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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Proof that the image of an Itō integral is convex if the driving Wiener process is in a metric ball

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $A := \int_0^1 f(t)\,d W_t$ be the Itō integral of an $L_2([0,1])$ deterministic function $f$ with respect to the Wiener process $W$. ...
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2answers
55 views

Show that process satisfy given equation

I have to show that process (1) $$X_t=e^{-bt}X_0+\int_0^te^{-b(t-s)}\sigma dW_s$$ satisfies the following equation (2) $$dX_t=-bX_tdt+\sigma dW_t$$ My attempt: Multiply both sides of (1) by $e^{bt}$ ...
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31 views

Defining the Radon-Nikodym as a solution to an SDE

Can someone please clarify this to me: If I have the Radon-Nikodym $L_t=\frac{dQ}{dP}$, on $\mathcal{F}_t$, then I know that $L_t$ is a non-negative P-martingale. So in many textbooks they say it is ...
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2answers
340 views

Variance of Time-Integrated Ornstein-Uhlenbeck Process

I'm attempting to filter white noise from a deterministic, finite-power signal using a low-pass filter. This filter can be described using an exponentially-decaying response function: $$ h(t) = ...
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0answers
33 views

Equivalence between solutions to SPDE

Consider the SPDE \begin{equation}\tag{1} \frac{\partial}{\partial t}u_t(x)=\frac{\kappa}{2} \frac{\partial^2}{\partial x^2}u_t(x)+ b(u_t(x)) + \sigma(u_t(x)) \xi (t,x), \end{equation} where $(t,x) ...
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1answer
38 views

Solution to stochastic differential eqn [closed]

How do you solve this stochastic differential equation? Not sure how to start on this. Need some guidance.
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47 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
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1answer
67 views

Solution of two (first) SDEs.

I'm about to study SDE's for the first time and I'm kinda having troubles "guessing"/"finding" solutions. Also I don't really know how and when analogies to simple ODEs are allowed (e.g. to get a ...
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0answers
122 views

When does almost sure convergence of stochastic integral imply $L^2$ convergence?

Consider a probability space $(\Omega, \mathcal{F}, P)$ equipped with a Brownian motion $W$. Let $(\xi_n)_{n=1}^\infty$ be a sequence of adapted $\mathcal{F}(t)$-progressively measurable processes. ...
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1answer
44 views

Expectation of stopping times

Let B = (Bt)t¸0 be a standard Brownian motion started at zero, let $X_t$ be a non negative stochastic process solving: $dX_t=1/X_tdt+dB_t$ Compute $E[\sigma]$ when $\sigma=\inf \{ t\ge 0 : X_t= 1 ...
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1answer
52 views

What is wrong with my example where the Itô Integral and Riemann-Stieltjes Integral don't coincide?

I have an interesting question concerning those two integrals. Considering a Brownian motion $(B_t)_{t \geq 0}$ with start in $x$. We can choose an $\omega \in \Omega$ such that, $t \to B_t(\omega)$ ...
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2answers
50 views

Let $X(t)=(1-t)\int_{0}^{t}\frac{dB(s)}{1-s}$ I want find $dX(t)$ [closed]

Let $X(t)=(1-t)\int_{0}^{t}\frac{dB(s)}{1-s}$, where $0\le t < 1$.Find $dX(t)$. thanks for help.
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66 views

A stochastic integral computed using Itô's lemma

I need some help with this question: I have to check the following "identity" using Itô's lemma, but I can't see how to do it... ...
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2answers
67 views

Distribution of stochastic integral in small time

Let $W^1$ be a Brownian motion and $\sigma(\cdot)$ be a positive, bounded, continuous function. Define \begin{align*} V_t=\int_0^t\sigma(Y_s)dW_s, \end{align*} where $(Y_t)_{t\geq 0}$ is a ...
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1answer
74 views

Ito Integral surjective?

