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Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

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One question about exponential martingale inequality at a paper of Ann. of Math.

I saw a strange inequality about martingales: If $\operatorname{M}\left(s,t\right)$ is an continuous $L^{2}$ martingale start at $s$, $\left[\operatorname{M}\right]\left(s,t\right)$ is its quadratic ...
shanlilinghuo's user avatar
1 vote
0 answers
44 views

Stochastic Integral and Ito's Isometry: Sharp-bracket process [M] or Angle-bracket process <M>?

I am learning stochastic integral, and I have noticed that the Ito's Isometry is sometimes stated using the quadratic variation process $[M]$ (e.g., pg. 47, Eq. (27.3), vol. 2 of Rogers & William),...
Mingzhou Liu's user avatar
0 votes
2 answers
38 views

Expectation of Solution to SDE, logarithmic extension of Vasicek

I have the solution to the following model $$dr(t)=r(t)(\eta-a\log r(t))dt+\sigma r(t)dW(t)$$ which, through the Vasicek model and a change of variable, I found to be $$r(t)=\exp[Y(0)e^{-at}+\frac{\...
mtcicero's user avatar
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2 votes
0 answers
36 views

Is there a closed form solution to the distribution of the running maximum for Gaussian processes?

Let $X=(X_t)_{t\geq0}$ be a zero mean Gaussian process with variance $\sigma(t)=\mathbb{E}X_tX_t^\top$ and define $$ S_T=\sup_{t\leq T}X_t\quad T\in[0,\infty) $$ the running maximum of $X$. Question: ...
Daan's user avatar
  • 362
1 vote
1 answer
146 views

"Heuristic" vs. Rigorous Ito's Lemma

Assuming $X_t$ is a standard Brownian motion and $t$ is the time variable, I have learned to derive Ito's lemma for a function $F(X_t, t)$ using the following results (below, $X_t=X(t)$, and I ...
Jan Stuller's user avatar
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-1 votes
1 answer
27 views

Geometric Brownian Motion inequality. [closed]

I have $\int_0^\infty e^{-\beta t}(1+\mathbb E[|Y_t|]+\mathbb E[Z_t])dt \le \int_0^\infty e^{-\beta t}[1+\mathbb E[|Y_t|^2]+z\mu t]^{\frac{1}{2}}dt$- Here $Z_t = ze^{\mu t-\frac{\sigma^2}{2}t+\sigma ...
oxedex's user avatar
  • 27
1 vote
1 answer
46 views

Prove that solution to geometric brownian motion is correct (plug into SDE)

As many previous questions/answers point out, the solution to the geometric brownian motion stochastic differential equation (GBM SDE) $$ \left\{ \begin{array}{ll} dX_t &=& \mu X_t \,dt + \...
SebaGM's user avatar
  • 13
0 votes
1 answer
45 views

$X\in L^{\infty,p},Y\in L^{2,q},prove \int_{0}^{t} XY_{s}dB_{s}$ is martingale

$L^{\infty,p}$ is X is progressively measurable and $E[\max_{0\le t\le T} \left | X_{t} \right |^{p} ]<\infty$ $L^{2,q}$ is Y is progressively measurable and $E[(\int_{0}^{T}\left | Y_{t} \right |^{...
Yu GongLian's user avatar
0 votes
1 answer
27 views

Showing that $M^T(N-N^T)$ is a continuous local martingale for all stopping times $T$ and continuous local martingales $M,N$

Given is that a cadlag adapted process $X=(X_t)_{t\geq 0}$ is a martingale if and only if $\mathbb{E}X_T=\mathbb{E}X_0$ and $X_T\in L^1$ for every bounded stopping time $T$. Now let $M,N$ be two ...
Daan's user avatar
  • 362
0 votes
0 answers
42 views

SDE's and Frechet derivatives

I'm learning the basics of SDE's, and it's usually stated that an SDE like $$dX_t = \mu(X_t, t)dt + \sigma(X_t,t)dW_t \tag{1}$$ is an abbreviation for the stochastic integral equation $$X_{t+h} - X_t =...
user541020's user avatar
1 vote
0 answers
22 views

A question on the proof of the ratio limit theorem for Brownian motions.

I am trying the prove the following ratio limit theorem, and I think I got it but I am not entirely sure that the mathematical logic in the last part of the proof is correct, and so I am asking if ...
Daan's user avatar
  • 362
2 votes
2 answers
62 views

Is $\int_s^t f(W_u)\mathrm{d}u$ independent to $\mathcal{F}_s$ for all (Lebesgue) measurable $f$?

I am trying to prove (or disprove) whether $$ I[s,t]=\int_s^t f(W_u)\mathrm{d}u $$ is independent of the $\sigma$-algebra $\mathcal{F}_s$ for any $t,s$ with $t\geq s$, for $W=(W_t)_{t\geq 0}$ the ...
Daan's user avatar
  • 362
3 votes
1 answer
63 views

Distribution of a stochastic integral $\int_0^t W_1(s) dW_2(s)$ with independent $W_1$ and $W_2$.

