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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4
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1answer
128 views

Can all Local Martingales Be Represented using Only Brownian Motion and Finite Variation Processes?

This is a cross-post of my unanswered (more than a week) question on Math.SE. Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to try to ...
0
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0answers
14 views

Expected value of stochastic process [on hold]

How can i calculate the expected value of $$S_t= S_0 \exp\left( mt-\frac12\int_{0}^{t}e^{2Y_s}ds+\int_{0}^{t}e^{Y_s}dB_s\right)\quad $$ where $${Y_t}$$ is the solution of a sde and follows tha normal ...
-1
votes
0answers
22 views

Closed form formula for fill rate given a discrete distribution? [on hold]

I'm wondering whether there is a closed form way to obtain good estimates for fill rate given a discrete distribution of demand. I created a simple monte carlo simulation to see if I could see any ...
2
votes
1answer
88 views

Representation of support of Gaussian measure by kernels of no-variance functionals

Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for $$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$ ...
5
votes
0answers
86 views

Short time asymptotics for Brownian motion on a compact manifold

Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in ...
1
vote
0answers
75 views

Brownian hitting probability of a small body

Consider a Brownian motion $B(t)$ starting from the origin $0$ in $\mathbb{R}^n$. Consider the ball $B(0, r)$ and an open set $V \subset B(0, r)$. If it is known that the probability of the Brownian ...
2
votes
0answers
66 views

Brownian motion in perturbed (asymptotically flat) metric

Let $g_{\mathbb{R}^n}$ denote the usual Euclidean metric on $\mathbb{R}^n$ and let $B_g(t)$ denote the Brownian motion associated to a complete metric $g$ on $\mathbb{R}^n$. Consider a Brownian motion ...
0
votes
0answers
32 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
5
votes
0answers
77 views

Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs. I'm reading Stochastic Differential Equations in ...
0
votes
1answer
50 views

Girsanov theorem and the density of a process

I am coming across a paper ( Proposition $1.1$ from http://www.sciencedirect.com/science/article/pii/0304414987901840 ) that claims the following fact which I don't understand why: On a ...
4
votes
2answers
173 views

A Stochastic Taylor Expansion/Asymptotics

Question: Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth ...
3
votes
1answer
140 views

Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ where the coefficients are assumed to be Lipschitz continuous. I hope to ...
0
votes
0answers
38 views

Where can I find this article of Doléans-Dade?

I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade. I could not find a pdf version online, and my university library does not have a printed version. Thank ...
0
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0answers
25 views

Smoothness of Value function for SDE with discontinuous coefficients

Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous). I'm interested in the function $v:\...
2
votes
0answers
171 views

Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process $$ dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0 $$ where $a\in (-\infty,+\infty), b&...
1
vote
0answers
78 views

Construction of a random variable

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the statement:...
1
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0answers
56 views

A problem on Markov chains and Dirichlet forms

Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for ...
3
votes
1answer
143 views

How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here. Say we have a stochastic process described by a ...
0
votes
0answers
63 views

Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let \begin{equation} I = \int_D g(\textbf{x})d\textbf{x}, \end{equation} where $D \subset \...
0
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0answers
27 views

Fubini's Theorem for Lévy bases

Let $M$ be an infinitely divisible independently scattered and homogeneous random measure on $\mathbb R^d$ (ie a homogeneous Lévy basis). Let $\nu$ be a sigma finite measure on $\mathbb R^k$. Let $f:\...
2
votes
1answer
160 views

Question on Wiener processes not hitting 0

Let $W_t$ be a standard Wiener process, and $0\leq a < b$. Let $\hat{W}_t:=W_{a+t}-W_a$. Then $\hat{W}_t$ is also a standard Wiener process. I think that the following should be true: $$\mathbb P\...
1
vote
1answer
100 views

Limit (Convergence) of stopping times

Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
2
votes
0answers
82 views

Hypergeometric function

Suppose that $V$ follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$ I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that $F(V)$ must ...
0
votes
0answers
38 views

regularity of the conditional expectation: from Markov to Non-Markov

Let $B=(B_t)_{0\le t\le T}$ be a standard Brownian motion and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Let $\xi=\xi(B)$ be a bounded measurable functional. Now let's ...
0
votes
0answers
33 views

Distribution of stopped Brownian motion in $\mathbb R^2$

Let $B=(B^1_t,B^2_t)_{t\ge 0}$ be a standard Brownian motion in $\mathbb R^2$. Let $U=(U^1,U^2)$ be an independent random variable taking values in a circle $C_1\subset\mathbb R^2$ with uniform ...
0
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0answers
55 views

limit multiple integral

I want to know if $\lim_{T-> \infty}$ of this integral $$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\ \times \int\limits_{[0,T]^{4}}e^{\theta(t_{1}-s_{1})}e^{\theta(t_{2}-s_{2})}\left\...
1
vote
0answers
36 views

Are the elementary predictable processes dense in $L^2([M])$ for $M$ a local martingale?

