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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15 views

Is optimal stopping problem a special case of stochastic optimal control problem?

Let's say all the processes are Markov. Is optimal stopping problem a special case of stochastic optimal control problem, in the sense that, to choose an admissible stopping time in an optimal ...
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57 views

Numerical Solution of SDEs

I want to find a sample path of the following stochastic process: $$dx(t)=f(x(t))dt+g(x(t))dB(t)$$ where $B$ is the Brownian motion. Let $x_0$ be an initial condition. Can I discrete the process as ...
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1answer
17 views

Subordinator - conditions for the Levy triple

different books write Levy Khinchnin's formula in different ways and thus I have a problem with understanding the conditions that Levy's triple must satisfy for the process to be a subordinator. In my ...
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Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process

We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
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33 views

Deriving Euler discretisation of Vasicek model parameters

The Euler discretisation of Vasicek model is given with: $$ \Delta r = \alpha (\beta - r ) \Delta t + \sigma \epsilon_t {\sqrt \Delta t} $$ (see page 91 here) As I understand and looking at others (...
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Sample paths of a diffusion are almost surely non-constant?

Consider a scalar diffusion $X=(X_t)_{t\geq 0}$ given by $\mathrm{d}X_t = b(t,X_t)\mathrm{d}t + \sigma(t,X_t)\mathrm{d}B_t, \quad X_0 = \xi$ for sufficiently regular coefficients $b$ and $\sigma$ and ...
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A question on a degenerate SDE

This may be a basic question of stochastic differential equation. Let $U$ be the closed unit disk $U=\{x \in \mathbb{R}^2 \mid |x|\le 1\}$, where $|\cdot|$ is the standard Euclidean norm on $\mathbb{R}...
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Levy processes - infinitely divisible distribution

I am reading the following text but can't understand the last sentence (source: Andreas E. Kyprianou "Fluctuations of Levy Processes with Applications"): From the definition of a Levy ...
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79 views

Defining stochastic process at $t=0$

I have the following process: $$ Z_t=\frac{1}{t}\Big(\frac{X^2_t}{\sigma^2_X}-\frac{Y^2_t}{\sigma^2_Y})+2 \Big(\frac{\mu_X}{\sigma^2_X}X_t-\frac{\mu_Y}{\sigma^2_Y}Y_t\Big)+\Big(\frac{\mu^2_X}{\sigma^...
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28 views

Continuous stochastic process and symmetric differences.

Consider a real stochastic process $X=(X_t)_{t\geq 0}$ with continuous sample paths on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ together with an open set $U\subseteq\mathbb{R}$. Setting ...
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48 views

Autocorrelation for an Ornstein-Uhlenbeck Process

For an Ornstein-Uhlenbeck process of the form \begin{equation} dx=a(t)x(t)dt+N(t)dW, \end{equation} Suppose $y(t)=x(t+\Delta t)$. I am trying to calculate $\mathbb{E}[xy]$ using Ito's lemma: $$d[xy]=...
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46 views

Partial derivative of function of correlated Brownian motions.

I am facing the following problem. Provided with the following two processes existing under an arbitrary measure $$dX_1(t) = \sigma(X_1(t))dW_1(t),$$ $$dX_2(t) = \sigma(X_2(t))dW_2(t),$$ with $\sigma$ ...
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1answer
49 views

Limit in L^2-Sense (brownian motion)

Let B be a standard Brownian Motion. Define as random variables $$L(t,\epsilon) = \frac{1}{\epsilon}\int_0^t 1_{B_s \in ]-\epsilon, \epsilon[} ds.$$ Prove that $$\exists \lim_{\epsilon \downarrow 0} L(...
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Derivate of a function of the Brownian Motion times dt w.r.t t

I have the function $F(t,W_t) = f(t)W_t$. And by Itô’s formula $dF(t,W_t) = F'_t(t,W_t)dt + F'_W(t, W_t)dW_t + 1/2F''_{WW}(t,W_t)dt$ I try to understand why $dF(t,W_t)$ is equal to: $f'(t)W_tdt + f(t)...
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33 views

