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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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16 views

Stochastic integration rules

I stumbled upon a stochastic integration result in a book and I'm not sure how it is derived. It goes as follows: Suppose the process $Y^{(t,y)}$ has dynamics under the equivalent martingale measure $$...
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0answers
9 views

Deterministic Integral of a Predictable Process is Predictable

I was reviewing a proof of existence of solutions to stochastic evolution equations which takes the form of a fixed point argument on the space of predictable processes such that $$ \sup_{t\leq T}\...
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18 views

Application of Gram-Charlier expansion for Swaption pricing with drift extension

I am doing a project for university and I'm stuck at some point. The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension. I already found out how ...
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15 views

Derive a SDE for process X [closed]

Question: Define a process $X$ by $X_t = W_t^2 - 2\int_0^t uW_udu$. Derive a SDE for $X$. I know to solve this problem I need to apply Ito. However, how can I deal with the integral $2\int_0^t ...
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1answer
31 views

Stochastic differential equation with respect to general stochastic process

I am a beginner with stochastic processes and I want to clarify some ideas that I just learnt. Say I have a stochastic process: $$X_t = X_0+ \int_0^t \sigma_s dW_s $$ where $W_s$ is a standard ...
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24 views

Show that stochastic integral is Gaussian by independency of increments

Let $h \in L^2[(0,1)]$ and consider the process $(X_t)_{t \in [0,1]} = \big( \int_0^t h(s) \text{d}B_s \big)_{t \in [0,1]}$, where $B_s$ is Brownian motion. By construction of the integral I know ...
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17 views

Apply the usual version of Itō formula to get an expression of the stochastic process {$|X_t-x|,t∈[0,t]$} as an Itō process?

Consider an Itō process $X_t=x+\int_0^tφ(s)dBs+\int_0^tφ(s)ds$. Could you apply the usual version of Itō formula to get an expression of the stochastic process {$|X_t-x|,t∈[0,t]$} as an Itō process? ...
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17 views

How would one solve for a process where a stochastic random variable is divided by a deterministic random variable?

I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its ...
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1answer
90 views

Show a Continuous Local Martingale is a Martingale

Let $B = (B_t)_{t≥0}$ be a standard Brownian motion started at zero, let $X=(X_t)_{t≥0}$ be a nonnegative stochastic process solving $$dX_t = 3 \, dt + 2\sqrt{X_t} \, dB_t \qquad(X_0 = 0)$$ and let $$...
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26 views

How to obtain the relationship of beta function [on hold]

Define $B ( \mu , v ) = \int _ { 0 } ^ { 1 } x ^ { \mu - 1 } ( 1 - x ) ^ { \nu - 1 } \mathrm { d } x = \frac { \Gamma ( \mu ) \Gamma ( \nu ) } { \Gamma ( \mu + \nu ) }$. For $u,v>0$ and $c>1$, ...
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28 views

Covariation of two processes

Consider the processin $[0,T]$, $Y_t=\int_0^t f(s,\omega)dW_s$, where $\int_0^t E[f(s,\omega)^2]ds<\infty$ and $W$ is a BM. Consider also, $X_t = \int_0^t M(s,\omega)ds$, where $M$ is some ...
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20 views

Integral representation of a stochastic differential equation

If I have a stochastic differential equation gives as $$ dX_t = aW_t^2dt +bW_t^3dW_t$$ where $w_t$ is a wiener process and $a,b$ are real numbers. How can I reach the integral representation of $X_t$? ...
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1answer
58 views

Maximal inequality for the Itō integral

Let $W$ be a Brownian motion and $X$ be a predictable process with $$\operatorname E\left[\int_0^t|X_s|^2\:{\rm d}s\right]<\infty\;\;\;\text{for all }t\ge0.$$ Now, let $p\ge2$. How can we show ...
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47 views

Is the strong solution of a SDE adapted to the filtration generated by the driving Brownian motion?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $\xi$ be an $\...
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2answers
51 views

Markov semigroup for normal distributed kernels

Let $\alpha,\sigma^2>0$. I want to show that the kernels defined by $$K_t(x,\cdot):=\mathcal{N}\big(xe^{-\alpha t},\frac{\sigma^2}{2\alpha}(1-e^{-2\alpha t})\big)\quad\text{for}\quad t>0$$ $$...
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22 views

If we want $\int_0^⋅XdM$ to be a martingale, do we need to assume $E[\int_0^tX_sd[M]_s]<∞$ for all $t$ or even $E[\int_0^∞X_sd[M]_s]<∞$?

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

Ito's formula proof - why can we assume $u(t,\omega), v(t,\omega)$ are elementary?

