This tag is used for questions about stochastic integrals - especially for calculations . For questions related to more theoretic aspects of stochastic integrals such as its construction. Stochastic-analysis may be a more appropriate tag.

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Stochastic integral: Interchanging the order of expectation and integration

Let $B$ be a standard Brownian motion and $$ X_t=\int_0^t f_s ds+\int_0^t g_s dB_s, $$ where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$. Is it true that $$ ...
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75 views

Local martingale iff each component is a local martingale?

This is probably an easy question: A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a ...
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64 views

Condition for existence of a stochastic differential equation

With $B$ a standard Brownian motion, write $$ dX_t=f_tdt+g_tdB_t. $$ What are the conditions on $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ for $X_t$ to exists? I think ...
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300 views

Help understanding the Feynman-Kac formula

From wikipedia: Suppose we wish to find the expected value of the function $e^{-\int_0^t V(x(\tau)) d\tau}$ in the case where $x(\tau)$ is some realization of a diffusion process starting at $x(0) = ...
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1answer
256 views

Integration of Wiener process: $\int_{t_1}^{t_2} dB(s)$

We all know that $\int_0^t dB(s) = B(t)$, where $B(t)$ is a standard Brownian Motion. However, is the following identity true? Also, why or why not? $\boxed{ \displaystyle \ \ \int_{t_1}^{t_2} ...
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1answer
134 views

Integrating a Brownian Bridge conditioned above a linear boundary

The Setup: A Brownian Bridge $B$ is a Brownian Motion on time interval $[0, 1]$ conditioned such that $B(0) = B(1) = 0$. I have a function $f(t) = mt+b$ with $m, b$ set such that $C(t) \le 0$ for $t ...
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1answer
392 views

Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
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1k views

Itô Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itô integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
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1answer
77 views

Rephrasing a Stochastic Process as a Stochastic Differential Equation

I have a continuous-time stochastic process $X$, described as follows: (1) If the process is at $x_0$ at time $t_0$, then the function $f(t_f, x_f \, | \, t_0, x_0)$ is a PDF in the parameter $x_f$ ...
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1answer
733 views

Applying Ito formula to the Brownian bridge

Let $B$ be a standard Brownian motion and $$ W_t=(1-t)\int_0^t \frac{1}{1-s}dB_s $$ be a Brownian bridge. Calculate $dW_t$. To apply Ito formula define $$ f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s $$ ...
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259 views

Analogue of Leibniz Rule for Stochastic Integrals

Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w}$$, where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ ...
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1answer
122 views

Differentiability of hitting time of Brownian motion

I have been trying to prove the following conjecture for a while, but so far to no avail. Would be very grateful for some tips! The conjecture is the following; Think of an $n$ dimensional Brownian ...
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1answer
55 views

$E \left\{ \left( \sum_{i=0}^{n-1} \left[ B_{c_i} \left( B_{t_{i+1}} - B_{t_i}\right)\right] \right)^2 \right\}$, where $c_i \in [t_i, t_{i+1}]$

Let $B$ be a standard Brownian motion and $\{t_i\}_{i=0}^n$ a partition of $[0,t]$. Define $c_i= (1-c)t_{i+1}+ct_i$, for some $c \in [0,1]$. Write $B_i$ for $B_{t_i}$ and $$ S_n=\sum_{i=0}^{n-1} ...
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41 views

First-Exit time in 2-dimensional problem

Could someone recommend me some books or papers related to this problem?
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66 views

an example indicating the relation between Brownian motion and PDE

I have a question: Let $(B_t)_{t\geq 0}$ be a brownian motion. Consider the following function $u(x)$ defined by ...
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43 views

a question about covariation in stochastic integration

Let H, K be bounded previsibe process. M, N be two local martingales. How can I prove $d<H.M, K.N>_t = H_tK_td<M,N>_t$ $<M>$ means the quadratic variation of M. Thanks
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1answer
69 views

Formulae to about Moment and Cross-moments of Stratanovitch Iterated Integrals

The title is a bit long but quite explicit, I am looking for a reference where the moments and cross moment Stratanovitch Iterated Integrals defined as : $E[J_n(1).J_p(1)]$ with $p\not=n$ With : ...
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1answer
49 views

Rewriting SDEs - “Multiplication on both sides”