The question is the one from the title. I know this is true when $M$ is an $L^2$ bounded martingale (which is often used in the classical approach to the construction of the stochastic integral) but I'...
3
votes
0answers
174 views

Proof of Feynman Kac formula

I am trying to write a complete proof of the Feynman Kac formula in the multi-dimensional case. My starting point was the proof of the univariate form on wikipedia, at https://en.wikipedia.org/wiki/...
1
vote
1answer
117 views

A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...
2
votes
1answer
64 views

Decouple system of SDEs / handle scaling problem

Consider $\begin{split} \newcommand{\d}{\mathrm d} \d x &= -yx \d t + x^2 \d B\\ \d y &= -2 y^2 \d t + 2xy \d B. \end{split}$ This is a system of two SDEs driven by the same standard ...
0
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0answers
34 views

Strong Markov vector-valued process from component strong Markov process and independence

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...
0
votes
1answer
70 views

Brownian motion increments

We know that if $W_t$ is a Brownian motion, $W_{t+t_0}-W_{t_0}$ is one too. Does the "converse" holds : Let $t_0$ be a positive number. I have a Brownian motion $W_t$ and I seek another Brownian ...
1
vote
0answers
108 views

Does the martingale property holds after changing filtration?

Let $\Omega$ be the space of continuous real-valued functions $\omega=(\omega_t)_{t\ge 0}$ starting at zero, i.e. $\omega_0=0$. Let $\Lambda=\Omega\times \mathbb R_+$ and denote by $\lambda=(\omega,\...
1
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0answers
68 views

Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$? For examples: are there: Ito Isometry(-...
2
votes
1answer
139 views

Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
0
votes
0answers
67 views

Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...
0
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0answers
38 views

Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
0
votes
0answers
76 views

Brownian motion - probability of hitting an open subset of the sphere

Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let $\mathbb{P}_t(A)$ be the probability of the particle striking $A \subset S^{n - 1}$ within time $t$, where $A = \{ (x_1, x_2,...
0
votes
1answer
126 views

Transition probabilities for the symmetric random walk on the integers

I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...
13
votes
0answers
111 views

Does a theory of stochastic differential algebras exist?

My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
0
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0answers
56 views

Circular process ergodic?

Let us define a continuous-time Markov process on a circle consisting of $m-$ equally spaced points, i.e. every point has two neighbours. Now, we define a space of functions $S:= \{-1,1\}^{\{1,...,m\}...
0
votes
1answer
86 views

Weak convergence of process

Background: I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist: Let $$L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) \...
1
vote
0answers
90 views

Malliavin differentiability of solutions to SDEs

In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...
-2
votes
1answer
67 views

Definition: Grigelionis Process?ch [closed]

Background I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and ...
1
vote
1answer
122 views

Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
2
votes
1answer
63 views

Quadratic variation and the variance of a semimartingales

I will describe an example that seemingly contradicts the following Theorem For a local martingale $M$, let $[M,M]_t$ be its quadratic variation at $t$. For any $t$, if $E[[M,M]_t]<\infty$, then $...
2
votes
0answers
53 views

Holomorphic solution to SDE

Consider the SDE $dZ_t = \mu(t,x) d_t + \sigma(t,x) dW_t$. Are there any known (necessary and) sufficient conditions on $\sigma(t,x)$ and on $\mu(t,x)$ guaranteeing that $f(T):=\mathbb{E}[\int_0^T Z_t ...
3
votes
2answers
175 views

Brownian motion in $\mathbb{R}^n$, probability of hitting a set

Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying ...
1
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0answers
84 views

Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable? This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion http://www.springer.com/gb/book/...
3
votes
1answer
123 views

Malliavin Calculus: directional derivatives of cylinder functions exist in what sense?

Denote by $P_0(\mathbb{R}^d)$ the sets of continuous paths over $[0,1]$ started at $x=0$ with values in $\mathbb{R}^d$, we equip this space with the sup-norm and make it into a probability space by ...