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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1answer
31 views

Integral of square of Brownian motion with respect to Brownian Motion

While trying to compute $\int_0^TB_t^2\ dB_t$, $B$ being the standard Brownian motion, I got stuck at showing the following. $$\sum_{i=0}^{n-1}B_{t_i}(B_{t_{i+1}}-B_{t_i})^2 \rightarrow \int_0^TB_t\ ...
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2answers
78 views

Integral of time with respect to Brownian motion

I am trying to compute $\int_0^T t\ dB_t$ where $B$ is the standard Brownian motion. To this end I define the sequence of simple predictable functions $$ f_n = ...
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0answers
35 views

What is the solution to these SDP?

I am in trouble with my homework, the quesetion is to solve a pair of stochastic differential equation. $dX_t^1 = X_t^2dt + \alpha dB_t^1$ $dX_t^2 = -X_t^1dt + \beta dB_t^2$ $\alpha \ and \ \beta$ ...
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0answers
32 views

Is the variance of an Ito process strictly increasing?

Consider the Ito equation: $dX_t = f(t, X_t) dt + G(t, X_t) dW_t$ where $f:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$, $G:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^{n\times m}$, $X_t \in ...
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0answers
39 views

Stratonovich SDE and generator in divergence form

Let $a:\mathbb{R}^d\rightarrow\mathcal{S}_{d\times d}$ be a smooth map that takes its values in the space of $d\times d$ symmetric matrices and suppose there exists a $d\times d$ matrix valued ...
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1answer
65 views

Prove that the quadratic covariation is a bilinear form

If we take $X,Y,Z$ to be square integrable martingales starting at zero, we want to show that for any $\alpha\in\mathbb{R}$ we have $\langle X + Y , Z \rangle = \langle X,Z\rangle + \langle Y, Z ...
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0answers
27 views

Preservation of the Markov Property for SDEs

Let $X$ be a continuous Markov process on $\mathbb{R}^d$ that is also a semimartingale. Let $V=(V_1,...,V_d)$ be a collection of suitably nice vector fields on $\mathbb{R}^d$ such that there exists a ...
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1answer
48 views

Compute the distribution of $\int_0^1 B_t dt$

I need an help with the following: let $(B_t)_t$ a Brownian motion. Compute the distribution of $X:=\int_0^1 B_t dt$. Integrating by parts we have that: $$\int_0^1 B_t dt=B_1-\int_0^1 t dB_t.$$ Now, ...
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0answers
27 views

Question on exponential martingale

I was reading the first proof here on exponential martingale, https://fabricebaudoin.wordpress.com/2012/09/27/lecture-23-time-changed-martingales-and-planar-brownian-motion/ It says that "Let $ ...
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0answers
59 views

Covariance of Stochastic Differential Equation

What is the general expression for the covariance $cov \left[ X_s X_t \right]$ of a stochastic process given by \begin{equation} dX_t = f(X_t,t)dt + g(X_t,t) dW_t \end{equation} for some general ...
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1answer
57 views

Integration by parts formula for Wiener integral

Hi I need an help understanding "integration by parts" in Wiener integral. I've defined this integral as in the following: let $T=[0,t]\subset \mathbb R$ we want to define $\int_T f(s) dB_s$ where ...
2
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1answer
90 views

Question related to Kolmogorov equations

Let $d X_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$ be an Ito diffusion. If we choose a continuously twice twice differentiable function $f$ with compact support and define $u(t,x) = E( f(X_t) | X_0 = x)$ ...
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2answers
153 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...
2
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0answers
36 views

Is there any standard way of analysing this integral?

I have a compound Poisson process $(X_t)$, with jump distribution $F$, which assigns mass only to $(0,\infty)$. In my working I have an expression of the following form: $$ \mathbb{E} \int_0^{\tau} ...
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2answers
64 views

Most General Theory of Stochastic Integration

I've learnt continuous stochastic integration using the classical books: - Revuz & Yor, - Karatzas & Shreve and - Oksendal. Now I want to learn general stochastic integration, i.e. possibly ...
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0answers
27 views

Pricing of Binary or Digital Options or Feynman-Kac Equation for $\mathbb E f(X_T)$ with diffusion $X$ and discontinuous function $f$.

