# All Questions

Tagged with stochastic-processes stochastic-calculus

381
questions

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**1**answer

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### Ito's Formula for functions that are $C^2$ a.e

In the conventional Ito's formula, it is required that $F$ is $C^2$ everywhere. However I've seen mentioning of a slightly weaker condition, where $F$ is $C^1$ everywhere but $C^2$ a.e.. Is there any ...

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**1**answer

43 views

### Convergence in mean and convergence in distribution

Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that
$$
0< ...

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102 views

### Reference: hitting time of Gaussian process

Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by
$$
Y_t = y+\int_0^t X_s ds + W_t,
$$
for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation ...

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64 views

### Wiener isometry for semimartingales

Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying
$$
\mathbb{E}\left[
\int_0^{\...

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59 views

### Mean field games approximate Nash equilibria

I am learning mean field games (MFG) through the notes by Cardaliaguet: https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.
I have a question about a step in theorem 3.8 on page 17. Let ...

**4**

votes

**1**answer

89 views

### Triangle inequality for Ito integral?

For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int_0^t f(s) \ ds\right\vert \le \Vert f \Vert_{\infty} \int_0^t \ ds$$
Now if ...

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99 views

### Random process - autocorrelation

I am currently working on random processes.
Let's consider the random process defined as
$u^s(x,t) = 2\sum_{n=1}^{N} \hat{u}^n \cos(\kappa^n\cdot x + \psi_n + \omega_n t )\sigma^n$
where $\hat{u}^n$,...

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**2**answers

73 views

### “Сross сubic variation” of two Brownian motions and interpretation of the simulation result

Consider two independent 1-dimensional Brownian motions $W_{t},B_{t}$, with an equidistant partition of the interval $[0,T]$, and $n\Delta≡T$.
How to calculate the expression below? Can we rewrite ...

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73 views

### The probability distribution of “derivative” of a random variable

Disclaimer: Cross-posted in math.SE.
Let me set the stage;
Consider a stochastic PDE, which has to following form
$$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$
where $H$ is a deterministic function, ...

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41 views

### Uniqueness of martingale problem for Levy type operator

Consider the following Levy type operator:
$$
L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d),
$$
...

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votes

**1**answer

206 views

### An application of Itô's formula to an SDE on a Lie group

I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...

**0**

votes

**1**answer

149 views

### Stochastic Calculus vs Stochastic Processes in Finance [closed]

I'm a second year student, interested in financial mathematics, who's trying to plan out his degree path currently. There's a stochastic processes unit offered in year 3 and a stochastic calculus unit ...

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51 views

### Distances between up and down crosses in Gaussian Processes

Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...

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73 views

### Sufficient conditions for taking limits in stochastic partial differential problems

Let's say we have a Cauchy problem:
$$
(1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t)=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$
$$(2) \hspace{0.5cm} u(x,0)=u_0(x),
$$
where $x \in ...

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42 views

### Maximal inequalities of stochastic integral with respect to Poisson random measures

Let $\Phi$ be a smooth convex function such that $\Psi(x)/|x|\to\infty$ as $|x|\to\infty$, and $\hat N(d z,ds)$ be a compensated Poisson measure on $R^d\times R_+$. Do we have the following inequality:...