Let $\Phi\in\mathcal{L}\left(M\right)$ if and only if $\Phi$ is a real predictable process and for every $\left\Vert \Phi\right\Vert_{2,t,M}:=\mathbb{E}\left[\int_{0}^{t}\Phi_{s}^2 d\langle ...
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1answer
45 views

Wiener process analytic expression from geometric brownian motion

The solution to the SDE $dx= -kx\ dt + cx \ dW$ is $x(t) = x_0 e^{(c - k^2/2)t}e^{-k W}$ with mean $\langle x(t) \rangle = x_0 e^{(c - k^2/2)t}$ where $W(t)$ is the Wiener process. Im ...
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1answer
95 views

Representing a stochastic integral as product of a unknown random variable and a standard normal random variable

Consider a probability space $(\Omega,\mathcal F, (\mathcal F_t)_{t\geq0},\mathbb P)$ where $\mathbb F=(\mathcal F_t)_{t\geq0}$ is generated by $B=(B_t)_ { t \geq 0}$ a standard brownian motion ...
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41 views

When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?
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1answer
61 views

Problem 3.2.28 of Karatzas and Shreve

It's the Problem 2.28 of Karatzas and Shreve on Page 147: Let $M=W$ be standard Brownian motion and $X\in\mathcal{p}$. We define for $0\leq s<t<\infty$ $$\zeta_t^s(X)\triangleq\int_s^t X_u ...
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2answers
66 views

Ito's Isometry for three factors

Ito's Isometry states the following: If $\{W_t\}_{t\ge0}$ is a Brownian motion and $\{\phi_t\}_{t\ge0},\{\psi_t\}_{t\ge0}$ are two non-anticipative piecewise-continous processes with $\mathbb ...
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1answer
161 views

Kolmogorov Backward Equation for Itô diffusion

Let $(X_t)_{t\ge 0}$ be the solution of the SDE $$ X_t = X_0 + \int_0^t \mu(s,X_s) \,ds + \int_0^t \sigma(s,X_s) \,dB_s, \quad t\ge 0 $$ where $\mu(s,x)$ and $\sigma(s,x) $ are Lipschitz continuous ...
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1answer
294 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
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1answer
36 views

How to calculate the Multiple Stratonovich Integral?

My question is about multiple Stratonovich-Integrals. I have the following Stratonovich-Integral $ \int \limits_{t_n}^{t_{n+1}} \int \limits_{t_n}^{s_1}1\,dW(s)dW(s_1).$ How can I calculate it? Is it ...
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1answer
23 views

A jump process as an integrand in Itô integral with respect to an Itô process

So, $X_1(s)$ is a jump process, $X_2(s)$ is another jump process, $X_2^c(s)$ is the continuous part of $X_2(s)$. And $\int_0^tX_1(s-)dX_2^c(s) = \int_0^tX_1(s)dX_2^c(s)$, is it because the ...
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0answers
38 views

Clarification on the definition of the îto integral

I have a question regarding the îto integral. In the definition of the integral we basically take the limit in probability of the sum $\Sigma H(t_i)\cdot(B(t_{i+1})-B(t_i))$ for suitable $H$ and a ...
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1answer
109 views

I want to calculate $\int B(t)^2 dB(t)$ where $B(t)$ is Brownian motion

Let $B(t)$ be Brownian motion. I want to calculate $\int B(t)^2 dB(t)$. definition.A process $\{X(t),0\le t \le T \}$ is called a simple adapted process if there exist times ...
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152 views

ito vs Stratonovich

I need to sum up the advantages of ito and stratonovich. I often heard, that the Stratonovich integral lacks the important property of the Itō integral, which does not "look into the future". Can you ...
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1answer
41 views

What is the analog Stratonovich SDE to WdW?

i have the Ito-SDE $\int \limits_0^t W(t) dW(t)$ But how can I change this SDE $\int \limits_0^t W(t) dW(t)$ into a Stratonovich-SDE? Normally I do $\underline f=f-\tfrac{1}{2}gg'$. Is the ...
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0answers
61 views

interchange stochastic and deterministic integration

If $f$ is a function in $L^2([0,1]^m)$, W is one-dimensional Brownian motion, $a,b \in [0,1]$, are the following two integrals equal? $$\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2} ...
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1answer
77 views

$\mathbb{E} \int_a^b W^3(t)\,dW(t)=?$

Is it true that $\mathbb{E} \int_a^b W^3(t)\,dW(t)=0$, for $a < b \in \mathbb{R}$ I know that for an adapted process $\Delta(t), t\geq 0$, the integral $\int_0^t \Delta(u)dW(u)$ is a ...
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2answers
74 views

How to show that $\mathbb{E}(\int_0^T t\mathrm \, dW_t) = 0 $?