Indistinguishability of jump size processes of stochastic integrals

I'm reading "Stochastic integration and differential equations" by Protter, and in the second chapter part 5 he claims that $\Delta(H\cdot X)_s$ is indistinguishable from $H_s(\Delta X_s)$. ...
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63 views

hello everyone please [closed]

who can explain the statement {contineouse image of a connected set} and give me the difinition and theorem and example about this, sorry for poor english.
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Is the book Stochastic Processes and Calculus from Hassler a good starting point?

I'm doing a PhD in economics, have done some courses in calculus, linear algebra, statistics and econometrics, so I have some understanding on the subject, but I never used a book to study ...
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1answer
44 views

Help with substitution

I am having troubles with the following transformation. I have: \begin{align*} Y_t^k=\int_0^t(t-s)^{\alpha-1}h(s)ds+\int_0^t (t-s)^{\alpha-1}\int_0^s(s-u)^{-\alpha}\sigma_{n_k}(X_u^{n_k})dB_u^{n_k}ds \...
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18 views

Application of optional stopping theorem on Stochastic integral with respect to Brownian motion from Ikeda Watanabe

This is a proposition from Chapter 2 of Ikeda Watanabe's Stochastic Differential Equations. Here, $I(\Phi)$ is the stochastic integral with respect to Brownian motion, for $\Phi$ that is measurable ...
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$\Phi'(t,\omega)= \limsup_{h \downarrow 0} \frac{1}{h} \int_{t-h}^t \Phi(s,\omega)ds$ is a predictable process equivalent to $\Phi$.

This is a statement I found from Ikeda Watanabe's Stochastic Differential Equations on Section 2.1. Let $\mathscr{L}_2$ be the space of all real measurable processes $\Phi = \{\Phi(t,\omega)\}_{t\ge 0}...
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What is the difference between stochastic process and random variable?

I am having a hard time grasping the core difference between a random variable and a stochastic process. A random variable assigns a number to every outcome of an experiment. A random process assigns ...
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Simulation of Cox Ingersoll process with general parameters

I am new to the maths exchange. My question is related to the simulation of the Cox-Ingersoll square root process for finance purposes. I intend to perform an order 2 discretization of the SDE to ...
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Filtration generated by diffusion processes on manifolds

This is a question that bothered me for a long time. For more than in one occasion I thought I found a satisfactory answer just to realize after few months I was wrong. Consider a compact Riemannian ...
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1answer
67 views

$ X_s^{0,\xi}(\omega) + \int_s^t b(r,X_r^{0,\xi})dr + \int_s^t \sigma(r,X_r^{0,\xi})dB_r(\omega)$ is $\sigma(B_r - B_s:r\ge s)$-measurable

I am reading the proof that the unique solution of the SDE is a Markov process from René Schilling's Brownian Motion. In the below proof, I have one question. When we define a functional $\Phi$ so ...
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26 views

Stationary distributions of diffusion $dX(t) = -\nabla U(X(t))dt + \sqrt{2}dB_t^d$, when $U$ is $C^1(R^d \to R)$ and $\nabla U$ is Lipschitz

Let $U:\mathbb R^d \to \mathbb R$ be a continuously differentiable function such that $Z_U:= \int_{\mathbb R^d}e^{-U(x)}dx < \infty$ and consider the Gibbs distribution $\pi(dx) := Z_U^{-1}e^{-U(x)}...
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61 views

Solve SDE Logistic Growth

Consider logistic growth with multiplicative noise, that is, the SDE $$dX_t = (\lambda X_t-X_t^2)dt + \sigma X_t dB_t$$, where $\lambda > 0$ and $\sigma > 0$ are fixed constants. How do I find ...
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Filtration Generated by Stochastic Process: Exercise 3.4.10 in Cohen-Elliott Stochastic Calculus

This is my first question here at the community, so I thank in advance anyone who is reading and, possibly, answering it. Recently, I have been doing some exercises from the book of Elliott and Cohen &...
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26 views

Change of measure

Let $(B_t)_{0 \leq t \leq 1} $ be a standard Brownian Motion under the probability measure $\mathbb{P}$. Let $\mathbb{Q}$ a probability measure with density $\frac{d\mathbb{Q} }{d\mathbb{P}} = e^{B_1-\...
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29 views

Integral from normal variable.

