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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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What mean $\mathbb P^x(X_t\in A)$ if $dX_t=\mu (X_t)dt+\sigma (X_t)dB_t$, $X_s=x$?

I know that $$dX_t=\mu(X_t)dt+\sigma (X_t)dB_t,\quad X_s=x$$ means $$\mathbb P\left(X_t=X_s+\int_s^t \mu(X_u)du+\int_s^t\sigma (X_u)dB_u\right)=1.$$ I denote such a process $X_t:=X^{s,x}_t$ Now, I'm ...
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Is a vector of independent Brownian motions a multivariate Brownian motion?

Given a filtered probability space $(\Omega, \mathcal{F}, \mathcal{F}_{t\geq 0}, P)$: If $B_1, B_2, \dots, B_m $ are all real $\mathcal{F}_t$ Brownian motions, jointly independent. Is the resulting ...
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Continuous function as difference of convex functions

Can every continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ be written as the difference of two convex functions? If not, can every twice continuously differentiable function $f:\mathbb{R} \...
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Motivation of Milstein scheme

Milstein scheme is motivated by improving the convergence speed sigma terms of Euler scheme. where sigma and mu is globally Lipschitz continuous$$t_i=\frac{T}{n}i\quad dX_t=\mu(X_t)dt+\sigma(X_t)dW_t$$...
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Meaning of $\int _0 ^T X_t dt$ when $(X_t)_t$ is a process

I am studying stochastic calculus (Ito integrals, to be precise) , and I am not sure if I got some things right. For instance, we have defined $\Lambda_B ^2 (a,b)$ as the space of progressively ...
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15 views

M/M/1 queue with two types of customers, distribution of the total number of customers

I have problems to derive the following: "A gas station offers two services. For each service customers arrive according to a Poisson process. On average 20 customers per hour for service 1 and 5 ...
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28 views

Stopped uniformly integrable process in discrete time is uniformly integrable?

I'm studying for work the book: "Stochastic Calculus and Application" by Choen and Elliot 2 ed. In section $4.2$ (pg. 91) it states the discrete version of the Optional Stopping Theorem for bounded ...
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24 views

Why does calculating the quadratic variation of a Brownian motion in this way not work?

This seems like a simple question, but I am stumped. I know the proofs for quadratic variation and cross variation, etc..., but for some reason can't understand why the following doesn't make sense to ...
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21 views

stochastically independence

I never took stochastic courses and need a proof for this task to continue my work at another problem. Can somebody help me out? Let $X$ and $Y$ be stochastically independent real discrete random ...
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38 views

Fokker-Planck equation

I'm struggling to proof the Fokker-Planck equation. Let $b:[0,T]\times \mathbb{R}^N\to\mathbb{R}^N$ and $\sigma:[0,T]\times \mathbb{R}^N\to\mathbb{R}^{N\times d}$ two measurable functions. Let $X=\...
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How can I show that $E[\sup_{0 \le t \le T} N_t^2]<\infty$ for all $T>0$

Given a Stochastic differential equation $dN_t=\sqrt{2\mu N}dW_t$ starting with a deterministic initial value $N_O$. How can I show that $E[\sup_{0 \le t \le T} N_t^2]<\infty$ for all $T>0$? I ...
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Using Girsanov Theorem Backwards?/ Obtaining Radon-Nikodym Derivative

On page 112/133 of Den Hollanders book on Large Deviations he wants to calculate the R.N derivative between two path measures : one is the path measure of the solution to an SDE $dX_t=H(X_t)dt+dW_t$ ...
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Paper of Jacques Azéma [closed]

I am currently looking for J. Azéma's works translated in English such as Quelques applications de la théorie générale des processus I. Kindly leave a link if you have one. Thank you! P.S. I do not ...
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25 views

Process of sum of flip coin is not uniformly integrable?

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $\left(Z_{n}\right)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables with $\mathbb{P}\left(Z_{n}=1\right)=\mathbb{P}\left(...
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Simulation of SDE

I need to write down a code to simulate an SDE of the following type: $$ dX_t = - ( \eta_t X_t + \chi_t ) dt + \sigma dW_t, \ X_0=v_0 \quad t \in [0,T] $$ where $\eta_t$ is a deterministic function, ...
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Equality of the stochastic integral under two probability measures

This questions is very short. Under the Girsanov Theorem assumptions we have two equivalent probability measures $\mathbb P$ and $\mathbb Q$ and a measurable space $(\Omega,\mathcal F)$, right? We ...
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1answer
43 views

Stochastic Question: $d \int B_s ds = ?$ [closed]

Stochastic Question: $d \int_0^t B_s ds = ?$ $B_s$ is the standard Brownian motion at time $s$. This is an Ito integral. Operator $d$ is defined in the standard Ito sense. For those who understands ...
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38 views

