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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Stratonovich integral of $\sin(W^2)$

I have to solve the following Stratonovich integral: $$\int_{0}^{t}\sin(W^2_s)\circ{dW_s}$$ First of all I use the conversion from Stratonovich to Ito, obtaining ...
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1answer
120 views

derivative of expected value of maximum of two stochastics variables (iid)

I need to optimize an expected value of a maximum value for $q$. The problem has three variables, $q$ is a constant and $D_1$ and $D_2$ are stochastic variables with pdf $f(x)$ and cdf $F(x)$. The ...
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0answers
152 views

How to solve this SDE ? stuck half way

Problem: $dX_t = \sigma X_tdB_t$, $X_0=x$. $dY_t=X_tdt-Z_tdt$ find $Y_t$, where $Z_t$ is a control and $B_t$ is standard Brownian motion. My attempt: From Ito's lemma, $\partial_BX_t=\sigma X_t$, ...
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1answer
45 views

Sufficient condition for time-changed quadratic covariation to vanish in probability

Let $(M_t^n)_{t \geq 0}$ be a sequence of continuous martingales of the form $M^n_t = \int_0^t X^n_s \, dB_s$ where $B_s$ is a Brownian motion. Let $\tau^n(t)$ be the time change associated to $M_t^n$ ...
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1answer
89 views

Independence of stochastic process $(dB_1t)(dB_2t)$=0?

What does it mean (definition) for two stochastic processes to be independent? like two independent Brownian motion $B_1(t), B_2(t)$. I come across this when I saw a solution of a problem says if ...
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24 views

Differential of $ \int_{0}^{t} e^{\int_{s}^{t} \sigma(\tau)dW(\tau)+(r(\tau)-\frac{1}{2}\sigma(\tau)^{2})d\tau} c(s)ds $

I think -- using the chain rule -- it's $$ e^{\int_{t}^{t}\cdots d\tau} c(t)dt \cdot e^{\int_{s}^{t} \sigma(\tau)dW(\tau)+(r(\tau)-\frac{1}{2}\sigma(\tau)^{2})d\tau}\cdot ...
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1answer
67 views

Is this a Brownian motion

I am learning SDE, and here is some basic things I have trouble with, Let $B(t)$ be a Brownian motion, and $F \in \mathcal L^2$ is any stochastic process and I know $\int_0^tF(s)dB(t)$ is Ito process ...
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1answer
286 views

Ornstein-Uhlenbeck process and Markov property

There isn't a similar question in the forum, so here it goes. Firstly, let the O-U velocity process be defined as $$ dV_t = -\beta V_t dt + \sigma dB_t $$ with $V_0 = v$, and $B = (B_t), t \geq 0$ a ...
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2answers
108 views

Expectation of this stochastic process

Let a stochastic process $X(t)= \int_0^t \operatorname{sign}(B(s)) \, dB(s)$, now how to show that $\Bbb E[B(t)X(t)]=0$ ? here $\operatorname{sign}(x)=-1$ for $x<0$, and $1$ otherwise. $B(t)$ is ...
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1answer
176 views

Poisson integral and discontinuous martingale (Ito-Levy formula)

Consider compounded Poisson process $P$ given by $P_t = \int_0 ^t \int _{\mathbb R}z~ N(dr,dz)$ where $N$ is a Poisson random measure of intensity $dt \otimes \nu$ and $\nu $ is a Levy measure. Why ...
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1answer
88 views

Show that $Y=\int_0^1f(s)B_s \, ds$ is normal and find $\text{var}(Y)$.

$B_t$ is a standard Brownian motion, $f(t)$ is a continuous function on $[0,1]$. $Y=\int_0^1f(s)B_sds$. How to show $Y$ is normal. And what is the variance? I know I can use characteristic function ...
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3answers
152 views

Show that $E(Y\mid X=x)$ is a linear function in $x$

Let $Y$ and $X$ be bivariate normal distributed with expectationvector $\mu=(\mu_Y,\mu_X)^T$ and covariance matrix $\Sigma=\begin{pmatrix}\sigma_Y^2 & p_{XY}\\p_{XY} & ...
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1answer
56 views

Elementary Malliavin Derivative question about definition.

