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17
votes
3answers
1k views

Do convex and decreasing functions preserve the semimartingale property?

Some time ago I spent a lot of effort trying to show that the semimartingale property is preserved by certain functions. Specifically, that a convex function of a semimartingale and decreasing ...
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 $\frac{1}{N}$: $$ ...
15
votes
3answers
3k views

Intuition and/or visualisation of Ito integral/Ito's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve. See e.g. Wikipedia:Riemann sum The Ito integral has due to the unbounded total variation ...
12
votes
4answers
964 views

Wiener process related counterexample

The Wiener process is defined by the three properties: 1. $W(0) = 0$, 2. $W(t)$ is almost surely continuous, and 3. $W(t)$ has independent increments with $W(t) - W(s) \sim N(0, t-s)$ (for $0 ≤ s ...
11
votes
1answer
686 views

Bochner integral of stochastic process = path by path Lebesgue integral?

After some helpful comments, I realized that I had to repost this question in a more systematic way. On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
11
votes
1answer
864 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 ...
10
votes
0answers
574 views

surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
10
votes
0answers
595 views

Karhunen–Loève approximation of Brownian motion and diffusions

The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows: $W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$ where $Z_n \sim ...
9
votes
1answer
1k views

Martingales in both discrete and continuous setting

I am wondering, polynomials like $S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
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 ...
9
votes
2answers
514 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 ...
9
votes
1answer
314 views

Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$: $$dX_t = f(X_t)\,dt + dW_t,$$ where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...
7
votes
1answer
253 views

When is it possible to construct a joint law from its two-dimensional marginals?

My question is much more specific than the title: Given a symmetric distribution $\Xi$ on $\mathbb R^2$, when is it possible to construct a sequence $\xi_1,\xi_2,\dots$ of random variables such ...
7
votes
2answers
259 views

Can every discrete martingale be embedded in a continuous martingale?

Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...
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 ...
6
votes
3answers
298 views

Solving SDE's on subsets of $R^n$.

I posted this on mathstackexchange to no avail. It is well-known (see for instance Oskendal's text) that if $T>0$ and $$b(\cdot,\cdot): [0,T] \times \mathbb{R}^n \rightarrow ...
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: ...
6
votes
3answers
546 views

A simple decomposition for fractional Brownian motion with parameter $H<1/2$

Background Let $X = \{X(t):t \geq 0\}$ be a (standard, real-valued) fractional Brownian motion (fBm) with parameter $H \in (0,1)$, i.e., a continuous centered Gaussian process with covariance ...
6
votes
1answer
517 views

Change of space-time in Walsh's stochastic integral

One can read about Walsh's construction of martingale integral in the paper (pp.16-23) www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf For $U,V\in \mathcal{B}(\mathbb{R}\times \mathbb{R}^+), ...
6
votes
0answers
253 views

Stochastic Integration via Skorohod Representation

I am interested to know if Ito integrals against Brownian motion can also be constructed via Skorohod representation. By this I mean the following: let $S_n$ be a simple random walk started at zero; ...
5
votes
3answers
476 views

A non-degenerate martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_t$ is a standard Brownian motion. Let $Y_t$ be a martingale given by ...
5
votes
1answer
433 views

Feynman-Kac for jump-diffusion

I'm looking for a more general Feynman-Kac formula that works in the case of jump-diffusion processes. I know that, given a pure diffusion process like $$dS_t=\mu_tdt+\sigma_tdW_t,$$ if $u(t,s)$ ...
5
votes
2answers
329 views

Itô-like calculus for $\alpha$-stable processes $\alpha \neq 2$.

After searching the net, I couldn't find a suitable reference to some extension of the Itô calculus that involves wilder sources of randomness than mere brownian motion. Motivation : Itô calculus is ...
5
votes
1answer
1k views

Martingale representation theorem for Levy processes

Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases?
5
votes
1answer
468 views

Exact simulation of SDE

Consider a one dimensional SDE of the form $dX_t = \mu(X_t) dt + \sigma dW_t$, where $\sigma>0$ is a constant. Under mild regularity assumptions on $\mu(\cdot)$, one can exactly simulate ...
5
votes
2answers
297 views

comparing diffusions

Consider a probability distribution $\pi$ on the real axis that has a density (w.r.t Lebesgue) proportional to $e^{-V(x)}$, where $V(\cdot)$ is a potential function. For any reasonable volatility ...
5
votes
2answers
457 views