Supposoe we have normal variable $ N(\mu_t, \sigma_t^2 ) $. Let's define X = $\int_0^T N(\mu_t, \sigma_t^2 ) dt$. Is X also a normal variable? It's simple to check that expectation of X is simply $\...
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117 views

Is it true that if $P(\int_0^T f^2(s) ds<\infty)=1$ then the exponential defines a density?

Let $f(t)$ be a progressively measurable process wrt Brownian motion $B(t)$ so that $$P\left(\int_0^Tf^2(s)ds<\infty\right)=1$$ Is it true then that the exponential $$\exp\left(\int_0^T f(s)dB(s)-\...
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51 views

Solve Langevin equation with blue noise?

I would like to solve the following Langevin equation $$\frac{d^2 x}{d t^2}+\omega_0^2x(t)=\eta(t),$$ where $\eta(t)$ is a blue noise signal given by $$\eta(t)=\int_{-\infty}^\infty \hat{\eta}(f)\exp(...
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1answer
30 views

What does $\sigma(v(t))$ in a Langevin equation refer to?

I am having a problem with this form of Langevin equation: $m d v(t)=-k v(t) d t+\sigma(v(t)) d W(t) \quad \cdots (1)$ where: $v(t)$ velocity of the particle $m$ mass of the particle $k$ coefficient ...
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33 views

Find the $\mathbb{E}[ W_{t} ^ {3} e^{W_1-\frac{1}{2}}]$

i am trying to compute the $$\mathbb{E}[ W_{t} ^ {3} e^{W_1-\frac{1}{2}}]$$ where $(W_t)_{t \geq 0}$ is a standard Brownian Motion. Any idea would be appreciated, thanks in advance !
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53 views

left-continuity / right-continuity of two processes with cycling definition.

Let's fix two processes, $\lambda$ and $N$. $N$ is defined as a point process (with some fancy conditions that are not important here. It shall be right continuity as processes are usually. On the ...
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1answer
27 views

Equivalent of spectral density function related to fBm: $\sum_{k\in\mathbb{Z}}\vert \lambda+2k\pi\vert^{-2H-1}$

Let $H\in (0,1/2)\cup (1/2,1)$. For $\lambda\in [-\pi,\pi]$ let $$f_H(\lambda)=\vert e^{i\lambda}-1\vert^2 \sum_{k\in\mathbb{Z}}\vert \lambda+2k\pi\vert^{-2H-1}.$$ I am reading a book that say "...
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mixing fractional Brownian motions

[first posted here] Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by $$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$ where $W_t^{2}$ and $Z_t$ are independent of each other. My ...
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20 views

Stochastic Differential Equation for Tensor Product of Diffusion Processes

Let $y$ and $z$ be given by the equations \begin{equation} \mathrm{d}y_t = A(y_t) \mathrm{d}t + B(y_t) \mathrm{d}W^1_t, \end{equation} and \begin{equation} \mathrm{d}z_t = C(z_t) \mathrm{d}t + D(z_t) \...
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55 views

Solution of SDE [closed]

I would like to find the solution of the following SDE: $$dX(t)=(X(t)+a)\sigma(t)dW(t)$$ with $X(0)=X_0$ and with $X_0$, $a$ are real numbers and dW is the classical Browninan motion. I tried using ...
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The integrand of Levy-Ito decomposition

Suppose that $W(t)$ is a Brownian Motion, $N(ds,dz)$ is a Possion random measure with intensity $ds\mu(dz)$,$\tilde{N}(ds,dz)$ is the compensated measure. Shall we call $$X(t)=W(t)+\int_0^t\int_0^{1}(...
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1answer
22 views

Reference request : existence and uniqueness of solution to a certain class of SPDE

Is there any papers/reference for the existence and uniqueness of the following type of Stochastic Partial Differential Equations (perhaps a much larger class of SPDES containing the following) $$ dv(...
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1answer
41 views

Why time derivative of random process always zero?