I am reading a book that defines the Malliavin derivative $D_tF$ as follows: If $F = \sum_{n=0}^{\infty} I_n(f_n)$ is the Wiener Chaos expansion. $F$ is in the brownian filtration and $F \in ...
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1answer
39 views

Derived Point Process

Can anyone give me some hint on the following problem? Thanks a lot! Let $\{T_n:n\ge 0\}$ be a point process and $\{N_t: t\ge 0\}$ be the corresponding counting process which admits a bounded ...
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1answer
469 views

Variance of Ito Integral

I want to find the variance of the Ito integral: $X(t)=\displaystyle \int_0^t\sqrt{s}WdW$ where W is a Brownian motion and s is the variable of integration. This is what I have done so far: ...
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1answer
69 views

Determining $dX_t$ for stochastic equations, and which of these are $\mathcal{F} $ - martingales?

I want to write down an expression for $dX_t$ for both: i. $X_t=t^2W_t^2-2\int_0^t(sW_s^2+s^2)ds$; and ii. $X_t=W_t^2-tW_t$ What is the process I would use for differentiating these stochastic ...
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0answers
70 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.
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1answer
64 views

$\mathbb{E}[B_t-B_s], \mathbb{E}[\exp(\sigma(B_t-B_s))]$ etc.

This may be a duplicate but I cannot find the corresponding question. I have been asked to show: $\mathbb{E}[\exp(\sigma(B_t-B_s))] = \exp\left(-\dfrac{\sigma^2}{2}(s-t)\right)$ As a side note I ...
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0answers
102 views

Mean-value like result for stochastic integrals

I'm working through Protter's book on stochastic integration; this is problem 16 from chapter 2. I can't seem to crack it--maybe someone here can give me a hint? Let B be standard Brownian and H be a ...
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1answer
100 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
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1answer
43 views

Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
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1answer
112 views

Ornstein-Uhlenbeck process written explicitly

I need to show that the Ornstein-Uhlenbeck process, $$ dX_t = -\theta X_tdt + dB(t) $$ Where $X_0=0$, $B(t)$ is Brownian motion and $\theta>0$ can be written explicitly as: $$ X_t=B(t) - \theta ...
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0answers
43 views

Weak stochastic integral

I recently encountered the following object, referred to as "weak stochastic integral" in the book of SPDE's by Prévôt/Röckner [PR07]: $$ \int_0^T \langle \Psi \,\mathrm dW(t), \Phi(t)\rangle $$ A ...
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1answer
43 views

Stochastic integral with respect to a stochastic integral

[From Bass R.F. Stochastic processes. Exercise 10.4] Let $N_t = \int_0^tH_sdM_s$ where $M$ is a continuous square integrable martingale and H is predictable and integrable and $L_t = \int_0^tK_sdN_s$ ...
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1answer
103 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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0answers
75 views

Ito formula for $f(X_t, Y_{t-s})$

I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function $f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}$. How do I apply Ito lemma in this case?(is Ito lemma still ...
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1answer
259 views

Use Ito's Lemma to show:

I am somewhat unsure how to go about showing this: Use Ito's Lemma to show for any deterministic differentiable function, $f$: $$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$ Where $B(t)$ ...
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1answer
106 views

Calculate Stochastic Integral

I found the following integral $\int_{0}^1 B_t t^{-1}dt,$ where $B_t$ is a standard Brownian motion. Using Ito formula with $f(t,x)=x\log(t)$ I achieved $$0=\log(1)B_1=\int_{0}^1 B_s s^{-1}ds ...
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1answer
154 views

Integrating brownian motion times exponential function

I am trying to calculate $$\int_0^tB_se^{\lambda s}ds$$ but I am unsure of how to start the computation. The motivation behind this is that I read (and am now trying to prove) that ...
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200 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
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1answer
60 views

question about martingale

In my lecture notes,I found the following problem: Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ...
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1answer
48 views

Quadratic Variation for $X_t= \int \sigma_s dW_s$ where $\sigma_s \in S$

Let $\sigma_s \in S$. Setting $X_t=\int^t_0 \sigma_s dW_s$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ such that $d_n=\max_i |t^n_{i+1}-t^n_i| \rightarrow 0$ as $n \rightarrow ...
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2answers
289 views

Conditioning on a random variable

The number of storms in the upcoming rainy season is Poisson distributed but with a parameter value that is uniformly distributed between (0,5). That is Λ is uniformly distributed over (0,5), and ...
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666 views

