer 81] (English translation to be found here) writes: "The class of processes of quadratic variation is clearly larger than the class of semimartingales: Just consider a deterministic process ...
1
vote
3answers
890 views

Föllmer: “Calcul d'Ito sans probabilités” in English or German?

Does anybody know a translation of Föllmer: Calcul d'Ito sans probabilités in English or German? It seems to be a very interesting text - Abstract: "It is shown that if a deterministic continuous ...
5
votes
2answers
245 views

how to sample a conditioned diffusion

there are several reasons why we could be interested in sampling conditioned diffusions: if we observed a diffusion at discrete time and want to do some kind of inference on the parameters of the ...
2
votes
1answer
909 views

Stationary Solutions of stochastic differential equations

When does the stationary density of an homogeneous Markov process exist?
3
votes
1answer
313 views

initial condition of a diffusion approximation

I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...
6
votes
1answer
2k 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: ...
4
votes
0answers
464 views

Dynamic programming principle (DPP)

In stochastic control problem, one shall use the measurable selection theorem to prove DPP. It was discussed in discrete time case in [Bertsekas and Shreve 1978]. Is there unified framework in ...
0
votes
1answer
431 views

Looking for a version of Itô's Lemma

Hi Everyone I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated. We are given a Brownian Motion $B_t$ and ...
1
vote
2answers
253 views

Problem with a Long Range Correlated Time Series

Consider a stochastic process $X_t$ , $t \in 1,2,3,..,N $. $X_t$ is a Bernoulli variable and $\Pr(X_t=1) = p$ for all $t$. The Autocovariance function $\gamma(|s-t|)= E[(X_t - p)(X_s -p)]$ is given ...
3
votes
2answers
362 views

Systematization of deterministic and stochastic integrals

With this question I try to build up a systematization of different kinds of integrals. The following table differentiates between deterministic and stochastic integrals, the summation processes ...
16
votes
5answers
2k views

Brownian motion and spheres

Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$: $$ ...
9
votes
2answers
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 ...
13
votes
2answers
1k views

Big Picture: What is the connection of Malliavin calculus with differential geometry?

I know that Paul Malliavin was heavily influenced by ideas from differential geometry while developing his calculus on Wiener space. But what are the concrete analogies between both areas of ...
6
votes
5answers
2k 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 ...
2
votes
1answer
326 views

Moment Generating Function: Pulling a term out of k-times differentiation

In Wiersema: Brownian Motion Calculus on p. 205 (in an Annex on Moment Generating Functions (mgf)) the following equation is being presented $${d^k \over d\theta^k} \left ({1\over ...
1
vote
2answers
495 views

Exploding Levy processes

Hi, probably this is a fairly newbie question, but is it possible that the a generic Levy process explodes (i.e. tends to infinity for finite time t with positive probability)? If yes, could you ...
1
vote
1answer
252 views

Extension of some feature of SDE Ornstein-Uhlenbeck type

Hi everyone, I am looking for some ideas (or references) in order to get an explicit SDE (if it exists) which would have a stylised property extending in some sense the mean-reversion property of SDE ...
0
votes
2answers
297 views

Units in Ornstein-Uhlenbeck(OU) process

Take an OU process characterized by X(0) = x dX(t) = - a X(t) dt + b dW(t) where a,b >0. The parameter a is usually interpreted a dissipative term, and b is a ...
10
votes
2answers
530 views

Inequality in Gaussian space — possibly provable by rearrangement?

The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem. Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in ...
1
vote
4answers
374 views

CAS for finding closed form solutions to PDEs and SDEs?

Are there any specialized Computer Algebra Systems (or packages for these) for finding closed form solutions to a) partial differential equations, b) stochastic differential equations? If yes, what ...
3
votes
2answers
472 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 ...