My question is about a simplification made in the proof of Ito's formula. In the proof of Ito's formula in my textbook it says that since $\int_0^T f dBs$ is defined as $$\int_0^T f dB_s = \lim_{n\...
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34 views

Bessel process time change

Let us consider a Bessel process $$dX_t = \frac{\mu - 1}{2 X_t} dt + dW_t$$ where $\mu > 1$ is constant (dimension) and $W_t$ is a standard Wiener process. Then I denote the additive functional $Y ...
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0answers
31 views

Find constants $a$ and $b$ such that $X(t)$ is a Brownian Motion

Let $B(t)$ be a Brownian Motion. Find all constants $A$ and $b$ such that $X(t)=\int_0^t(a+b\frac{u}{t})dB(u)$ is also a Brownian Motion. First we know that if $f \in L^2[a,b]$ then $\int_a^bfdB(u)$...
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1answer
23 views

Dynkin's Theorem and expectation

Suppose I have the following SDE. $dX_t=-k\cdot X_tdt+\sigma\sqrt{X_t}dB_t$ If I want to find a bound at any $t$ ofthe expectation of $X_t^2$, given $X_0=0$, is it legitimate to do the following? I ...
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29 views

Dynkin formula and expectation

I have a question about the use of Dynkin's formula. Suppose I have a stochastic process $X_t$, from an SDE that I cannot solve analytically. I want to find the expectation $\mathbb{E}X_t$. By Dynkin ...
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1answer
40 views

How to solve non-linear stochastic differential equations

In general, we use Ito's formula to solve linear stochastic differential equations. Consider for instance the geometric brownian motion: $$dX_t = \alpha X_t dt + \beta X_t dW.$$ My question is: ...
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1answer
47 views

Show that stochastic integral w.r.t. Brownian motion is normal distributed

I want to show the following claim: Let $B$ be a one-dimensional Brownian motion and let $$I(\phi):=\int_0^1 \phi(s) \text{d}B_s.$$ Show that $\mathbb{E}(I(\phi))=0$ and $\mathbb{V}(I(\phi))=\int_0^1(...
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0answers
37 views

Differential of Stochastic process

Given a stochastic process $$Z_t = e^{4t} \int_0^{t} e^{-2s} \, \,dB_s$$ where $B$ denotes the standard Brownian Motion. Determine $dZ_t$. I tried to make use of Ito's rule, seeing that $Z$ is a ...
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1answer
25 views

Finiteness of exponential moments of a stochastic integral implies finite moments

Background: Hello everybody. I'm reading the paper Optimal Portfolio under Fractional Stochastic Environment by Fouque and Hu. The proof of Proposition 2.2 contains an estimation (Abschätzung) of an ...
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1answer
74 views

Why is $dW^2=dt$ in stochastic calculus?(do not use Ito’ Lemma),

I'm trying to calculate the following integral: $$\int_0^t W_tdW_t$$ Without using Ito’ Lemma. I am confused about how $dW^2=dt$ when Ito's Lemma is not used. Hint Let W be a standard Wiener ...
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1answer
44 views

Why is an elementary Ito integral necessarily continuous?

So I am working/reading through a proof that general Ito integrals have continuous versions. So that $$I_f(t) = \int_0^t f(s,\omega)dB_s(\omega)$$ Has a continuous version in $t$. The proof I am ...
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0answers
24 views

filtration of a stochastic integral

1) Consider the Ito-integral: $S_t = \int_{0}^{t}f(s)dW_S$, where $f$ is a Borel bounded function and $W$ is a Brownian motion. Is the $\sigma$-algebra, $\sigma(S_t) = \sigma(W_s, s\leq t)?$ For ...
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1answer
77 views

Solve the SDE $dX_t=\sqrt t(X_t+\sin t)dW_t$

I am new to stochastic differential equation and ran into a question of solving $$dX_t=\sqrt t(X_t+\sin t)dW_t$$ where $W_t$ is the standard Wiener Process and $X_0 \equiv K\in \mathbb R$. I know Ito'...
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12 views

Conditional expectation of correlated processes

Consider the known $C^1$ functions $f^1, f^2$ and the continuous semimartingales $X^1,X^2,S^1,S^2$ (solutions of a non-linear SDE). Suppose that $X^i$ is correlated to $S^1$ and $S^2$ with correlation ...
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14 views

Brownian motion in random landscape

I have some trouble with an exercice: Take $(W_t)_{t\ge0}$ and $(B_t)_{t\ge0}$ two independent standard Brownian motions started from 0 and define the process $$X_t=\int_{\mathbb{R}}\sqrt{L_t^x(B)}...
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1answer
24 views

Stochastic integral with Poisson random measure

The following is what I read in paper and I am confused by some parts. We consider a one-dimensional Itô semimartingale $X$ which is defined on some probability space $(\Omega,\mathcal F,\{\mathcal ...
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0answers
24 views

Interpolation of martingale

Consider a non-constant martingale $(M_n)_{n\ge 0}$ with the filtration $(\mathcal{F}_n)_{n\ge 0}$, where $\mathcal{F}_n=\sigma(M_m:m\le n)$. Define for $t\in\mathbb{R}_{\ge 0}$: $$X_t:=M_{\lfloor ...
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27 views

Stochastic caculus

Suppose that $dX_{t} = -aX_{t}dt + \sigma dW_{t}$ ,find the following limit of expection $lim_{t \to \infty }E[X_{t}]$
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43 views

Covariance of two Ito / Diffusion processes

Let $B_t$ denote the standard Brownian motion process. $X_t$ and $Y_t$ are Ito diffusions with the following SDEs: \begin{align} dX_t &= \mu(t,X_t) \; dt + \sigma(t,X_t) \; dB_t \\ dY_t &= \mu(...
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0answers
42 views

Find the distribution of $\int _ 0^t W(s)\cos (t-s) ds$ where $W$ is a Brownian motion.

