# Tagged Questions

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.

11 views

99 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/...
629 views

### A difficult integral [closed]

Is there any analytical result on the following integral? $$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+e^{-(x-\mu)}}dx$$ Thanks a lot!
287 views

### Generalized Ito's lemma

I have the following quantity: $$g(t)=(f(t))^{M_{t}},$$ where $M_{t}$ is a jump process which is neither Markovian nor Levy, and $f(t)$ is a positive, increasing but limited, right-continuous ...
493 views

### weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
53 views

### Stochastic process with discontinuous drift

While studying a portfolio optimization problem, I came across the process $$dX(t) = X(t)\,\Big(\,\big(\mu - \alpha\,1_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big)$$ which has a discountinuous ...
66 views

165 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 ...
77 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 ...
35 views

If this is too basic for MathOverflow... say the word and I shall move it to Math.SE First consider this system of ODEs. Say I have two variables $u$ and $a$, following $$\dot u = -u + f(a)$$ $$\... 1answer 149 views ### Can all Local Martingales Be Represented using Only Brownian Motion and Finite Variation Processes? This is a cross-post of my unanswered (more than a week) question on Math.SE. Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to try to ... 0answers 102 views ### Strong law of large number for semimartingale I just want to know if for semimartingale X we have \lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0 or when it is possible. I know it is true for Brownian motion. Thanks 1answer 95 views ### Representation of support of Gaussian measure by kernels of no-variance functionals Let \mu be a Gaussian measure on a separable Banach space X and q is the covariance operator of \mu. I am reading a proof for$$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$... 0answers 92 views ### Short time asymptotics for Brownian motion on a compact manifold Consider a compact Riemannian manifold (M, g). Choose a ball B(p, r) inside M, and a quasi-isometric ball B(q, s) in \mathbb{R}^n, in the image of a coordinate chart containing B(p, r) (in ... 0answers 80 views ### Brownian hitting probability of a small body Consider a Brownian motion B(t) starting from the origin 0 in \mathbb{R}^n. Consider the ball B(0, r) and an open set V \subset B(0, r). If it is known that the probability of the Brownian ... 0answers 67 views ### Brownian motion in perturbed (asymptotically flat) metric Let g_{\mathbb{R}^n} denote the usual Euclidean metric on \mathbb{R}^n and let B_g(t) denote the Brownian motion associated to a complete metric g on \mathbb{R}^n. Consider a Brownian motion ... 0answers 82 views ### Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs. I'm reading Stochastic Differential Equations in ... 0answers 38 views ### System of stochastic equations I want to know if this system of SDE:$$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$... 2answers 176 views ### A Stochastic Taylor Expansion/Asymptotics Question: Let B(t) be the standard Brownian motion, \mu(t,x) and \sigma(t,x) are continuous functions, and$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$(\mu,\sigma) obeys the linear growth ... 1answer 53 views ### Girsanov theorem and the density of a process I am coming across a paper ( Proposition 1.1 from http://www.sciencedirect.com/science/article/pii/0304414987901840 ) that claims the following fact which I don't understand why: On a ... 1answer 163 views ### Question on Wiener processes not hitting 0 Let W_t be a standard Wiener process, and 0\leq a < b. Let \hat{W}_t:=W_{a+t}-W_a. Then \hat{W}_t is also a standard Wiener process. I think that the following should be true:$$\mathbb P\...
40 views

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 ...
176 views

79 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:...
157 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 ...
1k 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 ...
58 views

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 ...
195 views

### Question about the stochastic integral of martingales

Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $\mathbb{R}$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e....
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'...
191 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/...
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 ...
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 ...