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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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1answer
22 views

Variance of a Laplace Transform

I have a function $F(s)$ which is the Laplace transform of $f(t)$ (which is in itself a normally distributed random process), but I don't know what $f(t)$ is (this comes from solving a differential ...
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1answer
25 views

Quadratic Variation and Brownian Motion

Let $(X_n,F_n)$ be a martingale with $X_n \in L^2(\Omega,F,\mathbb{P})$. The quadratic Variation $(<X>_n)_n$ of the process $(X_n)_n$ is defined as $$ <X>_n := \sum\nolimits_{i=1}^{n}(\...
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0answers
15 views

How to I use Ito's formula to compute quadratic variation?

Let B be Brownian motion. Use Itos formula to compute the quadratic variation of $\left[X_t^i\right]$ for $\left[X_t^1\right]=e^B_t$, $\left[X_t^2\right]=ln(B_t^2+1)$ and $\left[X_t^1\right]=sin^2B_t+...
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0answers
8 views

Expectation of the product of Brownian processes (higher powers)

I have recently sat an exam that had elements of stochastic calculus, but I am now feeling like I might have gone wrong in some questions of it like the following. I am trying to evaluate $\mathbb{E}(...
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18 views

Double Wiener-Ito Integral of a Seperable Deterministic Function in $L^2([a,b]^2)$

Suppose $f,g \in L^2[a,b]$. Show that the double Wiener-Ito integral of the function $f(t)g(s)$ is given by $$ I_2(fg) := \int_a^b\int_a^bf(t)g(s)dB(t)dB(s)=I(f)I(g)-\int_a^bf(s)g(s)ds $$ where $B_t$ ...
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0answers
13 views

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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0answers
26 views

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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1answer
27 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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0answers
29 views

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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13 views

Expectation of local martingale

For a driftless diffusion process satisfying the SDE \begin{equation} \mathrm{d}X_t = \sigma(t,X_t) \mathrm{d}W_t \end{equation} I want to calculate the expectation \begin{equation} \mathbb{E}\left( \...
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0answers
18 views

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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1answer
40 views

Why is it more interesting to define Itô integral rather to use $f(t)B_t$?

SDE's are normally of the form $$dX_t=f(X_t,t)dt+\sigma (X_t,t)dB_t,$$ but the sense of that is $$X_t=X_0+\int_0^t f(X_s,s)ds+\int_0^t \sigma (X_s,s)dB_s,\quad a.s.$$ Now, I was wondering why is it ...
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1answer
103 views

“Conditional distribution” of Brownian sample paths

I would like to consider the "conditional distribution" of the Brownian sample paths conditional on certain sample path functionals, in a similar way that one considers the Brownian bridge. For ...
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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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1answer
39 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 ...
2
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1answer
39 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 ...
2
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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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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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0answers
16 views

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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0answers
31 views

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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1answer
25 views

Proof for identity involving joint probability and conditional probability. [closed]

How do you prove the following identity? $$\mathbb{P}(X \in A, Y \in B) = \int_B \mathbb{P}(X \in A| Y = y)\mathbb{P}_Y(dz)$$ Additionally, what assumptions on $X$, $Y$, $A$ and $B$ are needed?
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1answer
35 views

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 $$\mathbb E\int_0^T H_s^2ds=\int_0^T \mathbb E[H_s^2]ds,$$ ...
2
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1answer
43 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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1answer
54 views

Compute integral about cosh

How to simplify following form? $$(2\pi)^{-d/2} \det{(\Sigma)}^{-1/2} e^{-\mu^T \Sigma^{-1} \mu} \int_{x \in R^d} e^{-\frac{1}{2} x^T \Sigma^{-1}x } \cosh(\mu^T \Sigma^{-1} x) \ln(\cosh(\mu^T \...
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0answers
56 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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0answers
24 views

Computing the expected time for a stochastic process to hit a boundary [duplicate]

Let $X_{t}$ be a Wiener process starting at the point $x$. Compute $\mathbb{E}[\tau_{a, b}]$, where $\tau_{a, b}$ is the minimum time $t$ at which the process $X_{t}$ is equal to $a$ or $b$. That is, ...
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1answer
35 views

Solution to $\int_0^t dW_s $

This question is so basic that I can't find the answer anywhere. I'd think it would be just $W_t$ but since there are additional rules in stochastic calculus, I'm not 100% sure. I've only seen ...
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0answers
31 views

(Proof) If $M$, $N$ are two orthogonal local martingales, and $S$, $T$ stopping times, then the stopped local martingales $M^S$, $N^T$ are orthogonal.