Question: Let $W$ a Brownian motion. If $X(t) = \int_0^t W(s)\cos (t-s) ds$ compute $$P(X(t) \le x)=??$$ Attempt: Since that Brownian motion has continous paths, the Riemann integral $X(t)$ is well ...
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0answers
31 views

Expectation for a $\chi^2_n$ distributed random variable.

Consider $X_1,\ldots, X_n\sim\mathcal{N}(\mu,\sigma^2)$ iid, where $\mu$, $\sigma^2$ are unknown and $c>0$. Define $$S_n^2:=\sum_{k=1}^n\Big(X_k-\frac{1}{n}\sum_{i=1}^nX_i\Big)^2$$ Now I want ...
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0answers
26 views

Comparison of variance of stochastic and non-stochastic integrals of the Brownian motion

Given that $B_t$ is the standard Brownian motion, I need to contrast the mean and variance of the stochastic integral $\int\limits_{0}^t B_s dB_s = \frac{1}{2}(B_t^2 - t)$ with the non-stochastic ...
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1answer
37 views

Brownian motion on the n-sphere

From course notes on SDE's. We consider a Stratonovich equation. $dX_t=\left(I-\frac{1}{|X_t|^2}X_tX_t^T \right)\circ dB_t$ With $X_t\in \mathbb{R}^n$ and $\{B_t\}$ being n-dimensional brownian ...
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1answer
33 views

Conditional expectation of exponent to the power of BM, representation theorem

Let $T > 0$ and $M(t) = E[e^{W(T)}|F(t)]$ where $\{F(t) : t \geq 0\}$ is a natural filtration generated by W and $t \leq T$ I need to show that M(t) is a martingale and I also need to find a unique ...
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0answers
34 views

Law of Ito Integral

Let $\sigma$ be an $\mathcal{F}_t$-adapted caglad stochastic process. Let $W$ be a Brownian motion independent of $\sigma$. Let $r>0$ be a strictly positive real number. How can I prove that $$ \...
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20 views

When does the semigroup corresponding to a stochastic process has the Feller property?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(X_t)_{t\ge0}$ be a real-valued process on $(\Omega,\mathcal A,\operatorname P)$ $E$ be a closed subspace of $\left\{f:\mathbb R\to\...
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1answer
31 views

When is the local martingale in the Itō formula a (strict) martingale?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $W$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal ...
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1answer
24 views

$\mathbb E[S_n'^4]\le C\cdot n^2$

Let $X_1,...,X_n$ be iid random variables on $(\Omega,\mathcal A,\mathbb P),\ \ \mathbb E[X_1]=\mu\in\mathbb R,\ \ \mathbb E[X_1]^4<\infty,\ \ X_i':=X_i-\mu,\ \ S_n'=X_1'+...+X_n'.$ Prove that $\...
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0answers
41 views

Stochastic integral involving Wiener process

Suppose $W_t$ is the standard Wiener process. I am wondering whether I can use the independent increment property of the process as well as the fact that $W_t$ is zero-mean Gaussian to evaluate ...
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0answers
12 views

Order in probability of a ratio between two integrals

Suppose that $\mu$ is an adapted bounded stochastic process and suppose that $\sigma$ is an adapted bounded left-continuous stochastic process. I want to prove that $$ \frac{\int_0^{\Delta}\mu_s\,...
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0answers
61 views

Show that the solution of an autonomous SDE is a time-homogeneous Markov process

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space and $$\mathcal N:=\left\{N\in\mathcal A:\operatorname P[N]=0\right\}$$ $(W_t)_{t\ge0}$ be a Brownian motion on $(\Omega,\...
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1answer
45 views

Ito integral exponent of Brownian Motion

Let $F=e^{B_t}$, find such a process $f_s$ that $F=E[F]+\int_0^tf_sdB_s$. I have started with $$e^{B_t}=E[e^{B_t}]+\int_0^tf_sdB_s$$ We know that $E[e^{B_t}]=e^{t/2}$ and it gives us $$e^{B_t}=e^{t/...
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0answers
20 views

Simple example for a predictable process that is not in $L(X)$ for a semimartingale $X$

I am supposed to find some simple and basic examples for semimartingales $X$ and predictable processes $H$ such that $H\notin L(X)$, that means $H$ is not integrable with respect to $X$. Do you have ...
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0answers
33 views

Ito Lemma and identifying martingale parts

Suppose that $X_t$ is a càdlàg semi-martingale with decomposition $$ X_t= X_0+ B_t + M_t. $$ I know that using the Ito lemma for any $C^2$-function $f$, $$ f(X_t)= f(X_0)\\ + \int_{0^+}^tf_x(X_{s-})...