Consider two independent 1-d Wiener processes $W_1(t)$ and $W_2(t)$ with $W_1(0) = W_2(0) = 0$. I would like to know the distribution of the process $$Y(t) = \int_0^t W_1(s)dW_2(s). \tag{1}$$ I reason ...
MonteNero's user avatar
  • 367
4 votes
1 answer
102 views

Quadratic variation of the square of Brownian motion

Let $B_t$ be the Brownian Motion. Find the quadratic variation of a martingale $ M_t = B_t^2-t$. My solution: By Ito's formula for $f(t, x) = x^2-t$, we know $$d(B_t^2-t) = 2B_t dB_t$$ thus $\langle M ...
nessy's user avatar
  • 582
5 votes
1 answer
61 views

Showing bounds of Stochastic Process

Suppose that we have the SDE: $$ dZ_t = 2Z_t(1-Z_t)dt + 4Z_t(1-Z_t)dB_t $$ With $Z_0 = \frac{1}{3}$. How can I show that $0 \leq Z_t \leq 1$. I have tried solving the 'alalogous' differential ...
Lehmann's user avatar
  • 331
2 votes
1 answer
74 views

Quadratic covariation of martingale transforms to simple processes.

Let $X$ and $Y$ be simple processes, that is $X_t=\sum_{n=0}^\infty\xi_n{1}_{(t_{n},t_{n+1}]}(t)$ for a uniformly bounded sequence $(\xi_n)_{n\in\mathbb{N}}$ of random variables so that $\xi_n$ is $\...
Daan's user avatar
  • 362
4 votes
1 answer
66 views

Expressing a continuous local martingale as an integral against a Brownian motion

I'm interested in the following problem. Suppose $X$, $X_0=0$ is a continuous local martingale with quadratic variation $$ [X]_t = \int_0^t A_s\mathrm{d}s $$ for a non-negative previsible process $(...
raj's user avatar
  • 311
2 votes
0 answers
48 views

Can every $C^2$ function defined on a closed set in $\mathbb{R}^d$ be extended to $C^2(\mathbb{R}^d)$?

When reading Page 147 of the book "Continuous Martingales and Brownian Motion" by Daniel Revuz & Marc Yor, I am confused with the Remark $3^\circ$ of (3.3) Theorem (Ito's formula).In ...
Jesen's user avatar
  • 21
2 votes
1 answer
78 views

Find a PDE for $f$ satisfying $f(t,Y_t) = \exp(- \frac{\gamma^2}{2} t + \gamma W_t) E[\exp(\frac{\gamma^2}{2} T - \gamma W_T) F(Y_T) | \mathcal{F}_t]$

I am studying a course on Stochastic Calculus for Finance and am struggling with the following question: Given $dY_t = b(t,Y_t) \, dt + \sigma(t, Y_t) \, dW_t$ where $\gamma \neq 0$, and $$f(t,Y_t) = ...
FD_bfa's user avatar
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1 vote
0 answers
33 views

Stratonovich multiplication

I’m currently studying a paper Brownian heat engine with active reservoirs, and I have encountered some problems regarding Stratonovich multiplication. This is a part of the content in the article: $$...
Joker's user avatar
  • 49
0 votes
0 answers
12 views

Related to integration transformation from Cartesian plane to Polar coordinates

I am working in area of wireless communication that involves extensive use of probability, random variables and stochastic geometry. My system model is as follows: It consists of a main base station (...
Heretolearn's user avatar
1 vote
1 answer
50 views

Variance of an Itô Integral

Question Using Itô Isometry and the fact that $$\int_{0}^{t} Z_s \ ds = \int_{0}^{t} (t-s) \ dZ_s,$$ show that $$Var \left(\int_{t}^{T} Z_s - Z_t ds \right) = \frac{(T-t)^3}{3}$$ where $Z_s, Z_t$ are ...
Hmmmmm's user avatar
  • 333
1 vote
2 answers
130 views

Prove that $(t+1) X_{\frac{t}{t+1}}$ is a Brownian Motion using Levy's characterisation where $X$ is a Brownian Bridge

The following result is well documented and is a result of the Stochastic Processes course I am following. Below the result, I present the standard proof presented in my course (Method 1) and my ...
FD_bfa's user avatar
  • 4,343
0 votes
0 answers
24 views

Variant of Gronwall’s inequality

Gronwall inequality typically is used to bound a function $u(t)$ if it satisfies $u(t)\leq \alpha(t)+ \int_{0}^{t}\beta(s) u(s)ds $ with the condition that $\beta$ is non-negative. I want to use these ...
Tiramisu's user avatar
0 votes
1 answer
35 views

