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Questions tagged [stochastic-calculus]

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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Solution of SDE with additive noise

I have a question about stochastic differential equations with additive noise. My question is: Is the solution of a SDE with additive noise almost surely equal to the solution of the corresponding ...
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11 views

Multi-Dimensonal Burkholder Davis Gundy ineuqality

Let $A_t$ be a stochastic process of symmetric positive definite $n \times n$ matrices and let $B_t$ be the standard $n$-dimensional Brownian motion. Define the martingale $X_t =\int\limits_0^...
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Itô-Doeblin lemma for non-continuous semimartingales

On wikipedia there are some results on Itô's lemma applied to non-continuous semimartingales (sometimes called Itô-Doeblin lemma) I have looked up the books by Malliaris, Oksendal and Doob, but they ...
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32 views

Why is the set of continous paths of a browian motion not measurable?

Øksendal states in his book "stochastic differential equations" (Defintion 2.2.1 iii)) that the set $H = \{\ \omega \mid t → B_t (\omega)\ \text{is continuous}\ \}$ is not measurable with respect ...
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Construction of Independent Brownian Motions

Let $(\Omega, \mathcal{F}, \mathbb{F} = (\mathcal{F}_t)_{t \in [0,T] },P) $ be a filtered probability space. Can we construct for all $n\in \mathbb{N}$ a family $ \{B^1, \dots, B^n \} $ of ...
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Is a vector of Brownian motions a Brownian motion? [on hold]

Let $B$ be a Brownian motion. Is the $\mathbb{R}^n$-valued process $M = (B, \dots, B)$ a Brownian motion or a Levy process?
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19 views

Stochastic differential equation drive by dependent Brownian Motions

Let $b : [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n, \sigma : [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^{n \times m}.$ Consider an SDE $$ dX_t = b(t,X_t)dt + \sigma(t,X_t) dW_t$$ Then ...
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covariance of the integral of brownian motion over disjoint intervals

Let $W_i = \int_{t_{i-1}}^{t_i} B(t) \,dt$ be the integral of Brownian motion over the time $t \in [t_{i-1},t_i]$. I'm reading a paper which says that $W_i$ and $W_j$ are independent Gaussian random ...
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Why is $P^{\mu}[B_0 \in \Gamma]=\mu(\Gamma)$

Why is $P^{\mu}[B_0 \in \Gamma]=\mu(\Gamma)$ where $P^{\mu}(F)=\int_{\mathbb{R}^d}P^{x}(F)\mu(dx) $ and $P^x$ is defined as $$ P^x(F)=P^0(F-x) $$ and $F-x=\{\omega \in C[0,\infty)^d:\omega(.)+x \in F\}...
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Equivalent formulations of stochastic HJB equation

I have some trouble understanding stochastic HJB equations. There are basically two forms of this equation that I have encountered in books, lecture notes etc... (one-dimensional case) 1) $rv(x)=\pi (...
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Convergence in $L^2( [0,T] \times \Omega )$ implies uniform convergence

Let $X^n$ be a family of continuous stochastic processes such that $E[ \sup_{t \in [0,T} \mid X^n_t \mid ^2 ] < \infty $ for all $n.$ We assume that $$ \lim_{n \to \infty} E\left[ \int_0^T \mid X^...
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Time change and stochastic differentials

If $X$ is a $C$-continuous semi-martingale then $$ \int_0^{C_t} H_s\, dX_s = \int_0^t H_{C_s}\,dX_{C_s}. $$ As far as I'm aware, this and its consequences are the only relation between stochastic ...
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Let $W=\{W_t:t\geq0\}$ be a Brownian motion. Find $\operatorname{Var}(W_1^3)$

sorry for my newbie question. I'm learning stochastic process on my own and I got confused about a lot of concepts and notations. I need your help with this small exercise: Let $W=\{W_t:t\geq0\}$ be ...
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33 views

Paradox Brownian Motion : P(first passage time < infinite) = 1 yet E(first passage time) = infinite?

I am studying the bible of stochastic calculus for finance by Shreve aka God. But in the section "first passage time to level m" for the Brownian Motion there is a paradox : 1) P(first passage time ...
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Integration of the Jump part of SDE

Let $L_t^{\alpha}$ be the alpha stable Levy motion, I am not sure how to compute the integral $\int_{t_{i-1}}^{t_i} dL_t^{\alpha}$.
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A question about American option

I was reading Stochastic Calculus for Finance I written by Shreve. In the chapter 5.2 value of portfolio hedging an American option. He used the word "comsumed" to describe his portfolio $X$. Here is ...
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1answer
15 views

Malliavin derivative of a gaussian

Let $W$ be an $H$-isonormal Gaussian process and $H$ is a real separable Hilbert Space. Set $$X=f\big(W(h_1),\ldots, W(h_n)\big) $$ for $f$ an infinite differentiable with their partial derivatives ...
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Dealing with different definitions of the Ornstein–Uhlenbeck process