I just want to know why $\mathbb{E}\left(\int_0^T t \,\mathrm dW_t\right)=0$. I know it's got something to do with the Gaussian distribution but I don't really know what.
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1answer
53 views

stochastic integral and brownian motion

I'm trying to solve a problem similar to Stochastic Integral. I have to evaluate $$ \mathbb{Var}\left(\int_{0}^t ((B_s)^2 + s) \mathrm{d}B_s \right)$$ I have split the problem in two parts: 1) $ ...
2
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1answer
58 views

Optimal Investment Strategy

I am not sure to solve the following investment problem: I have an investor which receives an income $I_n\ge 0$ at the start of year $n$. The investor chooses a proportion $p_n\in[0,1]$ of this in ...
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16 views

quantile of Ito integral when integration limit goes to zero.

I woud like to calculate the Value at Risk of an Ito Integral in the following form in the limit! $$\lim_{\Delta t\to 0}\frac{1}{\Delta t}VaR_{q,t}\left[\int_t^{t+\Delta t}b(s,y(s))\pi_y^c(s,y(s))d ...
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1answer
150 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
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1answer
201 views

Holder continuity of Ito integral

Let $\sigma(t,\omega)$ be a progressively measurable function and $\mathbb{E}[\int_0^T \sigma_t^2\mathrm dt] < \infty$. Can we say that the Ito process $\int_0^t \sigma_s \mathrm dW_s$ is Hölder ...
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1answer
46 views

Cadlag process integration

Let $A,B$ be non-decreasing cadlag processes such that $A_0 = B_0 = 0$ and limits $A_\infty = \lim_{t \to \infty} A_t$ and $B_\infty = \lim_{t \to \infty} B_t$ are finite. I am trying to prove that ...
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54 views

Question about Ito integral

I was wondering if Ito integral: $\int_0^T B(t)dB(t) $ is Gaussian (in which B(t) is Brownian Motion)?? Thank you so much, I appreciate any help ^^
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1answer
220 views

Ornstein - Uhlenbeck Process

I'm considering the Ornstein - Uhlenbeck process $$ V_t = e^{\lambda t} v_o + \int_0^t e^{-\lambda (t-s)} dB_s $$ with $ \lambda > 0$, $v_0 \in \mathbb{R}$, and $B$ a brownian motion. I want to ...
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1answer
122 views

$L^1$ convergence of a sequence of stochastic integrals and convergence of their quadratic variations

On a filtered probability space $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$ containing a Brownian motion $W_t$. Let $\sigma^n_t$ be a sequence of square intergable adapted processes and consider: ...
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1answer
65 views

Weird equality of expectations involving stochastic integral

First of all, sry for the title. I just couldn't figure out any better description for this weird problem: Let $X$ be a bounded real r.v. and $(A_t)_{t\geq0}$ an increasing bounded process (hence ...
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1answer
75 views

Brownian motion and stochastic integration

How do I compute the following expectation? W(T) is a standard brownian motion (i.e.) W(T)~N(0,T) $E\left[ W(T)\int _{ 0 }^{ T }{ sdW(s) } \right] $ I know that Brownian motion of disjoint time ...
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51 views

Canonical semimartigale truncation function meaning

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: $H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
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1answer
79 views

Can an Itō integral be $\infty$?

In other words, can $\int_0^t f(s)dW(s)$ = $\infty$? Thanks!
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119 views

A mean square derivative

I'm doing an exercise where I have to check some properties about these two stochastical processes: $X(t)=At+B\;\;$ and $\;\;Y(t)=\frac{1}{t}\displaystyle\int_{0}^{t}X(\tau)\;d\tau$, $t>0$. ...
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1answer
125 views

Riemann integral of a function of the Wiener process

I'm trying to solve this exercise: $\bullet$ Find mean and variance of the next stochastical process, and prove it is a second order stationary process: ...
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48 views

Atypical exponential martingale

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} ...
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1answer
55 views

Approximation of Stochastic integral with Stieltjes integrals

Let $V^n(t,\omega)$ be a sequence of continuous, adapted and bounded variation processes such that with probability 1, $V^n$ converges to $B$ uniformly on compact intervals of $[0,\infty)$ ($B$ is ...
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1answer
106 views

Applying Ito To Geometric Brownian Motion

I'm trying to understand the example problem on the Wikipedia page for Ito's Lemma and need it dumbed down a little bit. $$dS = S(\sigma dB + \mu dt)$$ $$ f(S) = log(S) $$ Given Ito's lemma, ...
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1answer
180 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...