**1**

vote

**1**answer

359 views

### Change of time variable in Wiener process

I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
$...

**0**

votes

**1**answer

374 views

### Version of Ito's lemma applied to a stochastic function

The Ito's formula stated in most books in stochastic calculus is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and $...

**5**

votes

**2**answers

225 views

### Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds

I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a ...

**0**

votes

**1**answer

288 views

### Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where $\tilde{...

**2**

votes

**1**answer

193 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....

**2**

votes

**0**answers

90 views

### Sobolev Bundle on Wiener Space

Right now I am learning about analysis of stochastic processes and the Malliavin calculus. It seems though, that most of the theory works for Brownian motion in $\mathbb{R}^n$, and it seems non-...

**3**

votes

**2**answers

340 views

### Analytic Solution to SDEs

Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form:
\begin{equation}
dX_t = f(...

**3**

votes

**1**answer

122 views

### Certain construction of the Itô integral on manifolds

Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral
$$ I(X) = \int_0^T \langle X(t, ...

**2**

votes

**2**answers

384 views

### Uniqueness in martingale representation theorem

Dudley's martingale representation theorem states that if $W=\{W_t,\mathcal{F}_t;0\le t<+\infty\}$ is a standard one-dimensional Brownian motion, $0<T<+\infty$ and $\xi$ is $\mathcal{F}_T$-...

**6**

votes

**1**answer

390 views

### Itô's article “A measure-theoretic approach to Malliavin calculus”

Apart from citations all over the internet, the following paper appears to be off-the-grid.
K. Itô, A measure-theoretic approach to Malliavin calculus, in 'New Trends in Stochastic Analysis', Proc. ...

**3**

votes

**1**answer

99 views

### Could quadratic variation determine distribution?

Let $M=\{M_t,\mathcal{F}_t;0\le t<+\infty\}$, $N=\{N_t,\mathcal{F}_t;0\le t<+\infty\}$ be two continuous local martingales with $M_0=N_0=0\text{ a.s.}$. If $\langle M\rangle=\langle N\rangle$, ...

**6**

votes

**2**answers

418 views

### A version of Wald identity

Let $W$ be a standard one-dimensional Brownian motion. Let $T$ be a stopping time with $\mathbb{E}\sqrt{T}<+\infty$. Then
$$\mathbb{E}W_T=0\quad \mathbb{E}W^2_T=\mathbb{E}T$$
I can prove these ...

**0**

votes

**1**answer

372 views

### A question on Ito integral

Let $W$ be a standard one-dimensional Brownian motion and $0<T<+\infty$. Then
$$\lim_{\beta\to+\infty}\sup_{0\le t\le T}|e^{-\beta t}\int_0^te^{\beta s}\mathrm{d}W_s|=0\quad \text{a.s.}$$
Could ...

**2**

votes

**0**answers

67 views

### Existence of 1-1 mapping/homeomorphism

Let $B$ be a standard 2-D Brownian motion, and $\sigma: \Omega\times \mathbb R^{+} \mapsto \mathbb R^{2 \times 2}$ is an $\mathcal F_{t}$ adapted process satisfying, for some constants $0<\lambda&...

**3**

votes

**0**answers

60 views

### What is the probability of B.M. hitting two disjoint spheres $(d\geq 3)$?

The hitting probability for spheres centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$, where $|x|>r$.
1)So I was wondering how can one compute $P_{x}(T_{B_{r}(...

**3**

votes

**1**answer

299 views

### Onsager-Machlup function and most probable path of a diffusion process

Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation
\begin{equation}
dX_{t} = f(X_{t})dt + dW_{t},
\end{equation}
where $f \in C_{b}^{2}(R)$ is a ...

**0**

votes

**0**answers

90 views

### What is the sigma field of the derivative of a process?

When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...

**1**

vote

**1**answer

120 views

### Perturbation of a Bessel process of dimension 2

Bessel process of dimension 2 is defined to be solution of
$$
dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0
$$
where $B$ is a standard 1-dimensional Brownian motion.
$X$ can be viewed as the norm of a ...

**3**

votes

**2**answers

413 views

### Stochastic methods for solving very high-dimensional PDE

I am looking for stochastic methods for solving a very high-dimensional PDE (with one time dimension and very large number of spatial dimensions), which would reduce it to a lower-dimensional problem, ...

**1**

vote

**1**answer

129 views

### Functional representation of adapted jointly measurable stochastic processes

It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO.
Let $X_t : \Omega \to E, \ t \geq 0$ be ...

**3**

votes

**0**answers

189 views

### Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...

**3**

votes

**0**answers

175 views

### Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular,
Have the numerical implications, if any, of the results in ...

**3**

votes

**3**answers

218 views

### Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit:
$$
dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC
$$
where $W$ is a standard Brownian motion ...

**1**

vote

**1**answer

268 views

### relationship of SDE in Langevin equation form and Ito form

A formal SDE can be written in a way as (ito form):
$dx(t)=ax(t)dt+dw(t)$
where $w(t)$ is brownian motion.
Another way is to write the SDE (Langevin equation form) is
$\frac{dx(t)}{dt}=ax(t)+w(t)$
...