System of Stratonovitch SDEs $dX = \sigma X \circ dW$ to a system of Ito SDEs

I'm aware of several related stack questions, but my case is a bit different because I assume that the system of SDEs is multiplied by 1-dimensional increment $dW$. Suppose $\sigma$ is an $n \times n$ ...
MonteNero's user avatar
  • 367
0 votes
1 answer
47 views

Brownian bridge satisfied SDE

I am trying to solve the following problem: given, as usual, a Brownian motion B and Brownian bridge $Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s}$, prove that it satisfies the SDE $...
Noli's user avatar
  • 13
0 votes
1 answer
47 views

Normal distribution and conditional expectation

Why is $E(X|X>a) \left( 1- \Phi\left( \frac{ln(a)-\mu}{\sigma} \right) \right)=E(X) \left( 1-\Phi\left( \frac{ln(a)-\mu}{\sigma} -\sigma \right)\right)$? Maybe it's easy but I do not see it.
Frodo361's user avatar
  • 319
0 votes
0 answers
25 views

What's the definition of stochastic integral process at time zero? Is it always zero?

Question Let $(M_t)$ be a continuous local martingale and let $(H_t)$ be a predictable process such that $\int_0^t H_s^2d[M]_s<\infty$ for all $0\leq t\leq \infty$. Let $I_t$ be the process defined ...
Mingzhou Liu's user avatar
5 votes
0 answers
84 views

How to solve the SDE $\mathrm{d}X(t)=-X^{3}(t)\mathrm{d}t+\mathrm{d}B(t)$?

Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.4, the author claims that the problem $$\mathrm{d}X(t)=-X^{3}(t)\mathrm{d}t+\mathrm{d}B(t),X(0)=...
R-CH2OH's user avatar
  • 357
2 votes
1 answer
35 views

Exercise 9.1 in Introduction to stochastic processes by Lawler

This is not for any assignment or homework. I am studying Chapter 9, Stochastic Integration, in Introduction to Stochastic Processes by Lawler. Exercise 9.1 states Since I am self-studying, I'm ...
toronto hrb's user avatar
7 votes
1 answer
736 views

Are differentials on their own in stochastic calculus just an abuse of notation?

In stochastic calculus, it is often standard to write a DE in differential form, such as $\mathrm dY = H \, \mathrm dX$ for the stochastic integral $$\displaystyle\int\limits_0^t H \, \mathrm d X := \...
Markus Klyver's user avatar
1 vote
0 answers
37 views

ARCH-Vasicek model closed-form solution

I understand how we can obtain the solution of Vasicek model $dr_t=\alpha(\mu-r_t)dt+\sigma dW_t$: $$ r_t=r_0e^{-\alpha t}+\mu(1-e^{-\alpha t})+\sigma\int_0^te^{-\alpha(t-s)dW_{s}} $$ This easily ...
KiNest's user avatar
  • 11
1 vote
0 answers
27 views

Mean, Variance and Correlation Function of a quadratic SDE

I am struggling with the following nonlinear SDE: $ ds=dt(-\Omega s^2(t)+\alpha s(t)+\beta) + d\xi(t)(\gamma (1-s(t))) $ $ d\xi = dt(-\frac{1}{\tau} \xi(t)) + \sigma dW(t) $ Where $\alpha$, $\Omega$, ...
duodenum's user avatar
1 vote
2 answers
51 views

Book recommendation for stochastic integral wrt local martingales

I am looking for a book (or a chapter of a book) for stochastic integral wrt local martingales. The book should contain a rigourous introduction to the definition. It should also contain proofs for ...
Mingzhou Liu's user avatar
0 votes
0 answers
14 views

Continuous Local Martingales

Let $T^n$ be increasing stopping times such that $T^n \rightarrow \infty$ and let $M$ be a process such that $M_{T^n} \in \mathcal{M}_{\text{loc}}^c$ for all $n$. Show that $M \in \mathcal{M}_{\text{...
minkowski's user avatar
1 vote
1 answer
68 views

Computing the quadratic covariation $\langle B,B^2\rangle_t$ of a Brownian motion and its square

Problem Let $B$ be a Brownian motion. Compute the quadratic covariation of $B$ and $B^2$, i.e. $\langle B,B^2\rangle_t$ for every $t\geq 0$. My attempt Integration by parts yields: \begin{align*} &...
Wilfred Montoya's user avatar
2 votes
2 answers
54 views

Is the stochastic integral process $\left(\int_{0}^{t}e^{-\lambda\left(t-s\right)}\mathrm{d}B_s\right)_{t\geq 0}$ a martingale?

Up to now, I was only presented with stochastic integral processes of the form $\left(\int_{0}^{t}\phi_s\mathrm{d}B_s\right)_{t\geq 0}$ and the general way to show that such a process is a martingale ...
Wilfred Montoya's user avatar
1 vote
0 answers
33 views

Are processes derived from deterministic and square-integrable functions martingales?