Symmetric Feller processes and Dirichlet Forms

Let $(G, \mathcal D)$ be a densely defined operator on $C_0$ (continuous functions vanishing at infinity on some nice topological space) whose closure $\bar G$ generates a Feller semigroup and let $X$ ...
5
votes
2answers
439 views

Stochastic Green-Gauss Theorem

Is there a stochastic analog for the Green-Gauss theorem? I'm looking for an expression that relates the flux (or statistical moments of the flux) through a random surface to the divergence of the ...
5
votes
1answer
315 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 ...
5
votes
2answers
232 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 ...
5
votes
1answer
298 views

On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)

Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...
5
votes
0answers
360 views

When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb ...
5
votes
0answers
886 views

Levy jump measure vs. Levy measure vs. sum of jumps

This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone ...
5
votes
0answers
714 views

Brownian Motion Winding Number

Take a simple random walk $\gamma$ in the complex plane conditioned to start at point $a$ and end at point $b$. For this random walk, we can define the winding number $W_\gamma(a,b)$ around $b$ in the ...
4
votes
2answers
449 views

law of iterated logrithm

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_t$ is a standard Brownian motion. By law of iterated logarithm, one has ...
4
votes
3answers
399 views

Continuity in intial state of Brownian Motion

$ B = (B_t, \mathcal{F}_t; t\ge 0 ) $ is a 1-d Brownian family on a measurable space $(\Omega, \mathcal{F})$ with a family of probability measures $\{\mathbb{P}^x\}$, i.e. $\mathbb{P}^x(B_0 = x) = 1$, ...
4
votes
2answers
199 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
4
votes
2answers
177 views

Probability of winding number of 2D Brownian Motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
4
votes
1answer
270 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: ...
4
votes
1answer
342 views

Converse to Girsanov's theorem?

Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...
4
votes
2answers
938 views

Generalized Ito's formula

Consider classical statement of Ito's formula: Let $X$ be a continuous semimartingale and $F \in C^2(\mathbb{R}^d, \mathbb{R})$; then $F(X)$ is a continuous semimartingale and $$F(X_t) = F(X_0) + ...
4
votes
2answers
820 views

Weierstrass' function and Brownian motion

Is there a known connection between Weierstrass' function $W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$ and Brownian motion? Specifically, when $\alpha = 1/2$, the Weierstrass ...
4
votes
2answers
241 views

Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator

Let $\xi_t$ be a zero-mean gaussian process on $[0,1]$ with covariance operator $C$. I would like to better understand the relation between the covariance operator and the regularity of the ...
4
votes
1answer
447 views

Fractional Brownian motion and Laplacian

Having read this link on math stackexchange, I would like to submit to your wisdom the following questions. Is it possible, mutatis mutandis, to repeat the same reasoning for a fractional Brownian ...
4
votes
1answer
718 views

Distribution of running maximum of a local martingale

Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_t)$ be a given probability space with usual conditions, on which $W$ is a standard Brownian motion. For $x \ge 0$, consider $$X(t) = x + \int_0^t ...
4
votes
1answer
221 views

What is the optimal growth of the constant in BDG?

Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} ...
4
votes
1answer
342 views

Time-integral of a smooth, vector-valued function of a planar Brownian bridge

I'm looking for information on how to compute the distribution of the random vector $$Z = \int_0^t f(B_s) ds$$ where $t>0$ is fixed, $B_s$ is a 2D Brownian bridge with $B_0 = 0$, $B_t=b \in ...
4
votes
1answer
335 views

Homogeneous linear stochastic DE with noncommuting coefficients

The system I am studying can be reduced to a Stratonovich vector stochastic differential equation $dX = A X \; dt + \sum B_k X \circ dW_k$ with $W_k$, $k=1..m$ the Brownian motion in $m$ dimensions, ...
4
votes
0answers
68 views

Stochastic calculus for several inputs

In "On the Gap Between Deterministic and Stochastic Ordinary Differential Equations," The Annals of Probability, Vol. 6, No. 1 (Feb., 1978), pp. 19-41, Hector J. Sussmann showed that a stochastic ...
4
votes
0answers
155 views

Malliavin calculus w.r.t $G$-Brownian motion

I wonder if it is possible to define a Malliavin calculus w.r.t $G$-Brownian motion defined on a Sublinear Expectation Space available on this link. G–Brownian motion has a very rich and interesting ...