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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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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48 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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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
67 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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42 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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2answers
50 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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48 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
74 views

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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29 views

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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29 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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27 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
94 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
41 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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44 views

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
42 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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0answers
40 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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34 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
36 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
18 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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17 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 $\xi$ ...
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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 ...
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1answer
28 views

Will Ito's Isometry result in $E\left(\int_0^t \cos(u)\,dB_u \int_0^t \sin(u)\, dB_u \right) = E\left(\int_0^t \cos(u) \sin(u)\, du \right)$?

If I have two integrals, $X_t = \int_0^t \cos(u)\,dB_u$and $Y_t = \int_0^t \sin(u)\, dB_u$ , where $B_u$ is a Wiener Process and I am trying to find: $$ E\left(\int_0^t \cos(u)\,dB_u \int_0^t \sin(u)...
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67 views

Rephrase a multiparameter SDE indexed by time and space as an infinite dimensional SDE indexed by time

Let $\mathcal V_t\subseteq\mathbb R^3$ be the bounded space occupied by a closed particle system at $t\ge 0$ and $$[0,\infty)\ni t\mapsto X_t\in\mathcal V_t\tag 1$$ be the movement of a fixed particle ...
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1answer
44 views

Coefficient matching proof that $e^{\alpha x-\frac{1}{2} \alpha^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\alpha^n$, where $H_n(x)$ are Hermite poly.?

Hermite polynomials can be defined as (from wikipedia): $$ H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}. $$ I am trying to show that: $e^{\alpha x-\frac{1}{2} \alpha^2}=\sum_{n=0}^{\infty} \...
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1answer
43 views

Prove Wald's identities for Brownian motion using stochastic integrals

The question is as follows: Let $W$ be Brownian motion and $T$ a stopping time with $\mathbb{E} T < ∞$. Show (use stochastic integrals) that $\mathbb{E}W_T = 0$ and $\mathbb{E} W^2_T = \mathbb{E} ...
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24 views

Ito Formula for increments of Ito Processes

Let $X_{t}=X_{0}+\int_{0}^{t}a_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}$, $W_{t}$ is a standard BM. How can I apply Ito formula to $(X_{t}-X_{s})^{2}$? Should I use a multidimensional version?
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1answer
55 views

Definition/Construction of Wiener Measure

I want to make sure I understand this rigorously: Assume we already know that Brownian motion $B_t$ on $[0,\infty)$ exists/how to construct it. Every $\sigma$-field considered is implicitly assumed ...
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1answer
65 views

SDE Integration: Normal-Mean Reverting Process - Question

I am trying to figure out how a particular SDE can be integrated. The SDE is the normal mean-reverting model: $dX_t = \theta(\mu - X_t)dt + \sigma dW_t$ (1) Where $W_t - N(0,t)$. So far, I have ...
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1answer
41 views

Integral w.r.t. a Martingale

Consider the stochastic integral $$ Z_t = 1+\int_0^tZ_{s^{-}}\,dX_s $$ where $X$ is a Martingale. In the textbook by Shreve (see here pages 493-493) it is said that since $Z_{s^{-}}$ is left-...
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1answer
50 views

How can we identify $\omega\in\Omega$ with a path of Brownian motion $t\rightarrow B_t(\omega)$?

In the Stochastic Differential Euqations written by Oksendal(see page 12), As we shall soon see, the paths of a Brownian motion are (or, more correctly, can be chosen to be) continuous, a.s. ...
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51 views

$X_t=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{n}})dB(u)$ is a Brownian motion for suitable non-zero constants $a_0,\ldots,a_n$

Let $B(t)$ be brownian motion. Show that for any integer $n \geq 1$, there exist nonzero constants $a_{0},\ldots,a_{n}$ such that $X_{t}=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{...
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29 views

Multi-derivative of standard normal CDF

I am trying to solve the following $m$th-derivative of standard normal cdf, $$\frac{\text{d}^m}{\text{d}a^m}\Phi \left(\frac{a+\mu u}{\sqrt{u}}\right),$$ where $m> 0$ is an integer , $\mu>0$,...
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30 views

Laplace transform of survival probability for stochastic diffusion

Let $Y_t$ be a killed process defined by \begin{eqnarray} Y_t = X_t \quad \mbox{if } t<\xi,\\ Y_t = 0 \quad \mbox{if } t\geq\xi. \end{eqnarray} where $\xi$ is a random time such that $$ \xi=\inf \...
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1answer
31 views

Geometric Brownian motion with random drift and diffusion

One of my finance professors claims that the following is a meaningful SDE. $$dX_t = \delta_t\mu X_tdt + \delta_t\sigma X_tdW_t$$ Here $W$ is BM and $\mu$ and $\sigma$ are positive real constants. $(...
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1answer
36 views

Clarification on Stochastic Exponential

Consider a $d$-dimensional Brownian motion $B=\left(B_1,...,B_d\right)$ whose components are independent and let $A$ be a $d\times d$ squared matrix such that $\sum_{i=1}^dA_{ii}^2=1$. Define the $W=...
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1answer
28 views

Stochastic Integration

I am fairly new to stochastic calculus and am having problems solving this equation.. $$X(t)=\oint_0^TL(t)(\mu \, dt + \sigma \, dW_t)$$ Now, here $L(t)$ is a constant $k$. And I have to find $X(t)$...
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50 views

Radon-Nikodym derivatives $\frac{d\mathbb{P}_1}{d\mathbb{P}_0}$ and $\frac{d\mathbb{P}_2}{d\mathbb{P}_0}$

$\Omega$- is the interval [0,1], $\mathbb{P}_0$ is Lebesgue measure, $\mathbb{P}_1$ is the probability measure given by $\mathbb{P}_1([a,b])=\int_a^b 2\omega d\mathbb{P}_0(w)$ and $\mathbb{P}_2([a,b])=...
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15 views

Reference help for “stochastic integrals”

Please suggest some references for the regularity theorems in the book Stochastic Equations in Infinite Dimensions by Da Prato and Zabczyk.
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1answer
35 views

Change of variable in $\varphi(s) = t$, effect in $\mathbb{d} W_t$

I'm a little confused here. If I have the stochastic integral $$ \int_0^T f(t)\,\mathbb{d} W_t $$ and perform the change of variables $t = \varphi(s)$, how will $\mathbb{d} W_t$ transform (where the ...
4
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1answer
122 views

Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
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0answers
53 views

Variance of Brownian Integral when the end point is specified

Consider the Brownian $W_u$. Suppose you are only considering realizations of this brownian that verify both $W_0=0$ and, for a specific (given) $t$, $W_t=a$. Under these specific conditions, what is ...
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1answer
49 views

Calculate $E[\exp(iu\int_0^ts \, dB_s)]$ for a Brownian motion $(B_t)_{t \geq 0}$

Since $X_t:=\int_0^ts \, dB_s$ is a process with independent increments, its distribution is infinitely divisible and its variance is $c_t=\frac{1}{3}t^3$. I think, its characteristic function $E[\...