I've run up against a wall in reconciling two different definitions of the Ornstein–Uhlenbeck process, and would appreciate some help. On the one hand, as discussed here, we can define an Ornstein–...
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1answer
20 views

brownian motion - covariance in two independent brownian motions

$\text {Let } W \text { and } \widetilde W \text { be two independent Brownian motion and } \rho \text { is a constant } \in (0,1).$ $\text {For all } t \geq 0 , \text { let } X _ { t } = \rho W _ { t ...
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1answer
69 views

Distribution of random variable given by SDE at some point in time

Suppose we have an stochastic differential equation given by, $$\mathrm{d}X = N(X,t)\,\mathrm{d}t + M(X,t)\,\mathrm{d}B,$$ where $B$ is a brownian motion. As far as I understand we can think of this ...
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Integration by parts on Kushner equation

I read the filtering theory chapter of A Tutorial Introduction to Stochastic Analysis and Its Application (I.Karatzas), where it says, we have the Kushner equation $d\pi_t(f)=d\pi_t(\mathcal{A}f)dt+(\...
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1answer
40 views

Calculate a multiple Ito integral

Consider the following multiple Ito integral, $$I(n)=\int\limits_0^t \int\limits_0^{t_n} \int\limits_0^{t_{n-1}} \dots \int\limits_0^{t_2}dW_{t_1} \, dW_{t_2} \, \dots \, dW_{t_n},$$ where $W_t$ is ...
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Solution of the differential equation $\ddot{x}(t)+\sin(\omega t)x(t)=cos[\eta(t)]$

The differential equation: $$\ddot{x}(t)+\sin(\omega t)x(t)=cos(\eta t)$$ has an analytical solution involving Mathieu functions. This is valid if both $\omega$ and $\eta$ are constant. Suppose the ...
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2answers
57 views

brownian motion - expected value

Let $B$ be an brownian motion and let $s \leq t$. Compute $\mathrm { E } \left[ B _ { s } B _ { t } ^ { 2 } \right]$. I know, that the answer is $0$, but I can't see how this ends being $0$. My ...
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17 views

Time to first jump for doubly stochastic poisson process (cox process)

Let $N_t$ denote a doubly stochastic Poisson (Cox) process with intensity process $\lambda_t$. Lando (1998) defines the time to first jump $\tau$ as: $$\tau = \inf\{ t: \int_{0}^{t} \lambda_u du \geq ...
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convolution of two maximums

This is a problem that has applications in auction theory. Consider convolution of two random variables in the following form: Z1 = max(X1,X2) + Y3, Z2 = max(Y1,Y2) + X3, where X1, X2, X3 are i.i....
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Joint law of supremum and infimum of a Levy process with symmetric distribution?

I have a stochastic process $X$ at hand, which enjoys some nice properties: it is a Levy Gaussian process with symmetric distribution. My question concerns whether there is a formula for the law of $\...
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$X\geq0, Y\geq0$ independent; $Z = X - Y$. Is $E[X|Z\in[\underline{z}, \overline{z}]] > E[X|Z = \overline{z}]$ possible?

Let $X\geq0, Y\geq0$ be two independent, real-valued scalar random variables. Let $Z = X - Y$. Does there exists an interval $[\underline{z},\overline{z}]$ such that $E[X | Z \in [\underline{z},\...
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How to solve this stochastic differential equation, $dx_t=\mu dt+(\sigma-\bar{\sigma}x_t)dW_t$?

I have this stochastic differential equation which I would like to solve: $$ dx_t=\mu dt+(\sigma-\bar{\sigma}x_t)dW_t $$ It is similar to the mean-reverting process, but variance-reverting kind. No ...
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1answer
25 views

Convergence of “AA” Subsequence of sequence strongly convergent of order $\alpha$

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and consider a sequence $X = (X_n)_{n\in \mathbb N}$ of integrable random variables, let $X_\infty\in L_1$ and $\alpha > 0$. Say that $...
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Calculate the dynamics of ZCB given forward-rate dynamics

Suppose the forward rate is given by: $$df(t,u)= \frac{\partial}{\partial u} \bigg( \frac{\sigma(t,u)^2}{2} \bigg)dt - \frac{\partial}{\partial u} \big( \sigma(t,u) \big)dW_t$$ where $W_t$ is a ...
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What mathematical prerequisites do I need to properly learn stochastic DEs?

I'm a PhD student in biology, and I'm trying to teach myself the necessary math to get into mathematical modeling of the phenomenon we study, because this is an area of inquiry that 1) I'd like to get ...
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1answer
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Expectation over mixing distribution

I have a following expression for expectation of some random variable $X$: $E(X(t)|\lambda)=\frac{1}{p}(1-e^{-\lambda pt}) \ $ We assume that $p$ is fixed and $\lambda$ has a following mixing gamma ...
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1answer
56 views

Show that the stochastic exponential is a true martingale

Let $W = \{W_t : t\ge0\}$ be a standard Brownian motion on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, and let $f$ be a deterministic function such that $$ \int_0^tf^2(s)\,ds<\infty $$ ...
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Is there any reference of application tensor norm on signatures?

