Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

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12 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 ...
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1answer
38 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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11 views

SDE with indicator function

I am studying an equation in the form of a mean-revert stochastic differential equation, which suggested for the population model at the gates of the subway station. well known mean revert SDE like $$...
2
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1answer
12 views

Solving an Ornstein-Uhlenbeck style SDE

Solve the SDE $dN=a N \log(\frac MN)dt+\sigma NdB_t,$ where $M,a \in \mathbb{R}$. Apply Ito's formula to $ f = \log(N)$ gives \begin{align*} df &= \frac{\partial f}{\partial N} dN +\frac{1}...
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1answer
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Why is $\int_S^T f dB_t$ (Itô integral) $\mathcal{F}_t$-measurable?

In Oksendal's book, in Theorem 3.2.1, the author states that $\int_S^T f dB_t$ is $\mathcal{F}_T$-measurable. In the proof he says that since it holds for elementary functions, when we take the limits ...
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21 views

Stochastic integral $X_t=\int_0^{\pi} \sin(\lambda t)\,dZ(\lambda)$

I am new to statistics and time series. I was learning about orthogonal increment process. I am given a process $Z_t$ which is defined as $$ Z_t=\begin{cases} W, & \lambda \le U, \\ 0, & \...
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1answer
31 views

Summarize coefficients of $dX=f(X)dt+g_1(X)dW_1+g_2(X)dW_2$

Consider $X$ a solution of $$dX=f(X)dt+g_1(X)dW_1+g_2(X)dW_2$$ whereas $W_1$, $W_2$ are independent standard brownian motions and $f,g_1,g_2:\mathbb{R}\rightarrow\mathbb{R}$ bounded lipschitz-...
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1answer
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Using approximations of approximations to define the Itô integral

In Oksendal's book of Stochastic Differential Equations, the author does a reasoning similar to what's presented below. The follwing three points are proved: I can find $\{\phi_n\}$ such that $E(\...
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1answer
34 views

Unique strong solution of $X_t=t+\int_0^tX_sdW_s$

Let $(W_t)_{t\ge0}$ be a standard brownian motion. Using Ito's lemma, I was able to show that $$X_t:=\exp(W_t-\frac{1}{2}t)\int_0^t\exp(-(W_s-\frac{1}{2}s))ds$$ solves $$X_t=t+\int_0^tX_sdW_s$$ ...
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Joint convergence in distribution based on different Wiener processes - asking for reference

I have two estimators and want to prove their joint asymptotic distribution: First start with an example, that if we have that $$\Omega_1 = \int_{0}^{1} \delta_1(s) (w_1(s)-w_1(1))ds+o_p(1)$$ $$\...
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Is there any relationship between Doob-Meyer Decomposition and Martingale Representation theorem?

Doob-Meyer Decomposition Theorem decomposes a supermartingale into a martingale and an increasing adapted process. Martingale Representation seeks to decomposes a random variable into Ito's Integral. ...
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1answer
24 views

How to show $P(\inf_{t\geq0}\int_0^t e^{-s}\mathrm d B_s\geq -1)>0$?

Let $(B_t)_{t\geq 0}$ be a brownian motion, $P$ the measure of the probability space which satisfys the usual conditions and $\mathbb E$ the expected value. I like to show $$P\left(\inf_{t\geq0}\int_0^...
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30 views

Integral of a function of Brownian Motion

I am trying to integrate, $$\int_{0}^{t}e^{\alpha s - B(s)}ds$$ where, B(s) is the standard Brownian motion. My approach was to use integrate by parts, $$\int_{0}^{t}e^{\alpha s - B(s)}ds = e^{\alpha ...
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0answers
39 views

What is the popular way to simulate an Ito process?

Ito formula is defined as a process of the form $$ X(t)=X(0)+\int_0^t\delta(s)dW_s +\int_0^t\theta(s)ds $$ one obvious way to simulate it is to first generate a Brownian motion and then use the Euler ...
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Let $X$ be a semimartingale with $\triangle X \neq1$. Proof that $\mathcal{L}(\varepsilon(X))=X-X_0$.

Let $X$ be a semimartingale with $\triangle X \neq -1$. $\triangle X=X-X_{-}$ gives us the jumps of X. $Y=\varepsilon(X)$ is the solution of the equality $$Y=1+Y_{-}\cdot X.$$ Let $\mathcal{L}(Z):=\...
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9 views

Ito's Rule and Transition Densities for Time-Homogeneous Markov Process

I am currently working through a paper in the realm of stochastic processes, and I have to admit that I am somewhat stymied. My problem is as follows: I start out with an SDE given by $$dX_t=\mu(X_t)...
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17 views

Wiener integral as a special case of Ito integral

I feel like this might be really simple, but I have tried for some time and not been able to make any headway. If $f \in L^{2}( [0, \infty))$ and $\{B_{t},\{\mathcal{F}_{t}\}, t \in [0,\infty)\}$ is ...
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1answer
51 views

