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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Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv \int\...
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How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
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31 views

Lebesgue-Stieltjes integral and related topics

The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ...
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26 views

Integrating over random boundary

What are some correct stochastic integral notions or theories which make formal sense of the problem of "integrating a function over the boundary of random domain"?
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13 views

Solving the following SDE with a constant

Given is the stochastic differential equation: $\frac{dX(t)}{X(t)}=\mu+\sigma \theta dt+ \sigma dW(t)$, where $W(t)$ is the standard Wiener process and $X(0)=x_0$ I try to solve this by the Itos ...
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1answer
85 views

A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that $$\infty&...
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49 views

Can Stochastic Integration be Further Generalized?

Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ...
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1answer
125 views

Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral $$(H\...
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1answer
16 views

Question about law of substitution in stochastic integral

I need to compute some Integrals for my stochastic course. And i have the following problem: $$ \frac{\lambda^n}{\Gamma(n)} \int_0^{\infty} \exp(-\frac{\lambda}{y}) \frac{1}{y^n} dy = \star$$ so i ...
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2answers
82 views

Integral of Wiener Squared process

I don't have a background of stochastic calculus. It is known fact that definite integral of standard Wiener process from $0$ to $t$ results in another Gaussian process with slice distribution that ...
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1answer
38 views

Marginally Gaussian not Bivariate Gaussian - Ito Integral

Let $(W_t)_{0\leq t\leq 1}$ be a Wiener process defined up to time $1$ on some probability space. Consider the random vector $$\left(W_{1},\int_0^1 \operatorname{sgn}(W_s) \, dW_s\right)=:(W_1,X_1)$$ ...
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1answer
65 views

Covariance of stochastic integral

I have a big problem with such a task: Calculate $\text{Cov} \, (X_t,X_r)$ where $X_t=\int_0^ts^3W_s \, dW_s$, $t \ge 0$. I've tried to do this in this way: setting up $t \le r$ $$\text{Cov} \, (...
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1answer
25 views

Application of Ito's rule

I have that $\sigma$ is a piecewise continuous function on $[0,t]$, $W$ is Brownian motion, $X(t)=\int_0^t\sigma(s)dW(s)$, and $Z(t)= e^{iuX(t)},$ for some fixed $u\in\mathbb{R}$. It is then stated ...
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45 views

Mean and Variance Geometric Brownian Motion with not constant drift and volatility

I have to derive the Geometric Brownian motion (with not constant drift and volatility), and to find the mean and variance of the solution. $\quad \left\{\begin{aligned} & d X_t = \mu(t) X_t d t +...
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2answers
68 views

Discounted price process in Black-Scholes model is a martingale with respect to Q.

I have been presented a proof that the discounted price process in the Black and Scholes formula is a martingale, but there is something important omitted, and I am not able to fill in the gap. I will ...
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13 views

Partial Integration for Semimartingales

Let $X,Y$ be 2 continuous semimartingales. It could be shown that for every $t>0$, \begin{align} X_tY_t = X_0Y_0 + \int_0^t X_s dY_s + \int_0^t Y_s dX_s + \langle X, Y \rangle _t. \end{align} Let ...
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1answer
40 views

Quadratic variation of Stochastic Integral

Let $B$ be a Brownian motion and $M_t = \int_0^t \mathbb{1}_{B_s=0} dB_s$. It can be shown that $(M_t)_{t \geq 0}$ is a local martingale. Now, I want to calculate the quadratic variation of $M$. In ...
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24 views

Problem calculating the conditional expectation of an Ito process

let $X_t$ be an Ito process where $X_t = \int_{0}^t v_t dB_t$ where $v_t$ is a stochastic process, $B_t$ is a Wiener Process, $\mathcal{F}_t$ be a filtration: $\sigma\{B_t, 0 \leq t \leq T\}$, and $...
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1answer
32 views

the exact integrand space for stochastic integral?

I found it in Schilling, Partzsch's textbook "Brownian Motion": only consider in $[0,T]$, they define the Dolean's measure $\mathbb P\times\mu$, and the corresponding $L^2$ norm on $L^2(\Omega\times [...
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1answer
63 views

Itô-formula proof, remainder term.