This questions is related to https://en.wikipedia.org/wiki/Rough_path The signatures deriving from the path is used as features in machine learning areas. As I understand signature is equipped with a ...
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21 views

Distribution of Stopped Wiener Process with Stochastic Volatility

Let $(W_s)_{s \geq 0}$ be a Wiener process and $\tau$ be a random variable with an exponential distribution with parameter $\lambda$. Suppose that $W$ and $\tau$ are independent. In this question, we ...
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1answer
32 views

Predictable graph of a random set

Assume that we work on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with a filtration $\mathcal{F}_t, t\ge 0$ and stochastic basis $(\Omega, \mathcal{F}, \mathbb{P}, \mathbf{F})$ ...
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1answer
54 views

Prove that $\forall j,k \in \{1, …, N\}, \ \forall {n \in \mathbb N}, p^n_{j,k}=\langle T^n e_k,e_j \rangle$.

Let $(\Omega, \mathcal F,\mathbb P)$ be the probability space. Let $N \in \mathbb{N^*}$ and $(X_n)_{n \in \mathbb N}$ be a sequence of random variables with values in $\{1, ..., N\}$. Let $\mathcal ...
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64 views

Preservation of martingale property under absolutely continuous monotone time change

Suppose I have a martingale $M(t)$ adapted to some filtration $\mathbb{F}=\{\mathcal{F}_t\}_{t\geq 0}$ and a positive monotonically increasing time change $T(t)=\int_0^tv(s)\mathrm{d}s$ with an $\...
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2answers
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How can variance and mean of brownian motion be stated

I have seen in the construction of brownian motion that the Its mean is 0 and variance t. i.e $E(B(t))=0$ , $Var(B(t))=t$ Why in a problem sheet does it state $B_{j}(t)$ for $j=1,...,n$ are brownian ...
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1answer
40 views

Finding SDE for stochastic process

Let's assume that there is process $Z_t$: $$Z_t = \exp-(\sigma(T-t)W_t+\sigma \int_0^t W_sds + \int_0^Tf(0,u)du + \int_0 ^t\int_s^T\alpha(s,u)duds) $$ where $f(t,T)$ means instantaneous forward rate, ...
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34 views

Leibniz integral rule in SDE

My question refers to the last part of solution of this problem (I couldn't add a comment there due to lack of reputation): Dynamics of short rate in HJM I'm not sure I understand fully the last ...
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1answer
37 views

Expectation of the absolute of the difference between two B.M, $\DeclareMathOperator*{\E}{\mathbb{E}}|B(s)-B(t)|=\sqrt{\frac{2}{\pi}}|t-s|^{1/2}$?

I am going through Bernt Oksendal's (S.D.E an introduction with application sixth edition) book, I can do most of the questions without any issue, but I don't understand how (using my understanding of ...
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46 views

Gaussian process properties

I am reading Gaussian Processes (GP) for Machine Learning (http://www.gaussianprocess.org/gpml/). The GP definition is usually like this: "A Gaussian process is a collection of random variables, any ...
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1answer
33 views

Show that an Ito process is a martingale.

I wonder if someone can help me with this problem: Let $X_t= \exp\{ -\frac{1}{4c}(1-e^{-2ct}) + \int_{0}^t e^{-cs}dW_s\}$ where $W_s$ is a standard Brownian motion and $c>0$. I'm trying to show ...
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22 views

Discretization of stochastic differential equations for Kalman filter

I have some questions concerning the discretization of a stochastic differential equation (for the application of a Kalman filter). My starting point is \begin{equation*} \dot x(t) = Ax(t) + \nu(t), \...
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1answer
56 views

solving an optimal stopping problem

I am currently going through problems in Oksendal's intro SDE and stuck with this problem. I was wondering if I could get some help with it. I would sincerely appreciate if you would express out all ...
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13 views

A progressively measurable process.

Let $\mathcal{P}_2(\mathbb{R})$ be the space of measures on $\mathbb{R}$ with finite second moment. We equip this space with the Wasserstein metric $W^2.$ We recall tat this metric space is separable ...
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129 views

Equality in law of two stochastic processes

Let $\lambda(t)$ be a CIR process, i.e. the strong solution of the SDE $$ \mathrm{d}\lambda(t)=\kappa(\theta-\lambda)\mathrm{d}t+\sigma\sqrt{\lambda(t)}\mathrm{d}W^1(t) $$ The integrated CIR is ...
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1answer
21 views

Is the cross variation (of stochastic processes) bilinear?

I have not been able to find whether for three stochastic processes (adapted with respect to one constant filtration): $(X_t)_{t\geq 0}$, $(Y_t)_{t\geq 0}$ and $(Z_t)_{t\geq 0}$ we have the following ...