I am currently reading the first chapter on the general theory of stochastic processes in "Limit Theorems for Stochastic Processes" by Jacod and Shiryaev. On page 41 they state the following: 4.13 ...
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0answers
28 views

Write a summation to approximate the integral

Let $W(t)$ be a Brownian motion and let $I$ be the integral $I$ = $\int_0^T$ $W(t)dt$ $1)$ Given the partition $π$ : $0 = $$t_0$$ < $$t_1$$ < $$t_2$$ < . . . < T$, write a summation to ...
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0answers
22 views

What does non-degeneracy of of the diffusion coefficient in the context of a SDE mean?

In the introduction of a paper I was reading the author writes without elaborating that " In the case of a non-degenerate diffusion coefficient, Stroock and Varadhan , proved the existence of a ...
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2answers
75 views

Simple process in Itô calculus

For the definition of Itô integral, one uses simple stochastic processes. I have found two definitions for simple stochastic process, given a filtration $(\mathcal{F}_t)_{t\geq0}$, an interval $[0,T]$ ...
2
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1answer
53 views

Solving a Stratonovich SDE

I am trying to solve the following Stratonovich SDE $$dN_t=rN_tdt+\gamma N_t\circ dB_t$$ In my notes, the Stratonovich integral is defined as $$\int^t_0 N_s\circ dB_s=\int^t_0 N_sdB_s+\frac{1}{2}\...
2
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1answer
72 views

Is this random Lebesgue-integral well-defined?

Let $$ X : [0,T] \times \Omega \rightarrow \mathbb{R} $$ be an almost-surely continuous stochastic process. Then how is the random Lebesgue-integral $$ \omega \mapsto \int_{0}^{T} X_t(\omega ) dt \...
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0answers
27 views

Multidimensional Correlated Geometric Brownian Motion, finding exact form of the matrices

My goal is to understand the dimensions of the matrices involved, so I am initially writing things as column vectors, and defining all the dimensions. I am working with the following setup: ...
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1answer
19 views

Computing $dY^{-1}(t)$ using the SDE $dY(t)=\mu Y(t)dt+\sigma Y(t)dW(t)$

Let $\mu$ and $\sigma$ be constants and consider the SDE $dY(t)=\mu Y(t)dt+\sigma Y(t)dW(t)$ with $W(t)$ Brownian motion and $Y(0)=y_{0}$. Using the solution to the SDE, $Y(t)=y_{0}\exp[(\mu-\frac{\...
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3answers
88 views

Understand better stochastic integral through a.s. convergence

I know that $$\int_0^T f(B_t, t)dB_t=\lim_{n\to \infty }\sum_{i=1}^n f(B_{t_i^{(n)}},t_i^{(n)})(B_{t_{i+1}^{(n)}}-B_{t_i^{(n)}}),\quad \text{in }L^2,$$ where $\{t_i^{(n)}\}_{i=1}^n$ is a sequence of ...
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1answer
19 views

Bounded moments for solution of stochastic differential equation

Consider the following SDE: $$\mathrm{d} X_t = - \lambda X_t + \mathrm{d} B_t$$ with initial condition $X_0 = x$, and where $B_t$ is a standard Brownian motion. An application of Ito's formula gives ...
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1answer
42 views

covariance of two stochastic integrals

I'm trying to evaluate the covariance between two stochastic integrals such as $$Cov(\int_0^t g_udW_u, \int_0^t h_udW_u) = \int_0^t E[g_uh_u]du $$So I am trying to prove this and I thought I would do ...
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0answers
25 views