Show that $\operatorname P\left[\sup_{s\in[0,\:t]}\left(M_s-\frac\alpha2[M]_s\right)\ge\alpha\beta\right]\le e^{-\alpha\beta}$

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space and $(\mathcal F_t)_{t\ge0}$ be a complete filtration on $(\Omega,\mathcal A,\operatorname P)$. Let $(M_t)_{t\ge0}$ be a local ...
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1answer
26 views

Auxiliary result related to the exponential martingale inequality

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space and $(\mathcal F_t)_{t\ge0}$ be a complete filtration on $(\Omega,\mathcal A,\operatorname P)$. Let $(M_t)_{t\ge0}$ be a local ...
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1answer
36 views

Stochastic partial integration: Do the stochastic integrators have to be independent?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space, $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$, $\left(W_t^{(...
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22 views

Existence of functions which satisfy some non-integer moment condition

Problem Does there exist a non-degenerate positive continuous function $f$ such that: $$ \mathbb{E} \left[ \left( \int_0^1 t^{\alpha-1} f(t,B_t)\,dt \right)^{\frac{1}{\alpha}} \right] = c_1 \beta^{...
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38 views

Write the martingale part of this semimartingale as an Itō integral process with respect to a one-dimensional Brownian motion

Let $H$ be a separable $\mathbb R$-Hilbert space and $(X_t)_{t\ge0}$ be an $H$-valued semimartingale with $$\int_0^t\left\|X_s\right\|_H^2\:{\rm d}s<\infty\;\;\;\text{almost surely for all }t\ge0\...
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Stationary distribution of multivariate diffusions

Define stochastic processes $(X_t)_{t \geq 0}, (s_t)_{t \geq 0}$ such that $$ dX_t = b(X_t)dt + \sqrt{2}dB_t $$ $$ ds_t = \big(P(X_t)+f(s_t)s_t \big) dt $$ where $(X_t)_{t \geq 0}$ is on $\mathbb{R}^d$...
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56 views

Non-integer moment of some integral which is similar to the incomplete Gamma function integral

Problem. I want to solve the following: $$ \mathbb{E}\left[\left(\int_0^x t^{\beta -1} e^{-\gamma t + (1-t)B_t} \, dt \right)^{1/\beta } \right], $$ where $B_t$ is a standard Brownian motion, $0<\...
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15 views

Estimate for continuous Ito semimartingale

I'm reading an article in which one defines a continuous Ito semimartingale of the form $$\hat{V}_{i \Delta_n}' - \bar{V}_{i \Delta_n} = \frac{2}{k_n \Delta_n} \sum_{j=1}^{k_n} \int_{(i+j-1)\Delta_n}^{...
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1answer
26 views

Quotient of continuous local martingale with quadratic variation

Consider a local martingale $(M_t)_{t\ge 0}$ with continuous paths and $\lim_{t\rightarrow\infty}[M]_t=\infty$ a.s. I want to show, that $$\lim_{t\rightarrow\infty}\frac{M_t}{[M]_t}=0\quad\text{a.s.}$$...
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1answer
26 views

Use Ito formula to solve SDE

If $Z_t$ is a progressive process and $X_t$ is defined by $X_t = 1+\int_0^t Z_s dB_s$, then use Ito's formula to $Y_t=ln(X_t)$, to show that $$ X_t = \exp\Bigl(\int_0^t \frac{Z_s}{X_s} dB_s - \frac{1}{...
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0answers
114 views

What is an SDE for the process $dX(t)=f(B(t))\,dt+dB(t)$?

What is an SDE for the process $dX(t)=f(B(t))\,dt+dB(t)$? I'm wondering if I can turn this into an SDE $$dX(t)=\mu(X(t))\,dt+dB(t)$$ It is difficult to do. Any ideas? I tried applying Ito's formula ...
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33 views

On method of averaging of ODE's and SDE's

Consider the following ODE for the pair of scalars $(x(t),y(t))$ \begin{align} \dot{x} &= \epsilon f(x,y) \tag 1 \\ \dot{y} &= g(x,y) \tag 2 \end{align} where $\epsilon > 0$ is a small ...
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15 views

Solution form for a stochastic differential equation

Given the equation $dX(t)=rX(t)(k-X(t))dt+\beta X(t)dZ(t)$ letting the process $Z(t)$ be standard brownian motion. What will the solution of this SDE look like and can we use Ito's lemma for $Y(t)=1/X(...
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121 views

Find solution of stochastic differential equations

I need to solve this system of stochastic differential equations: $$\begin{cases} dX_1(t)=dt+dW_1(t), \\[2ex] dX_2(t)=X_1(t)dW_2(t), \end{cases}$$ where $W_1(t), W_2(t)$ are independent Wiener ...
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1answer
30 views

When is the Ito representation deterministic?

