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      1answer
      67 views

      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 ...
      -1
      votes
      1answer
      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< ...
      3
      votes
      0answers
      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 ...
      1
      vote
      0answers
      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^{\...
      0
      votes
      0answers
      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
      1answer
      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 ...
      0
      votes
      0answers
      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 $\hat{u}^n$,...
      2
      votes
      2answers
      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 ...
      0
      votes
      0answers
      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, ...
      1
      vote
      0answers
      41 views

      Uniqueness of martingale problem for Levy type operator

      Consider the following Levy type operator: $$ L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d), $$ ...
      4
      votes
      1answer
      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
      1answer
      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 ...
      1
      vote
      0answers
      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 ...
      2
      votes
      0answers
      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 ...
      0
      votes
      0answers
      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:...

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