# Questions tagged [stochastic-calculus]

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.

611
questions

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votes

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138 views

### Calculate Radon-Nikodym derivative

For the laws of two pure-jump Markov processes $\mu_1$ and $\mu_2$ on $\mathbb R^n$, which generators are
$H_1f(x)=\int h(x,dy) (f(y)-f(x))$
and $H_2f(x)=\int e^{-g(x,y)} h(x,dy) (f(y)-f(x))$ (...

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votes

**0**answers

22 views

### Can one change the dimension of a Bessel process by a Girsanov change of measure?

Recall that a (squared) Bessel process $X_t$ with the dimension $\delta_0>0$ is the solution of the SDE
$$d X_t = 2\,\sqrt{X_t}\,d W_t+\delta_0\,d t.$$
A naive application of the Girsanov Theorem ...

**5**

votes

**1**answer

84 views

### Scalar product of random unit vectors

Let $X,X'$ be two random vectors on the sphere $S^{d-1}$. What is the distribution of their dot product $X\cdot X'$ in the following cases:
$X,X'$ independent with uniform distribution on the sphere $...

**3**

votes

**1**answer

154 views

### Do Lyapunov functions imply exponential integrability of hitting times?

I have a question of some integrability of hitting times.
Let $X=(\{X_t\}_{t \ge0},\{P_x\}_{x \in E})$ be a diffusion process on a locally compact separable metric space $E$.
We assume that there ...

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votes

**0**answers

39 views

### Fourier transform of a general normal distribution [closed]

How can I calculate the Fourier transform of
this equation? Here is a known equation that may be helpful.

**-2**

votes

**0**answers

25 views

### Dominance convergence theorem to compute expectation of a sequence of random variables defined by their time derivatives

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Then consider a sequence $X_t^0,X_t^1,\ldots, X_t^n$ for which we get $Y_t^0,Y_t^1,\ldots, ...

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vote

**2**answers

168 views

### Exponential or sub-exponential ergodicity?

Consider the one-dimensional stochastic differential equation $$d X(t) = -sgn(X(t))dt + dW(t),$$ where $W$ is a standard Brownian motion, and $sgn(x) = 1$ if $x > 0$ and $-1$ if $x\le 0$. It can be ...

**0**

votes

**1**answer

27 views

### Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?

I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" ...

**-1**

votes

**0**answers

30 views

### Stopping times about Brownian motion with draft

Assumet $M(t) = B(t) + \mu t$ where $B(t)$ is a standard Brownian Motion. Denote:
$$T_a := \inf \{ t \geq 0, \, M(t) = a\}, \quad T_b := \inf \{ t \geq 0, \, M(t) = b\}$$
The question asks to ...

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vote

**1**answer

62 views

### The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$

In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...

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votes

**1**answer

58 views

### Find a conditional expectation of a difference of two independent Poisson process

Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the ...

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21 views

### Stochastic differential equations with correlated Brownian Motions

let's consider an sde of this kind:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t \\
X_0=x_0 \\
dY_t=B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^2 \\
Y_0=y_0
\end{...

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votes

**0**answers

16 views

### Bifractional Brownian motion admit a representation in the form of a stochastic integral?

good morning. You know the fractional Brownian motion, multifractional Brownian motion and sub-fractional Brownian motion, can be represented as a wiener integral ( moving average representation ). ...

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60 views

### Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$

I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that
If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...

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22 views

### Sufficient condition for weak existence of solution of a SDE

Please be adviced that I'm cross-posting this question from MSE since it's very likely it will remain unsolved, and I haven't been able to obtain an answer from my colleges/professors.
It's a well ...

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vote

**0**answers

207 views

### On the level of measure theory, what does it mean for a drift to be deterministic?

Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...

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19 views

### SDE conditional expectation

Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...

**0**

votes

**1**answer

66 views

### Question about the exit time of a time-homogeneous Itô diffusion

Consider a one-dimensional Itô diffusion:
$$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$
where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...

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44 views

### conditional expected value and in Stochastic differential equations

Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...

**0**

votes

**1**answer

58 views

### Stochastic invariant subset

Let us consider a stochastic differential equation (SDE),
$$
dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}%
$$
and a compact set $C\subset\mathbb{R}^{n}$.
Given a stochastic ...

**-3**

votes

**1**answer

59 views

### Does having the derivative in the limit suffice to solve the function at the limit? [closed]

Suppose that I have a function $f(x, \epsilon)$ and I know that
$$
\lim_{\epsilon \to 0} f'(x, \epsilon) = g'(x).
$$
Now let $g(x)$ be the function whose derivative appears above. How can I ...