Consider a functional on Wiener space $F\in L^2(\Omega, \Bbb R)$. Then by Ito's representation theorem we have that $$F=E[F]+\int_0^T\phi(s)dB(s)$$ for some nice $\phi$. Question, when is $\phi$ ...
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0answers
31 views

Application of Girsanov's theorem

Given a Brownian motion $W(t)$ under $P$ and the stochastic process $\hat{W}(t) := W(t) - \int_{0}^{t}\theta(s)ds$ which is a Brownian motion under an equivalent measure $Q \sim P$. The stochastic ...
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0answers
11 views

Stochastic differential equation with respect to several Brownian motions

I have the following SDE: $\frac{df(t)}{f(t)} = \sigma(dW_1(t) + dW_2(t)), $ where $W_1, W_2$ are independent Brownian motions, and $\sigma \in \mathbb{R}$. The issue is: How does one evaluate an ...
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0answers
45 views

stochastic differential equation linear diffusion coefficient

I have this Stochastic Differential Equation $$X_t=x+\int_0^t(\lambda X_s-X_s^2) \, ds+\int_0^t \sigma X_s W_s$$ with $x,σ,λ>0$ and $ W_t$ a brownian motion I considered it as a stochastic ...
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1answer
24 views

$E\int_0^T|X(t)|^2dt=0$ then $X=0 dP\times dt$

$X:\Omega\times [0,T]\to \mathbb{R}$ - process measurable with respect to the product sigma field such that $E\int_0^T|X(t)|^2dt=0$. Prove that then $X=0 dP\times dt$ Can anyone prove it?
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1answer
22 views

Does the Martingale Representation theorem hold both ways?

Can the Martingale Representation theorem be used to assume that the integral with respect to Brownian motion, $B(t,\omega)$, $$X=\int^{T}_{0}B^{4}(t,\omega)dB(t,\omega)$$ is a square integrable ...
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0answers
13 views

When does a Markov semigroup preserve differentiability?

Let $E$ be a $\mathbb R$-Banach space (for simplicity, assume $E=\mathbb R$, if you like) and $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$. I would like to know under which ...
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0answers
11 views

Constant stochastic integral

Let $M$ a local martingale continuous such as $M_0=0$ and $H$ a progressive process locally bounded. I have to show that almost surely for all real valued $0 \leq s <t$ the function $ u \rightarrow ...
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1answer
52 views

Question about Ito Process. Stochastic Processes

How to prove, that $W_{t/(1-t)}$ at $[0,1)$ is Ito Process ? (Have stohastic differential)
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26 views

Is stopped integral of predictable process predictable?

Assume that $H$ is a predictable process that is locally bounded with localizing sequence $(\tau_n)_n$. And assume that $\langle M, M \rangle$ is a increasing, right-continuous, predictable process. ...
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1answer
15 views

Variance of the difference of Brownian Motions

I have a question about the variance of the following formula: $Var(W(t) - \frac{t}{T}W(T-t))$. Where $W(t)$ is a Brownian motion. I tried the following: $Var(W(t) - \frac{t}{T}W(T-t)) = E[W(t) - \...
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0answers
11 views

Is stochastic integral subgaussian

Consider the solution $X$ of the stochastic differential equation $$ \mathrm{d}X_t = \sigma(X_t) \mathrm{d}W_t, $$ where $W$ is a Wiener process and assume the standard growth conditions ($X$ is a ...
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1answer
31 views

Solve an Itô Integral by Itô calculus

I saw an example where the following Itô integral was solved by Itô calculus: $\int^{T}_{0}W(t)dW(t)$. They say: let's take the stochastic process $X(t) = W(t)$, which means that $dX(t) = 0 dt + 1 dW(...
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0answers
31 views

Is it a good example for local martingale, but not for martingale?

If $B$ is a Brownian-motion in the $\mathcal{F}$ filtration, then the following process is a good example for being a local martingale, but not a martingale?$$S_{t}=\int_{0}^{t}\frac{1}{1-s}dB_{s},\;\;...
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0answers
20 views

Find continuous martingale, such that it's quadratic variation is a deterministic continuous function

Given a non decreasing continuous function $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ with $f(0)=0$. I want to find a continuous martingale $(X_t)_{t\ge 0}$ , such that it's quadratic variation is ...
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0answers
26 views

Integral Ito (stohastic integral) of Poisson process

I need to find $D\int_{0}^{t} N_{s}dW_{s}$, where $N_{s}$ is Poisson process and $N_{s}$ and $W_{s}$ are independent. How can i do it?
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1answer
38 views

Expected value and variance for Itô Integral

This is my first question and I hope it is ok :) Reading in a book I came along this answer to a question I did not understand and I would like to understand this very much: Problem: Calculate the ...
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0answers
16 views

Understanding conditional variance

I'm trying to understand conditional variance, but I'm struggling a bit. I've devised two examlples below: Example A: Suppose I have a Weiner process $W_t$ which is adapted to $\mathcal{F}_t$. What ...
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1answer
23 views

Integral of Stochastic Integral $ x(t) = \int_0^te^{as}dW_s $

I'm working through an exercise where I have got the following object: $$ x(t) = \int_0^te^{as}dW_s $$ for some constant $a$. Now, I need to find the distribution of $\int_0^t x(s) ds$. I'm ...

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