I have a question about the proof a a certain version of the Itô-formula. First the author defines an Itô-process and states the formula: My question is in regarding the proof. The proof uses ...
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26 views

showing a processes is martingale using ito's lemma

Let $Y(t) = t^2W_t - 2 \int_0^t sW_s \ ds$ where $W_t$ is brownian motion. I am trying to show it is a martingale by showing it is driftless. I set $Z(t,W_t) = t^2W_t$ and ito's gives $dZ = 2tW_t \ dt ...
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How to find the mean of $\int_0^t W_s ds$, where $W_s$ is a Wiener process?

am trying to find the expectation of $\int_0^t W_s ds$, with $W_s$ being the Standard Wiener process. I am trying to use Ito's formula, by decomposing as: $$ \frac{W_t^3}{6} = \frac{1}{2}\int_0^t B_s^...
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1answer
51 views

How to solve for the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$?

I would like to find the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$. My strategy is to use Ito's general formula with: $$ f(t, B_t) = f(0,0) + \int_0^t \frac{df}{dx}(s, B_s) dB_s + \int_0^...
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15 views

For stochastic differential equations, why do we care if the process is $L^2$ bounded?

I have been studying Stochastic Differential Equations, and one theorem relates to the existence of a solution to the SDE: $$ dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dB_t $$ with $X_0 = x_0$ and $0 \...
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2answers
76 views

Showing that this is a martingale.(4.13 in Øksendals SDE)

This is an exercise from Øksendals stochastic differential equations, where I get stuck. It is exercise number 4.13.(I simplified the notation a bit.) I have that X is an Itô-process where: $...
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46 views

Some Kind of Generalized Brownian Bridge

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not ...
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1answer
45 views

Martingale and local martingales

I have to show that $e^{B_t^1}\cos(B_t^2)$ is a martingale ($B=(B^1,B^2)$ is a two-dimensional Brownian Motion). I used Ito's formula and got $e^{B_t^1}\cos(B_t^2)=1+\int_0^t e^{B_s^1}\cos(B_s^2)dB_s^...
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51 views

Approximation of $\int_0^tF_x(s,X_s)Φ_0dW_s$ where $dX_s=φ_sds+Φ_sdW_s$ and $F_x$ is the Fréchet derivative of some $F:[0,t]×H→\mathbb R$

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ be equipped with the usual inner product $(\Omega,\mathcal A,\operatorname P)$ ...
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50 views

Martingal-property of stochastic Integral w.r.t. Brownian Motion

To Show that $(e^{B_t^1}cos(B_t^2))_{t \in \mathbb{R_+}}$ (where: $B=(B_s^1,B_s^2)$ is a 2-dimensional Brownian Motion) is a Martingal I used Ito's Lemma and showed that this is equal to: $ 1+ \int_0^...
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2answers
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Itō formula as presented in “Stochastic Equations in Infinite Dimensions” by Giuseppe Da Prato

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula: Given Hilbert spaces $(U,\langle\;\cdot\;,\;\cdot\;\...
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Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
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30 views

Integrability of a stochastic process

Let $x(t)$ be some random path $t\in[a,b]\subset\mathbb{R}$. I.e. $x:\Omega\rightarrow\mathbb{R}^{[a,b]}$ etc. When is $\int_a^b x(t)dt$ defined? If $x(t)$ is Brownian motion, I know it's ok. A ...
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30 views

Prove that a sum of random variables converges against an Itō integral

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be ...
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27 views

Derive an Itō formula for $f(t,X_t)$ where $X_t=X_0+tY+W_tZ$ and $f:[0,\infty)\times H\to\mathbb R$ and $H$ is a Hilbert space

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be Fréchet ...
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29 views

Itō isometry in Hilbert spaces

Let $U$ and $H$ be separable Hilbert spaces $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $\mathfrak L:=\mathfrak L(U,H)$ be ...
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1answer
96 views

Ito integral of average of the square of a Wiener signal?