I have a question concerning a calculus "trick" sometimes used in stochastic calculus (e.g. in the Book on Arbitrage Theory in Cont. Time of Bjoerk). There they do the following in the proof of Prop. ...
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1answer
364 views

Stochastic integral : $\int_0^T (W(s))^2dW(s)$

How to evaluate this integral $$\int_0^T(W(s))^2 \, dW(s)$$ where $W(s)$ is random variable associated with brownian motion. I am new to this .Thanks in advance.
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355 views

Integral of a gaussian with random variance

Assuming: $$X(x,\mu)=\frac{1}{\sqrt{(2\pi)\sigma^2}} \exp[-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}]$$ the integral of $X(x,\mu)$ from $-\infty$ to $+\infty$ is: $$S=\int_{-\infty}^{+\infty}dx ...
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1answer
122 views

Conditional expectation of a finite variation process

A simple question: Let $H$ be a cadlag, adapted process and $A$ a process of finite variation. Then also $\int_t^T HdA_t$ is a finite variation process (see "Limit Theorems... "Jacod&Shiryaev ...
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1answer
135 views

Confusion regarding Stochastic integral

I've a stupid doubt in the construction of stochastic integral of real scalar valued maps. Many times I've seen in books after the stochastic integral is defined in [$0,T$] for the integrand in $L^2$ ...
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2answers
181 views

Integral of a random function

How is it possible to evaluate the integral: $$I(\mu,\sigma)=\int_0^{2\pi}\sin(\omega t)^2dt$$ where $\omega$ is a random variable having a normal distribution $N(\mu,\sigma)$? What is the $pdf$ of ...
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127 views

Integrating the inverse of a squared bessel process - integrability

Let $X_t$ be a 4-dimension Squared Bessel Process (BESQ-4). Let $M_t$ be a continuous true martingale. Question: Does $\int_0^t \frac{1}{X_s}dH_s$ exist? If so, is it only a local or a true ...
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150 views

Ito's formula for irregular functions

Let's say we have \begin{align} Y_t=h(t,X_t) \end{align} and for simplicity \begin{align} dX_t=e\,dt+f\,dW_t \end{align} then by Ito's formula we have \begin{align} dY_t=\left(\frac{\partial ...
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79 views

a pair of Stochastic Differential Equations

I'm trying to complete a course on SDEs and I need to solve two stochastic differential equations. They are supposed to be easy, but I'm still a beginner and to be honest I'm quite stuck. The pair of ...
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70 views

Expectation of $\int_0^t X(r) \, dW(r)$ where $dX=\mu \, dt+\sigma \, dW$

I have a questionlike: if $dX=\mu \, dt+\sigma \, dW$, where $W$ is a standard B.m. Then, is this expectation still o,$\int_0^t X(r) \, dW(r)$ ? Thank you all.
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160 views

Existence of solutions to stochastic differential equations by the Banach contraction principle?

I've read a proof for existence of solutions to stochastic differential equation from a book of Ikeda and Watanabe and have a question. Is it possible to prove existence (and uniquness) by means of ...
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1answer
204 views

A question related to Novikov's condition

The well-known 'Novikov condition' says: Let $ L = (L_t)_{t \geq 0} $ be a continuous local martingale null at 0 and $ Z = \exp(L - \frac{1}{2} \langle L \rangle) $ its stochastic exponential. If ...
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168 views

Constructing Ito integral for adapted process

I am trying to construct Ito integral for adapted process. However, I am stuck at some point. Let $X^n(t)$ be a sequence of simple processes convergent in probability to the process $X(t)$. Then the ...
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1answer
446 views

Verifying Ito isometry for simple stochastic processes

It is known that stochastic integral must satisfy the isometry property which is $$ \mathbb{E}\left[ \left( \int_0^T X_t~dB_t\right)^2 \right] = \mathbb{E} \left[ \int_0^T X^2_t~dt \right] . $$ I am ...
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1answer
72 views

Covariation Paradox??

we can see that $\left\langle \int_0^t \! W_s \, \mathrm{d} s ,W_t \right\rangle_t = 0$ However if I am to use the expression $$\int_0^t \! W_s \, \mathrm{d} s= t W_t - \int_0^t \! s\, \mathrm{d} ...
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1answer
134 views