I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous by means of solving a PDE or ...
2
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0answers
55 views

Quadratic Variation and Semimartingales

It is clear that every (I am particularly interested in continuous) semimartingale has a well defined quadratic variation process. However, what can be said about processes that have a well defined ...
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1answer
32 views

Expectation of B(1) times stochastic integral? [closed]

I need to find the value of this expectation: $$\mathbb{E}\left(B(1) \int_0^1 f(t) dB(t)\right)$$ $B=(B(t))_{0\leq t\leq1}$ is a standard Brownian motion on $[0,1]$ and $f=(f(t))_{0\leq t\leq1}$ is ...
3
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0answers
167 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
1
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1answer
54 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it ...
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2answers
44 views

Predictable Processes in Brownian Setting

Maybe it's a silly question. I've been reading Protter's book on stochastic integration. And all the integrands are required to be predictable. But from what I can recall, in the traditional ...
2
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1answer
37 views

Bayes formula on a general $\sigma$-algebra

I want to prove the following Bayes formula: Let $\Omega$ be a sample space, $\mathbb A$ a $\sigma$-algebra over $\Omega$ and $\mathbb B$ $\sigma$-algebra of $\mathbb A$. For $A\in\mathbb A$ and ...
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0answers
5 views

Error from bias and noise in a linear operator

There's a result $S$ that depends linearly on some forcing $F$: $S=\int dt' G(t-t')F(t')$ Let's say I need to predict $S$, but can't measure $F$ exactly. I have both bias and noise in my ...
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0answers
41 views

martingale representation property

Suppose there are two independent Brownian motions $B_1$ and $B_2$ and the natural filtration of $B_1$ is denoted by $F^1_t$. Define ${g}:=\sigma\{B_2(t),t\in[0,T]\}$ and filtration $F_t$ is defined ...
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1answer
45 views

How to solve this question with Itô lemma?

Let $$M(t) = \int_{0}^t Y (u)dB(u).$$ where $$E \left[ \int_{0}^t Y^2(u)du\right] < \infty.$$ Use Itô’s rule to find the differential $dQ$ of the process $$Q(t) = M^2(t) − \int_{0}^t Y^2(u)du$$ ...
4
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1answer
75 views

Prove identity in law for stochastic process driven by Brownian Motion

Let $B = (B_t)_{t\geq 0}$ be a standard brownian motion started at $0$. Consider the two following stochastic equations: \begin{equation} \begin{split} dX_t &=& (13 + 2X_t)\,dt + (6 + ...
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1answer
36 views

Convergence properties of the Ito integral

I am currently going through the proof of the existence of a solution of the SDE \begin{align} dX_t = bdt + \sigma dB_t \end{align} where $B_t$ is a Brownian motion wrt a filtration ...
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1answer
39 views

The Itō Integral

In stochastic calculus and specifically for mathematical finance Ito's lemma is used for time varying processes I need to know intuitively why the Ito Integral is stochastic?
2
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1answer
62 views

Show that a process is gaussian

I need an help with the following exercise. I would like to know if what I've done is correct. Let $(X_t)_{t\geq 0}$ be the process define as $$X_t=e^{\lambda t} X_0-\sigma \Big(\lambda \int_0^t ...
0
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1answer
16 views

Conditional expectations one more time

Please someone verifies my results: 1) $E \Big( \int_0^3W_t^2dt|F_1\Big)=$(editing in progress) 2) $E \Big( \int_0^2 (tW_t+t^2)dt|F_1\Big)=E \Big( \int_0^2 tW_tdt|F_1\Big)+E \Big( \int_0^2 ...
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1answer
71 views

Integral on interval $[-\infty,W_t]$, $W_t$ is Brownian motion

Basicaly I have an expectation of an integral on the interval which contains Brownian motion and it look like this. $$ E\left[e^{W_t}\cdot\int_{-\infty}^{W_t} e^{-z^2}dz\right] $$ $W_t$ is Brownian ...
0
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1answer
49 views

How to decompose $X_t^2$ as an Itô process?