How do we evaluate the average of the square of a Wiener signal? Standard case: Typically, the signal average is $S(t)=\frac{1}{T}\int_{0}^{T}s(t)dt$, where we can write the integral in Ito form $S(...
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1answer
42 views

$tB_t$ Integral representation, Brownian Motion

I never learned stochastic differential equations, and so am trying to do some self study. I've arrive at this question: $tB_t\sim N(0,t^3)$? $B_t$ is standard brownian motion. $B_t\sim N(0,t)$, so ...
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Stochastic calculus with normal distribution

For $l=1,2......$ prove that $E[W^{2l+1} (t)]=0$ I am trying to find the ways of solving the task from Stochastic calculus, but it seems to be very difficult to start with. I am really appreciate ...
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1answer
37 views

Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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1answer
31 views

The Stratonovich Integral and its meaning as the limit in mean square of a sum?

I am studying the Stratonovich Integral and on wikipedia, Stratonovich Integral, it states that the integral, for a process $X:[0,T] \times\Omega \to \mathbb{R}$, as: $$ \int_0^T X_t \circ dW_t $$ ...
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1answer
44 views

Stochastic integral estimate

I'm trying to derive the estimate $$ E\left[\left|\int_{0}^{t}h_r\,dB_r\right|^4\right] \leq 3C^4t^2,$$ where $h_r$ is continuous, adapted (to the natural Brownian filtration up to time $t$) and ...
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1answer
17 views

Solving a simple, linear type SDE

I am a bit confused by SDE's. I am trying to solve the SDE $dX=(c-\mu X )dt+\sigma dB$, with $\mu,\sigma,c$ constants and $X_0=x_0$ deterministic. I already know the solution of $dX=fdt+gdB$ with $X(...
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1answer
33 views

Is the stochastic integral of the jumps process equal to zero for a continuous integrator?

Let $X$ be a continuous semimartingale and $H$ a progressively measurable process in $L(X)$. Assume $H$ has left limits almost surely. I claim that the jumps process of $H$, denoted by $\Delta H = H - ...
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48 views

Ito stochastic integral vs Skorohod integral

I'm new in stochastic calculus and I'm confused about specific, but interesting topic. Skorohod integral is an extension of Ito integral for non-adapted processes, but how should I think about this ...
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37 views

Application of Stochastic Calculus to Interest Rate Model (Ito's Formula)

Above is my question. Now, the setting is of mathematical finance, but the part that I'm stuck on isn't directly related to finance, but stochastic calculus (hence posting on this site). We have the ...
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1answer
41 views

Stochastic control HJB equation

I am trying to solve this optimal control problem : $ V(x,t) = inf( E[\int_{0}^{1}(x(t)^2 - \frac{1}{2}u^2(t))dt + x(1)^2])$ subject to $dx(t) = u(t)dW_t$ $x(0) = x_0 \in R $ $u(t) \in [-1,1] $ ...
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1answer
19 views

Finding a solution to the SDE of $dX_t = -2 (1-t)^{-1}X_tdt + \sqrt{2t(1-t)} dW_t$.

I am trying to find the solutions to the SDE: The solution of the following SDE $$dX_t = -2 \frac{X_t}{1-t} dt + \sqrt{2t(1-t)} dW_t, \quad X_0 = 0 $$ where $W_t$ is a Wiener process. I know that ...
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0answers
20 views

How to find the mean and variance of a stochastic integral?

If $B(t)$ is a standard Brownian motion, let $Z(t)= \int_{0}^{t} s^2 dB(s)$. I want to find the mean and variance of Z(t). It is given that $Z(t)$ is Gaussian process. My approach for finding the ...
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20 views

Backward stochastic differential equation

I am interested by this problem Find a solution to this backward stochastic differential equation : $\ y(t) = (ry(t) + az(t))*dt + z(t)dW_t$ with the terminal condition $y(T) = \xi$ with $\xi$ ...
3
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0answers
27 views

Finding the mean of $X_t = \int_0^t sW_sdW_s$

For the stochastic integral, where $W_t$ is a Wiener process, I am trying to find the mean of $X_t = \int_0^t sW_sdW_s$. I have read before that any stochastic integral with $dWt$ has mean zero, but I ...