It seems that whenever I am applying Ito's Lemma and am doing the time derivative bit it always turns out $$\frac{d}{dt}V=0$$ Take an example from a book $$\frac{d}{dt}\left(\frac{B^{2}}{2}\right) = 0$...
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80 views

$\operatorname{cov}\left(\int_0^tf(s,T_1)\,dW(s,T_1),\int_0^tf(s,T_2)\,dW(s,T_1)\right)=\text{?}$

Define a random field $W$ so that: $W(t,T)$ is a standard Brownian motion for every $T$. $dW(t,T_1) \, dW(t,T_2)=c(t,T_1,T_2)\,dt$ $W(t,\cdot)$ is a a continuous function for every $t$. Suppose $f$ ...
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1answer
35 views

show $E\int^\infty_0e^{-A_t}dA_t$ is bounded above

show $E\int^\infty_0e^{-A_t}dA_t$ is bounded above by constant, where $A_t$ is an increasing stochastic process of locally integrable variation starting from 0. $A$ is possible to be purely ...
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12 views

Boundary Conditions for Fokker Plank (Kolmogorov Forward Equation)

I have a stochastic process given by $$dX(t)=X(1-X)dtv+X(1-X)\sigma dW(t)$$ Once X reaches $0$ or $1$, the process stays there for ever. (the drift and diffusion terms goes to zero). I would have ...
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0answers
33 views

Why $\mathcal{M}^p([0,T],R^d)$ is a Banach Space?

Given a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq0},P)$. Let $1\leq p < \infty, $ a process $x(t)$ $\in \mathcal{L}^p([0,T],R^d)$ $\Leftrightarrow$ $\{ x(t), t \in [0,...
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0answers
53 views

Ito integral of a power of the stock price

An investor investing in a stock $S_t$, continually rebalances his portfolio, such that at any time $t$, the number of shares he holds is proportional to $ S_t^r$. Here $r$ is a constant real number. ...
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0answers
38 views

Radon–Nikodym derivative of two diffusion process

Let's consider two diffusion process $$dX_t = b(X_t) \, dt+\sigma(X_t) \, dW_t $$ and $$dX_t = c(X_t) \, dt+\theta(X_t) \, dW_t $$ with different drifts and diffusion coefficients. I would like to ...
3
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1answer
73 views

Ito's formula with $\log X_t$

I want to use Ito's formula on the following SDE: $$ dX_t= - \gamma (\log X_t - \theta) X_t d t + \sigma X_t d W_t $$ to obtain an expression for $ \log X_T $ where $T > t $ is some fixed time. ...
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1answer
28 views

Inverse Bessel Process as continuous local martingale

Let $B$ be a $n$-dimensional brownian motion. This question shows, that $$\Big(\sum_{i=1}^n\int_0^t\frac{B^i_s}{||Bs||}dB^i_s,||B_t||\Big)$$ is a weak solution of $dX=\frac{n-1}{2X}dt+dW$. Now I would ...
2
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1answer
60 views

Differentiating Laplace transform of $dX=cXdt+\sqrt{X}dW$

Consider a solution $X$ of the stochastic differential equation $$dX=cXdt+\sqrt{X}dW$$ For $\mathcal{L}(\alpha,t)=E[\exp(-\alpha X_t)]$ I want to show, that $$\frac{d}{dt}\mathcal{L}(\alpha,t)=(c\...

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