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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2
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1answer
112 views

Conditional expectation brownian motion

Somebody has an idea on how to tackle this quantity $$\mathbb{E}_{W_T}\left[ \frac{\int_0^T e^{\alpha W_t} dt}{\int_0^T e^{-\alpha W_t} dt + \int_0^T e^{\alpha W_t} dt} \right]$$ For $\alpha \in ...
0
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0answers
25 views

Evaluation of $\mathbb E[\int _{t_1} ^{t_2} f(s, X_s^{t,x} )ds \mid \mathcal F _{t_1} ]$ for a markovian SDE solution.

Given a probability space $(\Omega, \mathcal F , \mathbb P)$, a filtration $\mathbb F = (\mathcal F _t )_{t\geq 0}$ and $\mathbb F$-adapted brownian motion $W=(W_t)_{t \geq 0}$, consider $X^{t,x}= ...
1
vote
0answers
68 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
5
votes
1answer
90 views

Expectation of a stochastic integral

Let $M$ be a right-continuous local martingale, $s,t$ two times (stopping times, if you like). Under what conditions does the following hold: $$E\left(\int_s^t X \, dM\mid\mathcal{F}_s\right)\le 0$$ ...
2
votes
1answer
105 views

Oksendal SDE book mistake?

I am reading through Oksendals SDEs. I think there may be a mistake in question 5.18b and I can not find an errata so I was looking for some confirmation. The problem concerns the following SDE ...
1
vote
2answers
172 views

Prove that integral is a Gaussian random variable, compute its mean and variance

I have to prove that $X_t=\int_0^t W_s ds$ is a Gaussian random variable. I need also to compute it's mean and variance. My attempt: Let $W_t$ be a simple adapted process ...
1
vote
1answer
93 views

Limit of stochastic integrals?

Let $(W_t)t$ be a Wiener process. I want to find the limit for $\epsilon\to 0$ of $$\frac{W_t^2}{2\epsilon}\chi_{(-\epsilon,\epsilon)}(W_t)-\int_0^t ...
1
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1answer
37 views

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 $W$. ...
2
votes
2answers
57 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 $e^{bt}$ ...
0
votes
0answers
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 ...
1
vote
2answers
348 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) = ...
0
votes
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) ...
0
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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.
4
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0answers
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 ...
1
vote
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 ...
2
votes
0answers
124 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. ...
0
votes
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 ...
1
vote
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, $t \to B_t(\omega)$ ...
-1
votes
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.
0
votes
2answers
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... ...
2
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
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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 ...
3
votes
0answers
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?
1
vote
1answer
62 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 ...
2
votes
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 ...
2
votes
1answer
166 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 ...
5
votes
1answer
296 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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
2answers
153 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 ...
0
votes
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 ...
2
votes
0answers
64 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} ...
1
vote
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 ...
1
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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.
0
votes
1answer
54 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
votes
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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0answers
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 ...
4
votes
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 ...
5
votes
1answer
202 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 ...
0
votes
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 ...
1
vote
1answer
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 ^^
2
votes
1answer
233 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 ...
2
votes
1answer
123 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: ...
1
vote
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 ...
0
votes
1answer
77 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 ...
0
votes
0answers
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 ...