Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

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3
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1answer
469 views

Results for Hitting Times of (Not Stationary) Ito Processes

Let $W_t$ denote the Wiener process and let $$ dX_t = a(t, X_t) dt + b(t, X_t) dW_t $$ be an one dimensional Ito SDE (stochastic differential equation, see Ito stochastic calculus). A hitting time ...
6
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3answers
582 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 ...
7
votes
1answer
258 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 ...
17
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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 ...
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3answers
6k views

Expectation of time integral of Wiener process

I am trying to calculate $E(\int_0^T {W_s ds})$, where $W_s$ is a standard Brownian motion. Now two approaches I can think of: 1) Take a partition of $[0,T]$. Calculate $E(\sum {W_{t_i}(t_{i+1} - ...
6
votes
2answers
920 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 ...
1
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1answer
696 views

Need help understanding Mandelbrot and Van Ness Fractional Brownian Motion

I need help understanding the Mandelbot and Van Ness' definition of Fractional Brownian motion $ B_H( t , \omega ) - B_H( 0 , \omega ) = \frac{1}{\Gamma(H + \frac{1}{2})} \left\( \int_{-\infty}^0 ...
5
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1answer
328 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 ...
13
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4answers
1k 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 ...
3
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0answers
145 views

Characterizing polyhedron from Brownian particle collisions with a boundary

Please imagine that we have an ordinary 2-sphere, of radius $r_{sphere}$, and some three-dimensional polygon that has all of its points fixed at positions strictly internal to the sphere's surface. ...
4
votes
2answers
244 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 ...
2
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1answer
622 views

Taylor expansion to show that for Stratonovich stochastic calculus the chain rule takes the form of the classical one

As nobody seems to be able to give any kind of answer to that problem over there at math.stackexchange I post this question here: How can I show with a heuristic argument based on a Taylor expansion ...
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0answers
297 views

Change of Time in Stochastic Integral

Hi everyone, Let's be given $I(0,t)$ a Stochastic Integral with respect to a local martingale $ M_t$ of the form : $I(0,t)=\int_0^t h(s_-)dM_s$ with $h\in L(M)$ (for example $h$ is an adapted ...
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0answers
231 views

Density of Dolean exponentials in L2 and Wiener Measure

Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by ...
4
votes
3answers
404 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
1answer
836 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 ...
3
votes
1answer
378 views

Approximation of the law of a stochastic process

Hello Dear fellows, I thank you in advance for your help and ideas. I have just read an article and want you to help me understand the rational behind a part of it. We have two processes $v_t$ and ...
17
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6answers
5k 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 ...
3
votes
2answers
790 views

Is the truncated Brownian motion of the class DL?

Let $W$ be a standard Brownian motion under given probability space. For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time $T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = ...
2
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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
917 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 ...
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3answers
914 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
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2answers
252 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
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1answer
934 views

Stationary Solutions of stochastic differential equations

When does the stationary density of an homogeneous Markov process exist?
3
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1answer
314 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
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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
473 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
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1answer
433 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
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2answers
367 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 ...
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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 ...
14
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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
334 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 ...
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vote
2answers
496 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
257 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
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2answers
305 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
540 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
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4answers
379 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
475 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 ...