Find the compenstor of the standard ito integral

Let $B_t$ be a Brownian motion and let $\{\mathcal F_t : a<t<b\}$ be a filtration such that for each $t$ we have that $B_t$ is $\mathcal F_t$ measurable and for and $s<t$, the random variable ...
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38 views

Differential/derivative of time integral of a stochastic process, where the stochastic process depends on upper limit

For a standard Wiener Process/Brownian Motion, $W$, for the usual integrals $\int_0^t\sigma(u)dW(u)$ and $\int_0^tW(u)du$, I know how to manipulate them using Ito's Lemma/normal calculus rues like ...
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1answer
40 views

Gaussianity of a stochastic process

I am given the process $X_t = B_t -\int_0^t \frac{B_u}{u}du$ How can I show that it is gaussian, given a standard continuos Brownian motion $B$? As I know that $sB_{1/s} \rightarrow 0$ as $ s \...
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Bootstrap-Resampling: Why are the Bootstrap-Resamples independent again?

For the simple Bootstrap method, we consider n i.i.d. random variables with common distribution function F, which is estimated by ^F (parametric or non parametric). For the procedure we then generate ...
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1answer
49 views

Cox-Ingersoll-Ross Model

CIR model foresee (on the basis of structure of similar model) the following system: $\left\{\begin{matrix} \dot A(t,T)-a \gamma B(t,T)=0, A(T,T)=0\\ \dot B(t,T)-aB(t,T)-\frac{\sigma^2}{2}(B(t,T)...
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Interpolating process that follows geometric Brownian motion

So I have a set of time series data that follows (at least assumed) GBM but there are missing data. To interpolate, I'm thinking about simulation and create some sample paths. I have two options in ...
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1answer
47 views

Deriving an equation and boundary condition from an SDE

Let $\dot{X_{t}} = b(X_{t}) + \sigma(X_{t})\dot{W_{t}}$ be a stochastic differential equation, where $W_{t}$ is a Wiener process. Also, let $X_{0} = x \in \mathbb{R}$. Define $$u(x) = \mathbb{...
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probability density of peaks in stationary stochastic process [closed]

Suppose that a displacement time series of $x (t)$ recorded at a point in ground surface due to earthquake is a realization of an stationary stochastic process with power spectral density function of $...
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If $X$, $Y$ are continuous semimartingales and we have that $XdY = YdX$ can I conclude that $X = Y$?

If not true in general, are there any (mild) conditions on $X$ and $Y$ under which this is true? Sorry if this has already been asked!
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Hitting time expectation squared for Brownian motion

Let $\dot{X_{t}} = b(X_{t}) + \sigma(X_{t})\dot{W_{t}}$ where $X_{0} = x \in \mathbb{R}$ and $W_{t}$ is a Wiener process. Let $\tau = \min\{t \mid X_{t} \not \in G\}$, where $G = (M, N) \subset \...
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Why the convergence of $\sum_{k=0}^{n-1}B_{t_{i}}(B_{t_{i+1}}-B_{t_i})$ depend on the partition pointwise but not in $L^2$?

Let $(B_t)$ a Brownian motion. Let $0=t_0<t_1<...<t_n=T$ a partition on $[0,T]$. I know that $$\lim_{n\to \infty }\sum_{k=0}^{n-1}B_{t_{i}}(B_{t_{i+1}}-B_{t_i})$$ doesn't depend on the ...
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Uniqueness of Solution to Stochastic Integral Equation

Suppose that $N$ is an $(\mathcal{F}_{t})$-continuous local martingale, with $N_{0}=1$, $N_{t}\gt0$ a.s. for $t\geq0$ and $N$ satisfies: $$ N_{t}=1+\lambda\int_{0}^{t} N_{s}dB_{s} $$ Applying Ito's ...
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GBM within brownian bridge

I noticed that a brownian bridge between say a and b amy start at B0 and B1 where B1 can be greater than B0. I wonder if this path can be a GBM. If so, what’s the SDE like?
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Ito's Differential Problem

Let $W_t$ be the standard Brownian motion. Is the random process a martingale? - $Y_t = exp(\int_0^t sdW_s)$ (Find $dY_t$ using Ito formula in its differential form) Base on what I have learned we ...
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scaling invariance of brownian local time

I am studying Brownian local time processes and several references mentioned the scaling invariance of local time. For example, page 10 of this reference (https://hal.archives-ouvertes.fr/hal-00091335/...
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Continuous Modification of Stochastic Process Indexed with Compact Space

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a (complete) probability space, let $D$ be a compact, separable, metrizable topological space and let $$ S : \Omega \times D \rightarrow \mathbb{R}$$ be a $(\...
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44 views

Finite sequence of random step processes such that $\lim_{n\to\infty}E(\int_{0}^{\infty}|f(t)-f_n(t)|^2dt)=0$ for $f(t)=e^{-t^2/4}$