Develop a stochastic process into a stochastic integral

How can we write the following stochastic process $Xt=e^{-(Bt+t)^{2}}$ as stochastic itegral? Note that Bt is a standard BM. I'm guessing that we will need Itô's lemma to transform the expression, but ...
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0answers
14 views

How to represent a stochastic process as an Itô process

How can we write the following process $Xt=(Bt)^{5/2}te^{-t}$ as an Itô process? Note that $Bt$ is a standard BM. I'm guessing that we will need Itô's lemma to transform the expression, but I'm having ...
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1answer
33 views

Differentiating a stochastic integral

How do i differentiate the following stochatic integral? $$\frac {d}{dW_t} \int_{0}^t \frac{1}{1-u} dW_u$$ My guess is $$\frac {d}{dW_t} \int_{0}^t \frac{1}{1-u} dW_u = \left.\frac {1}{1-u} \...
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0answers
23 views

Milstein discretization of the CIR process

Given the CIR process: $$\ dX_t = (a − bX_t ) dt + \sigma \sqrt{X_t}dW_t$$ I want to show that its Milstein scheme is: $$\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\...
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0answers
36 views

How to solve $\mathop{dX_{t}} = \alpha X_{t} \mathop{dt} + \beta \mathop{dW_{t}}$ holds, where $X_{0} = x?$ [duplicate]

I'm learning about stochastic processes, and I want to solve $$\mathop{dX_{t}} = \alpha X_{t} \mathop{dt} + \beta \mathop{dW_{t}}$$ where $X_{0} = x$. I think that the solution uses Ito's Lemma; ...
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0answers
20 views

Show the derivability of a process using Itô's lemma

We have the process $Xt=log(\sqrt(Bt^2+Wt^2))$. Note that $Bt$ and $Wt$ are two independant standard browninan motions. Now, we need to find an expression with the form: $dXt=...$ I`m pretty sure we ...
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1answer
50 views

The transformation from Ito integral to Stratnonvich integral

In the book Introduction to SDE by Evans, it says that if $\mathbf{X}$ solves the Ito sde $$ \left\{\begin{aligned} d \mathbf{X} &=\mathbf{b}(\mathbf{X}, t) d t+\mathbf{B}(\mathbf{X}, t) d \mathbf{...
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1answer
38 views

How to show $X_{t}$ is a solution to a stochastic differential equation?

Prove that $$X_{t} = X_{0}\cdot \text{exp}((\alpha - \beta^2/2)t + \beta W_{t}) $$ is a solution to the equation $\dot{X_{t}} = \alpha X_{t} + \beta X_{t} \dot{W_{t}}.$ Moreover, calculate $\...
2
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1answer
40 views

Independent Increments of Time Integral of Brownian Motion

I am wondering if $\int_0^tW(s)ds$ is independent of $\int_t^TW(s)ds$, where $W$ is a standard brownian motion/wiener process, and for $0 \leq t \leq T$ Writing them as limits of Lebesgue Integrals, ...
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0answers
56 views

Stochastic integration by parts

My professor asserts However, Oksendal asserts in his textbook: $X_{t} Y_{t}-X_{0} Y_{0}=\int_{0}^{t} X_{s} d Y_{s}+\int_{0}^{t} Y_{s} d X_{s}+\int_{0}^{t}dXdY$ These are not equivalent - consider $...
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0answers
25 views

Prove bounded set implies $L^2$ integrable function?

On a complete and filtered probability space, $(\Omega, \mathcal{F}, \mathcal{P})$, I would like to show that a function $f(t, \omega_1, \omega_2): \mathbb{R} \times \mathbb{R}^2 \to A \in \mathbb{R}$ ...
2
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0answers
39 views

How can I simulate the Stochastic integral $\int X_sdW_s$ when X is a stochastic process and W is a Brownian motion?

How can I simulate the Stochastic integral $\int_0^1 X_sdW_s$ where $X$ is strong solution of of an SDE driven by a Brownian motion independent of $W$(the integrator above). I have already computed $...