Expectation of a Poisson Process

Cars pass a certain street location according to a Poisson Process with rate $\lambda$. An old lady and her trusty boyscout want to cross the street at this location. They wait until they can ensure ...
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0answers
48 views

Proving $(\int_0^t f(X_s) dW_s)_{t \in [0T]}$, $f$ a $k$-Lipschitz function, is a continuous martingale

Consider $X =(X_t)_{t \in [0T]}$ progressively measurable with $X_t \in \mathbb L^p, \forall t \in [0,T]$ for $p\geq 1$ and $f$ a $k$-Lipschitz function. I would like to show that $(\int_0^t f(X_s) ...
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1answer
56 views

Clarification about a very simple stochastic integral

I'm studying stochastic integrals right now and I feel like this question is incredibly easy but I'm not sure. I want to evaluate $\int_0^t sdB_s$. Using Ito's formula I get $tB_t$ by setting ...
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1answer
627 views

How to calculate this easy stochastic integral?

I have a relatively simple homework for stochastic calculus that I recently started to learn. I cannot seem to calculate the following integral: $$ \int_0^t s dW_s $$ In principle, it should be solved ...
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37 views

markov spectral radius independent of states?

Let $\Pi$ be a stochastic matrix of an irreducible markov chain. We define the spectral radius of $\Pi$ as: $\rho(\Pi) := \limsup_{n \to \infty} \left( \pi^{(n)}_{(a,b)} \right)^{\frac{1}{n}}$ Why ...
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1answer
73 views

Path Continuity and Stochastic Integration

In a book I'm working through there is a proof that $$\int_0^{\tau(\omega)\land t}f(\omega,s)dB_s(\omega) = \int_0^t f(\omega,s)1\{s\le \tau(\omega)\}dB_s(\omega)$$ The proof begins by claiming that ...
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1answer
135 views

Stochastic Integration and Ito Calculus

Before reading this I must not I think I am a little behind on some of the prereq for this topic but I really want to be able to understand it in a relatively meaningful way. I am having trouble ...
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1answer
145 views

Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...
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1answer
46 views

$\int_t^T 1_C\cdot A\;d\!X=1_C\cdot\int_t^T A\;d\!X$ for $C\in\mathcal F_t$?

Given a semi-martingale $X$ on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$, an integrand $A$ and a set $C\in\mathcal F_t$. Show: $$\int_t^T 1_C\cdot ...
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65 views

If two stochastic integrands are equal on some measurable set, will the stochastic integrals be equal on that set?

Given a $X$ semi-martingale on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$ I am trying to prove: For any $B\in\mathcal F_\infty$ and processes $a_1,a_2$ such that ...
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1answer
63 views

Stochastic Integrals and Martingales

I am attempting the following proof but two aspects of the solution confuse me: Given \begin{align} I^{n}_{t} = \int^t_0 \Delta_u^ndW_u = \sum_{j=0}^{k-1}\Delta_{t_{j}}(W_{t_{j+1}}-W_{t_{j}}) + ...
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0answers
152 views

Expected value of stopping time of Stochastic Process.

I am trying to solve the following problem: Let $X$ be the strong solution of the following Stochastic Differential Equation: $\mathrm dX_t = sign(X_t)dt + \mathrm dW_t, X_0 = 0$, where $W_t$ is a ...
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0answers
220 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
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0answers
60 views

Differential of the integral of a stochastic process

In the HJM model one considers the forward rates to be on the form $$\mathrm df(t,T) = \alpha(t,T)\,\mathrm dt + \sigma(t,T)\,\mathrm dW(t)$$ In the proof of showing the drift condition on $\alpha$ ...
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1answer
69 views

Computation of a simple stochastic integral

For $t \in [0,T]$. consider two stochastic integrals with a nonnegative constant integrand $c$ $$\mathbb{E} \left[ \int_0^{t(\omega)^* \wedge T} c \cdot dW_t \right]$$ where $t^*$ is random ...
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1answer
117 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
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1answer
21 views

Expectation of a stochastic integral conditioned on a particular σ-algebra

Suppose that $g$ is a simple process in the class $\mathcal{V}=\mathcal{V}[U,T]$. Using the notations $g_k=g(t_k)$, $\Delta B_k = B(t_{k+1})-B(t_k)$, and $\mathcal{F}_k=\mathcal{F}_{t_k}$, with the ...