I am given the stochastic process $X_t$ to be the unique process starting at $X_0$ and solution of the following SDE: $$dX_t = (a-bX_t)\,dt + \sigma \sqrt{X_t} \, dW_t,$$ where $W_t$ is a real ...
2
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0answers
46 views

Sufficient condition for martingale property

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space and $M=(M_t)_{t\geq 0}$ an $\mathcal{F}_t$-adapted stochastic process. If $$ \forall t<s, \ ...
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0answers
22 views

variance of total residence time in up state

Hello; I really appreciate it if someone help me about this problem
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0answers
20 views

Stratonovich integral of Wienere process [duplicate]

I need an help with the following exercise. Let $(W_t)_{t\geq 0}$ a Wiener process on $(\Omega, \mathcal E, \mathbb P)$ and let $I=[0,T]$ be an interval. We want to prove that the Stratonovich ...
2
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2answers
103 views

Product of stochastic integral and brownian motion

I am trying to compute the following expectation: $$ M_T = \mathbb E\left[W_T\int_0^T\,t\,d W_t \right] $$ where $0<t<T$ and $W = (W_t)_{t\geq 0}$ is a standard Brownian Motion started at $0$. ...
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1answer
62 views

Show that $E[X_t^2]<\infty$

Show that $E[X_t^2]<\infty$, where $$ X_t=e^{3W_t-\frac{3t}{2}}-3e^{W_t-\frac{t}{2}}\underbrace{\int_0^te^{2W_s-s}ds}_{A_t},\quad. t\geq0, $$ where $t$ is a fixed number and $W_t$ is Brownian ...
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0answers
29 views

How do you calculate optimal quantity if there is a fixed cost in the Newsvendor Model?

I have listed the problem below. I know that in order to calculate optimal quantity that must be ordered, you need to do F(d)>= (Cu-Cv)/(Cu+Co) and compare it to the probabilities of demand. However, ...
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0answers
105 views

The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma? Here are some unsatisfactory references to the Ito-Wentzell Formula: ...
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0answers
33 views

How can I solve $E[B^4_t B^3_t]$?

How can I solve the following expected value: $$ E[B^4_t B^3_t] $$ where $ B_t $ is a standard Brownian Motion.
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1answer
105 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
0
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1answer
29 views

Black Scholes Solution

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
5
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1answer
107 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
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1answer
58 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
3
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1answer
99 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
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1answer
69 views

Deriving the Doob Meyer decomposition of a Sub Martingale using Ito's

Given the standard brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$ and defining the sub-m.g.: $$X_t =B^6_t+2t$$ I would like to derive its Doob-Meyer decomposition: [Sub-m.g.]= [increasing ...
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0answers
43 views

Profit Maximization

I have listed a homework problem below that I have been working on. How do I get the expected number sold/expected number unsold/expected number lost if I do not have the pdf for the demand? Am I ...
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1answer
91 views

Expected Value of the exponential of a stochastic integral

What is the expected value of the following process: $$ e^{\int_0^t B_u\, du} $$ Thanks.
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2answers
35 views

Find the value of the real $\alpha$ for which exp($2B(t) - \alpha t$) is a martingale.

I tried to answer using the three conditions to be a martingale (measurability, integrability, and martingality), validating the integrability condition, which is $$ E |e^{2B(t) - \alpha t} | < ...
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0answers
32 views

Deterministic integrals involving a Brownian motion [duplicate]

I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$. The first expression is bewildering me a bit, since it seems like somehow the Itô ...