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 Leibniz rule Ito integral

Assume that $W$ is a Brownian motion and $f=f(t,u)$ is a function of 2 variables such that for all $t$, $f(t,\cdot)$ is adapted to the natural filtration of the Brownian motion and the Ito integral ...
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1answer
29 views

Integrated Ornstein-Uhlenbeck

Suppose we have an OU process given by the stochastic differential equation $dr_t = \kappa(\theta-r_t)dt + \sigma dW_t$. I think that the distribution of $D(t,T) := \int_t^T r_s\;ds$ is normal (I ...
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15 views

stochastic integration with respect to quadratic variation

I have been studying stochastic integral with respect to Brownian motion. At some point my professor generalized our approach such that we are able to integrate with respect to general Martingales. ...
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0answers
43 views

Expectation of e^(cX) if X is a geometric Brownian motion

(Edit:) The short version: Calculate $$E[e^{cY}]$$ if $c < 0$ and $Y$ is lognormally distributed, i.e. $\log(Y) \sim N(\tilde\mu, \tilde\sigma^2)$. The long version: I want to calculate ...
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1answer
18 views

Comparison between these Ito Lemma versions

According to wikipedia : I found another version : Please explain the difference for me.
2
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1answer
25 views

upper bound for Ito integral of deterministic integrand

It is well known that Ito integrals with respect to a Brownian motion cannot be defined pathwise because the Brownian motion has infinite 1st order variation. These integrals are defined as limits of ...
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0answers
12 views

Does an integrable IID continuous time stochastic process exist?

Let $t\in[0,T)$ where $0 < T \leq \infty$, and assume a stochastic process exists $Z_t$. The question is: does there exist an IID stochastic process for $Z_t$ such that $Z_t \perp Z_{\tau}$ for ...
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1answer
18 views

Ito integrals and the Euler scheme

I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ...
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1answer
39 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
4
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1answer
307 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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0answers
18 views

Milestein Scheme

Im struggling in the following schemes. I cant understand how the first scheme is equivalent to the second one. Can somebody help me? Thanks in advance. Moreover there is a typo error in the ...
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0answers
69 views

Expected value of correlated stochastic integrals

I do not understand the following result: Suppose $dz_\chi$ and $ dz_\xi$ are correlated increments of standard Brownian motion with $dz_\chi dz_\xi=\rho dt$ you have the following expectation ...
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49 views

Why the Ito isometry implies this equality? [duplicate]

If $${\rm Cov}[dW_t,dB_t]=\rho \, dt$$ then why $\mathbb{Cov} \left( \int_0^t \sigma_{1}(s) \mathrm{d} W_s, \int_0^t \sigma_{2}(u) \mathrm{d} B_u \right)$ $\stackrel{\text{Ito isometry}}{=} ...
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0answers
33 views

The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
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1answer
208 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
30 views

Evaluating Stratonovich integral from definition

I am struggling to evaluate the integral $\displaystyle \int^{T}_{0} B_t \circ dB_t $ from definition. So far I have that $\begin{align} \displaystyle \sum ...
2
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1answer
527 views

Stochastic Calc

(a) Consider the process $$ \mathrm d\sqrt{v} = (\alpha - \beta\sqrt{v})\mathrm dt + \delta \mathrm dW $$ Here $\alpha, \beta,$ and $\delta$ are constants. Using Ito's Lemma show that $$ \mathrm dv = ...
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1answer
513 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
19 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
3
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1answer
261 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
47 views

What is the distribution of this random variable? [closed]

Find the distribution of this random variable: $$X_t=\exp\left(t \int_0^t sdW_s\right)$$ knowing that $W$ is a Brownian motion in the filtered space $(\Omega, \mathcal{F},P,(\mathcal{F}_t)_{t\geq0} ...
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26 views

Matlab code for Simulation of SDE [duplicate]

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
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1answer
33 views

Solution to SDE using Itô calculus

So if I have the following generator and an initial condition: $$A(f)(x) = \alpha x f'(x) + f''(x) \\ X_0 = x \in \mathbb{R}^+$$ I've been asked to find $X_t$ and assume that $\alpha$ is a constant. ...
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2answers
111 views

Matlab Code to simulate trajectories of Ito process.

