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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How can I solve $E[B^4_t B^3_t]$?

How can I solve the following expected value: $$ E[B^4_t B^3_t] $$ where $ B_t $ is a standard Brownian Motion.
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0answers
29 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
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1answer
20 views

Black Scholes Solution

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
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1answer
45 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
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0answers
44 views

Expected value and Variance of a stochastic time integral of a deterministic variable (Standard Brownian motion)

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, define: $$E(e^{\int_0^tudB_u})=?$$ $$ Var(e^{\int_0^tudB_u})=?$$ Sidenote to be edited later: Here is my try, I'm not capable to ...
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1answer
28 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
2
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1answer
30 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
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1answer
41 views

Deriving the Doob Meyer decomposition of a Sub Martingale using Ito's

Given the standard brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$ and defining the sub-m.g.: $$X_t =B^6_t+2t$$ I would like to derive its Doob-Meyer decomposition: [Sub-m.g.]= [increasing ...
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0answers
20 views

Profit Maximization

I have listed a homework problem below that I have been working on. How do I get the expected number sold/expected number unsold/expected number lost if I do not have the pdf for the demand? Am I ...
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0answers
17 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
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1answer
300 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
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1answer
27 views
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2answers
23 views

Find the value of the real $\alpha$ for which exp($2B(t) - \alpha t$) is a martingale.

I tried to answer using the three conditions to be a martingale (measurability, integrability, and martingality), validating the integrability condition, which is $$ E |e^{2B(t) - \alpha t} | < ...
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1answer
34 views

Prove directly from the definition of the Ito's integral

I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question. Prove directly from the ...
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0answers
19 views

Deterministic integrals involving a Brownian motion [duplicate]

I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$. The first expression is bewildering me a bit, since it seems like somehow the Itô ...
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0answers
71 views

Expected value of a brownian motion times the deterministic integral of a brownian motion

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, $E (B_t \int_0^tB_s^3ds)$ = ? I try to turn the expected value into a double integral by rewriting the $B_t$ term as 1) $E(\int_0^t ...
3
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1answer
71 views

conditional expectation of some solution of SDE

Let $(M_t)$ be a nonnegative martingale in a probability space $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, \mathbb{P} )$ given by \begin{equation} dM_t = M_t \sigma_t dW_t \end{equation} for some ...
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0answers
27 views

Solve the stochastic differential equation

I have to solve the following SDE: $$dX_t=X_t dt+2W_tdW_t$$ Let $Y_t=X_t e^{-t}$. By Ito formula we have: $$dY_t=-X_te^{-t}dt+e^{-t}(X_t dt+2W_tdW_t)=2e^{-t}W_tdW_t$$ Thus ...
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0answers
70 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
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1answer
47 views

Ito's process and martingale [duplicate]

Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...
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1answer
42 views

$Cov(X_t,X_s)$ of martingales

Let $X_t = \int_0^t W_u^2dW_u$ martingale compute : $$Cov(X_t,X_s)$$ note that $$Cov(\int_0^T a(t)dWt,\ \int_0^T b(t)dWt)\ = E[\int_0^T a(t)b(t)dWt]$$ My attempts: $$Cov(X_t,X_s)\ = ...
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1answer
68 views

Eigenvalue problem in functional analysis?

How can I find the eigenvalues and eigenvectors of \begin{align} Ay(p):=\int_{0}^{\infty} k^2 \cos(pk)y(k)dk \end{align} $A$ is a Hilbert-Schmidt operator. Well actually, i came across this in ...
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49 views

Characterize the limit of an O-U process: $dX_t = -\tfrac{\mu}{\theta} X_t dt + \tfrac{\sigma}{\theta^{1/2}} dW_t$ as $\lim_{\theta \to 0}$.

Standard O-U Formulas: Take the Ornstein–Uhlenbeck process defined by the SDE $$ dX_t = -\frac{\mu}{\theta} X_t dt + \frac{\sigma}{\theta^{1/2}} dW_t $$ where $\mu > 0, \theta > 0, $ and ...
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1answer
18 views

Malliavin Derivative

Motivation : We know that, if the randomness in the system is due to Brownian Motion then any contingent claim with mean zero can be written as Ito integral. (Of course, we need to have boundedness ...
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1answer
67 views

Expected value and variance of a stochastic process

Having trouble finding expected value and variance of a stochastic process defined by SDE: $dX_{t} = a X_{t} dt + b dB_{t}$ $X_0 = x$, $a$ and $b$ are constant values, $B_t$~$N(0,t)$ Thank you for ...
1
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1answer
18 views

A question on integration wr.t to a local martingale

In a lemma in my graduate level course on financial mathematics uses the fact that integral of a progressive portfolio process(which is almost surely lower bounded i.e it is admissible) $\theta_t$ ...
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0answers
73 views

Find the relation between 2 stochastic integral

$g(s,t)(\omega)$ is an adapted stochastic process on $\mathbb R^2$ define: $$X=\int_0^1\int_0^1g(s,t) \,dW_s\,dt$$ $$Y=\int_0^1\int_0^1g(s,t) \,dt\,dW_s$$ Could we conclude that "$X=Y$ a.s"? I ...
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1answer
20 views