**3**

votes

**0**answers

265 views

### Generalization of Ito's formula

If $f:R\to R$ is a convex function then we have Ito-Tanaka formula. Now my question is that if we are given a function $u: R\times R_+\to R$ such that $u(s,\cdot)$ is smooth for every $s\in R$ and $u(\...

**2**

votes

**0**answers

69 views

### The distribution of maximum of fraction Brownian motion over finite time interval

Suppose that $\{B_t^H,\ t\geq 0\}$ is a fractional Brownian motion with Hurst exponent $H$, I wonder if there are explicit expressions for the joint distribution of
$(\sup_{0\leq t\leq T}B_t^H,B_T^H)$...

**3**

votes

**1**answer

241 views

### Unusual augmentation of a filtration

consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...

**2**

votes

**1**answer

149 views

### An identity for the exponential of a martingale

I am trying to understand a Lemma in Olav Kallenberg's book "Foundations of Modern Probability" (Lemma 26.19 in the second edition or 23.19 in the first edition).
The part of the lemma that I do not ...

**2**

votes

**0**answers

301 views

### Girsanov theorem with Geometric Brownian Motion

I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...

**4**

votes

**1**answer

225 views

### Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...

**5**

votes

**0**answers

122 views

### Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...

**0**

votes

**0**answers

116 views

### Probability that d-Brownian Motion ,$d\geq 3$, avoids a fixed set A

In other words, the probability that Brownian motion stays within $A^{c}$.
What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...

**4**

votes

**1**answer

472 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 $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

**0**

votes

**1**answer

233 views

### On the superior of generalized Ornstein-Uhlenbeck process

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde:
$dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $
$\partial_x X_t(0) = \partial_x X_t(1) = 0, $
$X_0 = 0, $
...

**2**

votes

**1**answer

147 views

### Numerical computation of Skorokhod integral

How can I numerically compute the Skorokhod integral of a non-adapted process? If it is adapted, that is easy since the integral is just an Ito integral.
I have found that computing the Malliavin ...

**1**

vote

**0**answers

241 views

### Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...

**8**

votes

**1**answer

696 views

### Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.
Here is what I mean exactly. ...

**3**

votes

**0**answers

89 views

### How can one do change of variables for solutions to a staochastic partial differential equation?

isHow can one do change of variables for solutions to a staochastic partial differential equation? For example, let us consider the following stochastic transport equation:
$$
dy(t,x) + y_x(t,x) + y(t,...

**1**

vote

**1**answer

160 views

### Is $\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{n p_n}{n})|^{n-k}- e^{- \lambda}|}{k!}=0$?

I am currently the convergence of different processes. Doing this, I ended up with this expression and was wondering whether it is true that$$\lim_{n \rightarrow \infty}\sum_{k=0}^{n} \frac{|(1-\frac{...

**1**

vote

**0**answers

63 views

### question related to Tanaka Formulae

Supposse $X=(X_t)$ is a cadlag martingale taking values in $\mathbb{R}$. If $f:\mathbb{R}\to\mathbb{R}$ is a convex function, then we have Tanaka Formulae. Now let $g: \mathbb{R}_+\times\mathbb{R}\to\...

**1**

vote

**1**answer

51 views

### question about the optimal decomposition of supermartingale

Given a filtered probability space $(\Omega, \mathbb{F}, \{\mathcal{F}_t\}_{0\le t\le 1}, \mathbb{P})$, let $X$ be a cadlag martingale and $V$ be cadlag supermartingale. Suppose $V$ has the following ...

**0**

votes

**1**answer

128 views

### Monte Carlo estimator with autocorrelated samples

Given an integration problem $I=\int{f(x)dx}$, we can construct an ordinary Monte Carlo estimator as
$E[I]=\sum\limits_i\frac{f(x_i)}{p(x_i)}$
where the samples $x_i$ are usually i.i.d. and drawn ...

**4**

votes

**0**answers

188 views

### Expectation of running maximum of diffusion processes

Let $X$ be a one-dimensional Ito diffusion $$X_t=x+ \int_0^t b(X_s)ds + \int_0^t \sigma(X_s)dW_s,$$ where $b,\sigma$ satisfy the usual Lipschitz continuity and linear growth conditions. Define the ...

**2**

votes

**0**answers

248 views

### Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type
$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$
converges to zero (in probability/distribution) for $h \...

**4**

votes

**3**answers

374 views

### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,
\end{equation}
where $b,\...

**2**

votes

**3**answers

77 views

### a special filtration satisfying $0$-$1$ law

Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let $\mathcal{F}_t=\sigma(\...

**0**

votes

**1**answer

79 views

### Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent.
1): $X$ is $L^p$-integrable.
2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} |...

**1**

vote

**0**answers

51 views

### a question about the modification of a supermartingale

Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process
$$X_{t}(\...

**4**

votes

**1**answer

239 views

### explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition:
$$V_t=V_0+\int_0^tH_sdX_s-K_t$$
where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with $...

**3**

votes

**2**answers

494 views

### Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$,
the sequence $(M^n)_n$ converges.
Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices $(...