Problem Let $f:\mathbb{R}_{\geq 0}\to\mathbb{R}$ be a deterministic and square-integrable function, i.e. $\int_0^\infty f(t)^2\mathrm{d}t < \infty$, and define the process $X=(X_t)_{t\geq 0}$ by $...
Wilfred Montoya's user avatar
2 votes
1 answer
32 views

Example of a process that yields a non-martingale

I'm attending a class on stochastic processes and as a “side quest” to understand a discrete martingale transform properly we were given a task: Provide/construct a not previsible adapted process $C$ ...
markovian's user avatar
  • 157
2 votes
1 answer
75 views

Converse of a martingale transform theorem

I read this question Understanding proof of martingale transform being supermartingale and was wondering if the converse of the statement in the problem is also true. To be precise my question is: Let ...
Jodasilva's user avatar
  • 131
2 votes
1 answer
42 views

Understanding stochastic integration with semimartingale as integrator: The locally bounded variation part.

Let $X_t = M_t + A_t$ be a semimartingale with continuous paths, and M is a local martingale, and A is a process with paths locally of bounded variation. The book I'm reading, states that we can ...
An old man in the sea.'s user avatar
2 votes
0 answers
58 views

Conditional expectations between Ito diffusions

If one has $dX_t = f(Y_t)\mu_1(X_t, t)dt + \sigma_1(X_t, t)dW_t$ $dY_t = \mu_2(Y_t, t) + \sigma_2(Y_t, t) dW_t$ driven by the same Brownian motion. Is it sensible to consider a conditional expectation ...
Theo Diamantakis's user avatar
0 votes
0 answers
48 views

Finding a general solution to the SDE $dX_t = (\mu_1 (t) X_t + \mu_2 (t)) dt + \sigma (t) dB_t$ [duplicate]

I am looking to find a general solution to the stochastic differential equation: $$dX_t = (\mu_1 (t) X_t + \mu_2 (t)) dt + \sigma (t) dB_t \tag{1}$$ I am happy to assume any regularity conditions ...
FD_bfa's user avatar
  • 4,343
2 votes
1 answer
65 views

What is the solution to the SDE $X^x_t = B_t + \int_0^t \frac{x − X^x_s}{1-s} ds$

In a course on Stochastic Calculus, I have seen the result (without proof) that: The solution to the stochastic differential equation: $$X^x_t = B_t + \int_0^t \frac{x − X^x_s}{1-s} ds \space \text{ ...
FD_bfa's user avatar
  • 4,343
4 votes
1 answer
71 views

Using Ito's Lemma to take a stochastic integral

I need to calculate the following stochastic integrals, assuming $B_{t}$ is a standard Brownian motion: $$\int_{0}^{T}{t}\ dB_{t}$$ $$\int_{0}^{T}{B_t^{2} }\ dB_{t}$$ I don't know how to go about this ...
two_hearted_river's user avatar
6 votes
1 answer
67 views

Stochastic integration: Computing $\mathbb{E}[\exp(- \lambda \int_0^{t∧T_\epsilon}\frac{ds}{B_s^2}) ]$

Recently I started taking Stochastic calculus class, and I am struggling still with computation of stochastic integrals and I would appreciate any help with the following example: We define: $$ Y_t = (...
variableXYZ's user avatar
  • 1,073
0 votes
1 answer
65 views

Stochastic integration and use of ito's lemma

Note to the moderators: This question has been solved, and is indeed a valid question. Please post a comment explaning what needs to be elaborated on. Thank you very much! - random0620 Proposition 6.7 ...
minkowski's user avatar
2 votes
1 answer
99 views

Weak Uniqueness of solution of SDE

The following is an excercise where we try to prove Lemma 5.3.1. Oksendal "Stochastic Differential Equations" Suppose that $b(t,x)$ and $\sigma(t,x)$ satisfy the hypotheses of the existence/(...
Irving Lee's user avatar
2 votes
0 answers
39 views

Understanding the definition of the Ito integral in Oksendal

I'm reading Stochastic Differential Equations (sixth edition) by Bernt Oksendal and I'm trying to understand the author's definition of the Ito integral. For a given probability space $(\Omega, \...
Leonidas's user avatar
  • 1,054
3 votes
0 answers
47 views

Proving that the expected value of an $L^2$ approximation of $\int_{0}^{T} B_t^2 \,dt$ by simple functions goes to 0 as $n \to \infty$

Let $B = (B_{t})_{t \geq 0}$ be a 1-dimensional Brownian motion on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ (i.e. that is, $B_t: \Omega \to \mathbb{R}$ for each $t \geq 0$), and let $T &...
Leonidas's user avatar
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