**0**

votes

**2**answers

110 views

### Transience of 3-dimensional Brownian motion

I'm attempting Exercise 5.33 of Le Gall's Brownian motion, Martingales and Stochastic Calculus.
Let $B_t$ be a 3-dimensional Brownian motion starting from $x$.
Part 6 asks me to show that
$$|B_t| = |...

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votes

**2**answers

94 views

### Convergence of fraction of expectation values

Let $X_1,...,X_n$ be iid normal random variables.
I am looking for a strategy to establish the following limit for fraction of expectation values
$$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i ...

**0**

votes

**0**answers

43 views

### Exponential functional of an Ito processes

Let $\sigma_t \in L^2 (\mathbb{R})$ an adapted square integrable process and $W_t$ a brownian motion.
Does the closed form of law of the following process $I_t$ existe?
$$I_t(\sigma_t) = \int_0^t e^...

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votes

**0**answers

29 views

### Associating “weak” solutions of stochastic differential equation on manifolds with real valued weak solutions

Let's say we are working on a (real) differentiable manifold $M$. For smooth vector fields $A_0,A_1,...,A_r$ on $M$ we define stochastic differential equations as $$dX_t = A_{\alpha}(X_t) \circ dB^{\...

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vote

**2**answers

159 views

### Continuity of Brownian motion constructed from Kolmogorov extension theorem?

I'm trying to construct Brownian motion using the Kolmogorov extension theorem.
I am happy with the construction of a process with the required FDDs as (the canonical process associated with) a ...

**-2**

votes

**1**answer

50 views

### Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?

I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...

**0**

votes

**1**answer

79 views

### Is the integral of an adapted, measurable process adapted?

Let $X_s(\omega)$ be measurable and adapted.
Under what conditions will the process $F_s(\omega) = \int_0^t X_s(\omega) \, ds$ also be adapted?
To me it seems that adaptedness and measurability ...

**0**

votes

**0**answers

25 views

### Gap between optimal policies for optimal control problems

Suppose that one is given a stochastic optimal control problem
$$
J(u)\triangleq \mathbb{E}\left[
\int_0^T L(t,u_t,X_t^u) dt + g(X_T)
\right],
$$
where $L$ and $G$ are convex, non-negative valued, and ...

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votes

**0**answers

59 views

### Martingale polynomial functions

If $B_t$ is a Brownian motion then using Hermite polynomials one can find that
$$1, B_t, B_t^2-t, B_t^3 - 3tB_t,...$$
are martingales.
If $X_t$ is a diffusion
$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)...

**8**

votes

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254 views

### Show that this process is not a martingale

I am cross-posting this question from MSE since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer.
The most surprising ...

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votes

**0**answers

26 views

### Reference for an infinite system of SDEs

Consider system of the following form,
\begin{align*}
\mathrm{d} X_k(t) = \big(AX(t)\big)_k\mathrm{d}t + B_k(X_k(t))\mathrm{d}t+\mathrm{d}W_k(t),\quad k\in\mathbb{Z},
\end{align*}
where $A$ is matrix, ...

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22 views

### Step verification in derivation of Ito formula

At the page 125 (see 4.90) (see images below, this proof is from "Statistics of Stochastic processes" by Lipster and Shiryaev) we consider function $u(t,W_t)=f(t,at+bW_t)$ and $a(\omega), b(\omega)$ ...

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45 views

### Representation of optimal controls as diffusions

In reading this post I couldn't help but wonder the following question:
Let $\sigma>0$ and suppose, as in the motivational post, we are given a stochastic optimal control problem:
$$
\begin{...

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votes

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34 views

### Reference request: Ito formula for function $G(t, x)$ when $G$ depend on $\omega$

There is proved Lemma in book : Let the function $G(t,x)$ is defined when $t\in [0,T], x\in(-\infty,\infty)$, $G$ has continuous derivative with respect to $t$ and twice continuously diferentiable ...

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votes

**1**answer

74 views

### What's the role of commutation relations in stochastic mechanics?

In a stochastic context, we can understand a term like
$$ \int_0^T \frac{d q(t)}{dt} dq $$
either as the (Ito) limit
$$ \lim_{N\to\infty} \sum_{i}^N dq(t_i) \frac{d q(t_i)}{dt} $$
or the (Anti-...