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
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1answer
92 views

Itô's formula and SDE

Solve this SDE: $dX_t=\frac{1}{2}\sigma(X_t)\sigma'(X_t)dt+\sigma(X_t)dW_t$ with $X_0=x_0$ My try is let $f(x)=\int_{x_0}^{x}\frac{dy}{\sigma(y)}$ and $(f^{-1})'=\sigma(x),(f^{-1})''=\sigma'(x)$ ...
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0answers
43 views

Stochastic Differential equations with $\sin(x^2)$ as drift.

Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance. ...
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1answer
39 views

$dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u $

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] $. As far as I can see though, Ito's formula says ...
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1answer
42 views

integral approximation (law of large numbers)

I am totally at a loss with this question and don't even know where to begin. Let $g:[0, 1]\rightarrow \mathbb{R}$ be a measurable and Lebesgue-integrable function. $U_1, U_2, \dots$ be a series of ...
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0answers
40 views

An exponential martingale

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
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2answers
65 views

How to solve SDE

SDE: $dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T$ $X_0=a$ answer Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$ and ...
3
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0answers
67 views

Multipe Ito Integrals

Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ...
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0answers
76 views

Write the Hamilton Jacobi Bellman equation

Consider the following stochastic optimal control problem. \begin{equation} V(t,x) = \max_{u}\,\, \log \left(\mathbb{E}\left[\int_{0}^{T} u^{2}(t)dt\right]\right) \end{equation} subject to the ...
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1answer
32 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
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32 views

partial derivative of stochastic variable inside an integral

Very simple question, is it correct to take a partial derivative of stochastic variable inside an integral. If not, why? is$ \frac {\partial}{\partial R} \int_q^Q R(v) dv = \int_q^Q dv$ ? where R is ...
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2answers
353 views

Variance of stochastic integral of brownian motion

How do i compute this integral? $ Var [\int_0^T W(t)dW(t)] $ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
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2answers
62 views

Show $E[h(X)] = \int_0^{\infty} h'(t)P[X>t]dt$ and the first two moments

Let $X\geq 0$ be a real random variable and $h:\mathbb{R} \rightarrow \mathbb{R}$ a monotonously growing, continuously differentiable function with $h(0)=0$. Show: $E[h(X)] = \int_0^{\infty} ...
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1answer
95 views

martingale and stochastic Integral

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

Area enclosed by 2-dimensional random curve

Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
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3answers
1k views

Itō Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itō integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
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0answers
22 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 ...
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0answers
20 views

Expected value of solution of SDE

Is there any way to find expectation of $X_t$ defined by the following SDE? $dX_t = -[\sin(2X(t)) + \frac{1}{4}\sin(4X(t))]dt + \sqrt{2}\cos^2 x dB(t), X(0)=1, t \in [0,\tau),$ where $\mathbb{B}$ is ...
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1answer
20 views

Question on Ito Isometry and bounds of integration

I am trying to find the variance of $\int_t^T(T-s)~dW_s$ I was wondering if this approach is correct: $$ Var~(\int_t^T(T-s)~dW_s~)=\mathbb E~[~(~\int_t^T(T-s)~dW_s~)^2~]=\mathbb ...
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1answer
35 views

Stochastic differential equation with trigonometric functions

I heard that the following SDE can be solved analitically by substitution: $dX(t) = - \left[ \sin (2 X(t) ) + \frac{1}{4} \sin (4 X(t) ) \right] dt + \sqrt{2} \cos^2 X(t) dB(t),$ $X(0) = 1, \; t \in ...
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1answer
160 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: ...
2
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0answers
90 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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2answers
134 views

Simple stochastic differential equation

Solve the following stochastic differential equation: $$ dX_t=X_t\,dt+dW_t. $$ Thank you very much for help! I even don't know where to start...
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5answers
321 views

Why do people write stochastic differential equations in differential form?

I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ...
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1answer
166 views

how to do such stochastic integration $dS = a S^b dt + c S dW$?

How to do stochastic integration $dS = a S^b dt + c S dW$, where $a$, $b$ and $c$ are constant, $b > 0$, and $W$ is the Wiener process. I know how to do integration for $dS = aS dt + cS dW$, or ...
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1answer
109 views

$dX_t=1_{X_t\not=0} dW_t$

Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $ how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss Indeed Consider the stopping time $$ ...
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0answers
22 views

Question about a Bessel process

Are there any explicit path solutions for a 3 dimensional Bessel process? E.g. the Ito SDE: $$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$ where $W_t$ is a standard Wiener process.