Let $$f(t)=e^{-t^2/4}, \ \ \ t \ge 0$$ I want to show that $f$ is in $M^2$ where $M^2$ denotes the class of stochastic processes $f(t),t\ge0$ such that $$E\left(\int_0^\infty|f(t)|^2dt\right)&...
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37 views

Density function related to Brownian motion

I am dealing with a question listed below. I am trying to use the running maximum of Brownian motion to deal with the problem, but it does not work out. Let $ \tau_{M}=\inf\{t;W(t)=M\},M>0,$ and ...
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Using Ito's lemma to determine $dY(t)$ when $Y=\sin(t+B_t), \ \ \ t\ge0$

Let $(\Omega, \mathcal F, P)$ be a probability space and $\{B_t\}_{t\ge0}$ a Brownian motion. Furthermore let $\{F_t\}_{t\ge0}$ be the natural filtration of $B$. Let $$Y(t)=\sin(t+B_t), \ \ \ t\ge0$$ ...
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1answer
41 views

When $ t\mapsto X_t(\omega)$ is continuous for almost all omega then does it imply$t \mapsto E[X_t]$ continuous?

When $ t\mapsto X_t(\omega)$ is continuous for almost all omega then does it imply that $t \mapsto E[X_t]$ continuous? If this is true how can go about proving it? If not what condition do I need? ...
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How to show that this is a stopping time [closed]

In the notes Basics of Stochastic Analysis proof of Theorem 2.23 p.62, it is asked to check the following statement. Let $(\mathcal F_t)$ be a stopping time, and let $\tau$ be a stopping time with ...
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Where is the problem in this logic regarding the supremum of the expectation being equal to the expectation of the supremum using control processes

This is just a question I have from stochastic control. I am totally new to analysis/stochastic processes, so I am unsure how controls are used properly/manipulated in expressions, but here's my ...
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1answer
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When $\mathbb E\int_0^T H_s^2ds=\int_0^T \mathbb E[H_s^2]ds$ is not true?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $(\mathcal F_t)_t$ a filtration. In all example I can see, we always have that $$\mathbb E\int_0^T H_s^2ds=\int_0^T \mathbb E[H_s^2]ds,$$ ...
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1answer
41 views

How to judge the solution process of an SDE to lie on the sphere?

Consider the following SDE on $\mathbf R^d$: \begin{equation}\tag{*} dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d, \end{equation} where $W = (W^1,W^2,...
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What is the connection between stochastic and ordinary differential equations?

(Note: I know nothing about stochastic differential equations) Given an ordinary differential equation, $$\dot x = f(x), x_o = x(0)$$ Is there some connection between this ODE and a stochastic ...
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Multinomial setup for Stochastic rounding

I am looking into stochastic rounding problems. The most common way to do a stochastic rounding would be to consider a binary case. Say, A number $x = 0.12$ to be rounded to a binary interval of $[a,...
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1answer
24 views

Independence of Solution of SDE $S^{(x_0, \sigma, \mu)}_t$ of Initial Information $\mathcal{G}_0$

Question Consider the following stochastic differential equation, given as an equivalent stochastic integral equation, where the multidimensional integrals are to be read componentwise: \begin{...
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1answer
29 views

Conditional expectation and the Dirac delta function

I'm looking for a rigorous proof of an identity I came across many times (in the context of Gÿongy's lemma). Suppose $X$ and $Y$ are two r.v. We know that $\mathbb{E}\left[X|Y\right]$ is $\sigma(Y)$-...
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36 views

Dynamics of the differences of an Ito process

I can't totally wrap my head around this (I am sure it is simple). If $$\frac{dX_t}{X_t}=\mu dt+\sigma dB_t$$ where $B$ is a Brownian motion, then what is the dynamics of $$Y_t=X_t-\alpha X_{t-\delta}...
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1answer
30 views

Time homogeneity of Ito diffusion

Consider a time homogeneous Ito diffusion satisfying a SDE, \begin{equation}\label{1} dX_t=b(X_t)dt+\sigma(X_t)dB_t, X_s=x \end{equation} $t\geq s$. The unique solution of the SDE is denoted by $...
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2answers
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Find the quadratic variation process of $\int f(s) \, dB_s$

Let $f \in L^2[a,b]$ and let $\displaystyle M(t)=\int_a^tf(s)dB(s)$. Find the quadratic variation process, $[M]_t$ , of $M(t)$. Here the quadratic variation process is the limit in probability of $\...
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54 views

How can I numerically compute a stochastic integral?

I am trying simulate a to solve a 2-dimensional stochastic process and $Y_t^1$ is a mean-reverting square root process which I simulated on a time grid using its known conditional distribution. I ...
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40 views

Simplifying a differential equation (basic algebra)

Given the CIR process: $$\ dX_t = (a − bX_t ) dt + \sigma dW_t$$ The Milstein scheme is: $$\ X_{i+1} = X_i + \mu \Delta+\sigma\epsilon_i +\frac{1}{2}\sigma\sigma_x(\epsilon_i^2-\Delta)$$ and can be ...