One correlated Stochastic Integral

If $${\rm Cov}[dW_t,dB_t]=\rho dt$$ then what is $$\mathbb{E} \left[\int_0^t\sigma_{1s}dW_s \int_0^t\sigma_{2s}dB_s\right]$$ where $\sigma_{1s}$ and $\sigma_{2s}$ are two deterministic functions ...
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1answer
567 views

Probability density function of the integral of a continuous stochastic process

I am interested in whether there is a general method to calculate the pdf of the integral of a stochastic process that is continuous in time. My specific example: I am studying a stochastic given ...
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249 views

Ito process generator and Dynkin's Formula

I have attempted to answer the following multi-part question but am especially having trouble with part iv) of the question. Any feedback would be greatly appreciated!. Consider the Ito process $X(t) ...
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1answer
779 views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
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72 views

Applicability of Itô's Lemma for $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}_0([0,1]^2)$

Let the domain be $[0,1]^2$. And let $W^x_t$ be the standard Brownian Motion started in $x\in [0,1]^2$ with absorbption on $\partial [0,1]^2$ and choose some $g\in \mathcal{C}^2((0,1)^2)\cap ...
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67 views

Are affine SDEs invertible?

If we have an a process $X_t$ with values in $\mathbb{R}^{n \times n}$ which solves a linear Stratonovich SDE $$ dX_t = A_t X_t dt + B_t X_t \circ dW_t $$ then the inverse of $X_t$ exists and solves ...
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187 views

property of the bracket process and stochastic integral

suppose $M$ is a continuous local martingale and $H$ is a predictable process such that $E[\int_0^{\tau_n} H_s^2d\langle M\rangle ]<\infty$, where $\tau_n$ is a sequence of stopping times ...
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367 views

Why is it true that the continuous local martingale with quadratic variation “t” is a square integrable continuous martingale?

I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $M_t$ be a continuous local martingale. On page 157, it wrote that "because $\langle M\rangle_t = t$, we have $M \in ...
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1answer
108 views

properties about stochastic integral

I have a question about stochastic / Lebesgue Stieltjes integrals. I'm following Revuz / Yor. The space $H^2$ is the space of all $L^2$ bounded continuous martingales. If $M\in H^2$ then they call ...
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1answer
149 views

Stochastic differential equation: Itô's formula?

I came across a problem with SDE and need your help once again: $$dX_t=tX_t \, dt+\exp \left(\frac{t^2}{2}\right)$$ and I'm supposed to solve this, in the way $X_t=f(t,W_t)$. So I use Itô's formula: ...
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1answer
132 views

The variance of bilateral filtered random variables

I am glad to have found this great site. There is a problem I am trying to solve for a while. I want to analyze the noise attenuation behavior of the bilateral filter. So given the unnormalized ...
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1answer
335 views

$\mathcal{F_t}$-martingales with Itô's formula?

I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar: ...
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1answer
223 views

Karhunen-Loève expansion of Poisson process

Let $X_t,t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the Karhunen-Loève expansion of $X$ in interval $[0, T]$. How about the KL expansion of the centered process $X_t−\lambda ...
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184 views

PDF for the integral of a Stochastic Process

My continuous-time, continuous step Stochastic Process P runs from time $t=0$ to $t=t_f$ and generates a path. I am able to observe its starting and ending position (so $P(0)=a$ and $P(t_f)=b$), but ...
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100 views

Why is this a martingale?

In our homework assignment, we are supposed to prove: If $ M $ is a countinuous local martingale and if for each $ T > 0, E[\sup_{t \leq T } |M_t|] < + \infty $ and $ H^T $ is a bounded ...
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1answer
228 views

Futures pricing and futures price process under the real world measure

This is something that keeps bothering me about the Benchmark approach of Platen, which (very) shortly is as follows: Compare the development of an economic value with a growth optimal portfolio. ...
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1answer
458 views

Scalar product of Gaussian process

Assume that $n(t)$ is a White Gaussian Noise (WGN) process with $E[n(t)]=0$, $E[n(t)^2]=\sigma^2$ and $x(t)$ a deterministic function defined in $[0,T]$. How can I compute from first principles the ...
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1answer
109 views

Why is $ N^\tau ( M - M^\tau ) $ a continuous local martingale if $ M $ and $ N $ are?

Working through my stochastic calculus script, I encountered the following identity, for which no proof is given: $ \langle M, N^\tau \rangle = \langle M, N \rangle^\tau $, if $ M, N $ are continuous ...