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votes

**0**answers

47 views

### Scaling Property Ito diffusion processes

It is well-known that a Brownian motion $W$ has the following scaling property
$$
c^{-1/2}W_{ct} \qquad \mbox{ for any $c,t>0$}.
$$
In particular this means that the increments of the processes $W$ ...

**15**

votes

**2**answers

503 views

### Why do stochastic integrals depend on the choice of partitioning points?

When we integrate a function, we must make some choice about how we approximate it before we take the limit.
In principle, we can choose $\tau_i$ to be any value between $t_{i-1}$ and $t_i$. But for ...

**3**

votes

**1**answer

311 views

### White noise vs. black noise

In this excellent lecture ("2d Percolation Revisited") Stanislav Smirnov mentioned the connection of the theory of percolation with the notion of the so called black noise—see at 29:42 (the notion ...

**1**

vote

**1**answer

82 views

### Explicit densities for Brownian motion hitting times

I'm looking for functions $g: \mathbb{R}_+ \to \mathbb{R}$ such that the hitting time
$$\tau := \inf \{t \geq 0 : B_t \nleq g(t) \} $$
has an explicit density with respect to the Lebesgue measure, ...

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votes

**0**answers

35 views

### differential of posterior probability distribution over the mean drift of brownian motion

Let $W_t$ be the Weiner process, and let $X_t = W_t + \mu t$, where $\mu$ is either 0 or 1. We wish to get information about $\mu$ by looking at $X_t$. Let $q_t$ be the probability we assign to $\mu=1$...

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28 views

### Applications of Kazamaki Conditions

I'm interested in applications of this theorem by Sekiguchi Kazamaki:
"Continuous Exponential Martingales and BMO" - Theorem 1.12:
Let $M$ be a continuous local martingale and $Z(M):= \exp(M-\frac{1}{...

**1**

vote

**1**answer

75 views

### A generalized Mercer's Theorem?

If $X_t$ is a mean zero, square integrable process with covariance kernel $k(s,t),$ Mercer's theorem states that there exists an orthogonal basis $\{\phi_i\}$ in $L^2$ and eigenvalues satisfying $$\...

**2**

votes

**1**answer

150 views

### Pathwise stochastic integral as a linear operator on continuous functions

Let $B$ be a Brownian motion. Definining a pathwise stochastic integral $I(f):=\int f~dB$ for certain classes of deterministic functions is straightforward: For instance if $f=\sum_ic_i1\{[t_i,t_{i+1})...

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**0**answers

150 views

### Proving that $dX_t=a_tX_td\tilde{B}_t$ is a martingale

Fix $T>0. $Consider the probability space $(\Omega,F,Q)$ and a Brownian motion $\{B_t\}_{t\leq T}$ and filtration $\{F\}_t$ generated by the Brownian paths. Suppose $a_t,\gamma_t$ are random ...

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votes

**0**answers

84 views

### Malliavin derivative of Ito process

Let $X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$ where $\mu$ and $\sigma$ are $C^1$ functions satisfying the usual growth restriction and $W_t$ is a $d$-dimensional Brownian motion. ...

**2**

votes

**2**answers

208 views

### Weak convergence in Skorohod topology

Let $D([0,T];R^d)$ be the space of càdlàg functions endowed with the usual Skorohod topology. $X_t(\omega):=\omega(t)$ denotes the usual canonical process. Assume that a family of probability ...

**0**

votes

**0**answers

34 views

### Derivation of a differential equation from a SDE

Suppose there is a non-homogeneous Markov process with state space $\mathbb{R}_{+}$
driven by this McKean-Vlasov-tipe SDE:
$$ dY_t = a \mathbb{E}[Y_t]\ dt - b\ Y_t\ dt - Y_t\ dN_{aY_t}$$
where $...

**0**

votes

**0**answers

66 views

### Eigenfunction expansion and ultracontractivity

I have a question on heat kernels and eigenfunctions of symmetric Markov processes.
Let $X=(\{X_t\}_{t \ge0 },\{P_x\}_{x \in M})$ be a $\mu$-symmetric Markov process on a locally compact separable ...

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votes

**0**answers

90 views

### About martingales induced by iterative processes

Suppose I have a discrete stochastic process $\{ X_i \}_{i=1,\ldots..}$ defined as, $X_{i+1} = X_i - \eta \nabla f(X_i) + \sqrt{\eta} \xi_i$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and $\xi_i \...