Conditional Ito's isometry

I am looking for a formal proof of the following (if true): $\mathbb E \left[ \int_0^1 g_1(s)\,dW_s \int_0^1 g_2(s) K_s\,dW_s \big|\mathscr F^K \right]=\int_0^1 g_1(s)g_2(s)K_s\,ds $, where ...
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1answer
16 views

Covariance of Wiener Processes on the same Brownian Motion

I am trying to solve $Cov(Tw_T,\int^{T}_{0}tdw_t)=\mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t]$, my attempt is as below: \begin{split} \mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t] & ...
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0answers
27 views

Integral of a non-linear step function on closed interval

I need to compute the following integral for a random variable $a$ with known support and CDF: \begin{equation} \int_{a^L}^{a^H} \left( \sum_{j=1}^{N} \begin{cases} B_j a \mbox{ if } a \leq a_j^*\\ 0 ...
3
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1answer
313 views

Ito's Lemma application [closed]

$Z(t) = \int_0^t g(s)\,dW(s)$, where $g$ is an adapted stochastic process. Find $dZ$ ?
4
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1answer
346 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
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1answer
649 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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0answers
18 views

A problem on Ito integral [duplicate]

Let $W$ be a standard, one-dimensional Brownian motion. Let $T\in(0,+\infty)$. Then $$\lim_{\beta\to+\infty}\sup_{0\le t\le T}\left|e^{-\beta t}\int_0^te^{\beta ...
3
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1answer
75 views

What is the explicit obstruction to almost sure convergence in stochastic integrals?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
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1answer
317 views

Stochastic representation formula

Consider the following boundary value problem in the domain $[0,T]$ x $R$ for an unknown function F. $\frac{\partial F}{\partial t}(t,x) + \mu(t,x)\frac{\partial F}{\partial x}(t,x) + \frac ...
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1answer
37 views

Integral of a Brownian bridge with respect to time

Let $(W_s)_{s\geq 0}$ be a Brownian motion and $t$ a fixed point in time. What is the distribution of $$\Big.\int_0^tW_sds\Big|W_t$$ i.e. the integral of a Brownian bridge with respect to time? Is it ...
0
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1answer
96 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
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1answer
31 views

Solution to truncated renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
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1answer
64 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [closed]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
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2answers
56 views

Verifying Property of Stochastic Integral

I am trying to verify this simple property for a stochastic integral. Given that f(t,w) is a bounded, nonanticipating function for a given Wiener process $W_t$ show that $E((\int_{0}^{T} f(s,w) ...
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1answer
49 views

Variance of integrated squared wiener process

So I'm trying to figure out the mean and variance of $X = \int_{0}^{1} W^2(t) dt $ where $W$ is the Wiener process. The mean I've worked out easily to be $\frac{\sigma^2}{2}$ but I'm having ...
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2answers
126 views

Variance of sum of two ito integrals

I don't really understand how to solve the following problem: Var(X) where X = $\int_0^2 2t dW(t) + \int_4^6 W(t) dW(t)$ If I use $E [(A+B)^2] = E(A^2) + E(B^2) + 2E(AB)$ I get to the point where I ...
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0answers
420 views

Solution to nonlinear Stochastic Differential Equation

$dX_t=\left(\sqrt{1+X_t^2}+\frac{1}{2}X_t\right)dt+\sqrt{1+X_t^2}dW_t, X_0=0$, where $W_t$ is brownian. I tried using $X_t=\sinh(W_t)$ but then when I apply Ito's lemma to it, I can't get the first ...
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0answers
27 views

Gibbs Sampler integral computeable

here is an example of a changepoint in a poisson world with the gibbs sampler, it is an bayesian approach. the data are assumed to follow this distributions : $\begin{equation} \nonumber Y_i \sim ...
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0answers
118 views

Condition for a process to be a supermartingale

I am struggling in this question: Let $W$ denote a Brownian motion. Given that $ X_t = e^{- \lambda t} X_0 + \int_0^t \sigma e^{- \lambda (t-s)} \,dW_s$ solves the SDE \begin{equation} dX_t = - ...
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1answer
23 views

Ito integral's zero mean

My Sto Cal prof gave a long proof for the fact that $E[\int_{0}^{t} f_s dW_s] = 0$ where W is Brownian and f is Borel x $\mathscr{F}$-measurable, adapted and satisfies some integrability condition. ...
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1answer
32 views

Brownian motion on the circle and Itô processes

Consider the differential system \begin{cases} dX_t &=& -\frac{1}{2}X_t dt - Y_tdB_t, \\ dY_t &=& -\frac{1}{2}Y_tdt + X_tdB_t, \end{cases} $X_0 = 1$, $Y_0 = 0$. Let $X_t$ and $Y_t$ ...
1
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1answer
21 views

Basic question on application of Itô's formula to a stochastic process

I am working on a problem where I now find myself wanting to apply Itô's formula to: \begin{equation} X_t = \exp(W_t -W_0-\frac{t}{2}+\int\limits_0^tX_sds) \end{equation} where $W_t$ is 1D Brownian ...
2
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0answers
42 views

Why is the pathwise integral of $\alpha_s$ w.r.t the Lebesgue measure continuous?

My class notes on stochastic calculus say that the if $(\alpha_s(\omega))_{s\in \mathbb{R_+}}$ is progressive then $\int_0^t \alpha_s ds$ is a pathwise continuous process? How does the joint ...