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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Expected value of geometric Brownian motion

So everyone knows that the expected value of GBM is given by: $X_0 \exp(\mu t)$ My question is that what does this say about such stochastic processes? Since $X_0$ and $\mu$ are within "my control" ...
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78 views
+100

How can we prove that $\langle\int_0^t\Phi_s{\rm d}W_s,x\rangle_H=\sum_{n\in\mathbb N}\int_0^t\langle\sqrt{λ_n}\Phi_se_n,x\rangle_H{\rm d}B_s^{(n)}$?

Let$^1$ $U$ and $H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U,H)$ be nonnegative and symmetric operator on $U$ with finite trace $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $...
4
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1answer
45 views

What Stochastic Calculi Other Than Ito And Stratonovich Exist?

When learning about stochastic calculus, you typically encounter Ito and Stratonovich calculi, usually in that order. There are many differences between the two (Ito processes have better martingale ...
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0answers
23 views

Stochastic Differential Equation Time-Independent

I know that a generic 1-D SDE can be written in Ito form as: $dX_{t} = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$. I was curious as to how such an SDE is written when modelling time-independent processes. I ...
4
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0answers
71 views

Is the integral of an Ito process still an Ito process?

Assume $r(t)$ is an Ito diffusion: $$dr_t = \mu_tdt + \sigma_tdW_t$$ Then, define the following process: $$X_t = \int_0^tr(s)ds$$ Is $X_t$ still an Ito diffusion?
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1answer
47 views

System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
2
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1answer
22 views

Square Integrable Martingales and the Unit Process

Let $X_t$ be a continuous square-integrable martingale. Then it is true (I think, please correct me if I am wrong) that: $$\forall t \in [0,\infty), \quad \int_0^t 1_{[0,t]}(s) dX_s = X_t - X_0$$ So ...
2
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1answer
57 views

Expectation of Product of Ito Integrals wrt Independent Brownian Motions

Let $W_1(t)$, $W_2(t)$, $W_3(t)$ be independent Brownian motions and $f$, $g$ smooth functions. I want to know if the following is true: $$ \mathbb{E}\left[ \left( \int\limits_0^t f(...
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0answers
23 views

When is a stochastic integral a martingale?

In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded ...
0
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1answer
31 views

Non standard stochastic integral

I don't know how to deal with the following stochastic integral: $\int_0^t \frac{1}{\sqrt{t-s+1}} d W(s)$ As you can see, the variable $t$ appears both as an endpoint of the interval of integration ...
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0answers
8 views

Moment bounds for solutions to SDEs

I've been looking at Kuo's book on stochastic integration, and I noticed that in Section 10.7, he proves some estimates for solutions to SDEs, written as $$dX_t = f(t,X_t)dt + \sigma(t,X_t)dB_t$$. ...
3
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0answers
31 views

Are martingales progressively measurable? (Application to square integrable martingales)

This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively ...
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0answers
6 views

What is the rate of convergence of Brownian motions Increments?

Would like to know what the rate of convergence of brownian motion is? I know each brownian motion increment is distributed with N(0,t) so do i need to apply a CLT?
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1answer
63 views

Differentiability of a simple value function driven by a diffusion

Consider a diffusion given by, $d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$ $X_0 = x$. Suppose the functions $\mu$ and $\sigma$ are as follows - $f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
3
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0answers
36 views

Computation of an Ito Integral [closed]

I have the following Ito Integral $$\begin{align}\int_{0}^{t}B_se^{-\sigma B_S}dB_s&& \end{align} $$ Here, $\sigma \gt 0$. Can someone please show me how to compute this Ito integral? ...
0
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1answer
27 views

Mean and Variance of SDE

How would I compute the mean and variance of the following SDE? $dX_t = \alpha X_t dt + \sigma dB_t$ I know $E[X_t]$ produces the mean and $E[(X_t)^2]$ produces the variance, but I'm not sure how to ...
3
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1answer
499 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf E\bigg[\exp\Big(\...
0
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0answers
17 views

Expected value of the exponential of a Geometric Brownian motion

I am trying to compute the following expectation: $$ E[ \exp (A_T)], $$ where $A_T = - C \int_{0}^{T} \exp( 2 \alpha W_t - \alpha^2 t) dt $, with $C$ and $\alpha$ positive constants, $W_t$ a standard ...
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0answers
23 views

JL Doob / KL Chung paper: Fields Optionality and Measurability

I was reading the following paper: https://www.jstor.org/stable/2373011?seq=1#page_scan_tab_contents I had a question about a proof that is part of the paper. Here are some of the definitions and ...
2
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1answer
76 views

Solving an Itô Integral

Can someone please show me how to solve this Itô Integral? $$\begin{align}\int_{1}^{t}\frac{dB_s}{B_s^2 + B_s^4} && \end{align} $$
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0answers
45 views

Why are we allowed to multiply the differential form of an SDE by a function?

Suppose for example that we have the following SDE: $$dX_{t} = a(X_{t})\,dt + b(X_{t}) \,dB_{t}. $$ What rigorous justification is there for then saying, for example, that we can multiply both sides ...
2
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0answers
29 views

Applying Ito formula to Ito process

I would like to simplify the expression $\left(\phi(s_{1})\cdot(X_{s_{1}}-X_{s_{2}})+\phi(s_{2})\cdot(X_{s_{2}}-X_{s_{3}})+\ldots+\phi(s_{n-1})\cdot(X_{s_{n-1}}-X_{s_{n}})\right)^{2}$ where $X_{t}$ ...
2
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2answers
51 views

Simulating a Stochastic Integral of OU process

The stochastic integral I want to simulate is $$\int_{0}^{1}J_c(s)dJ_c(s)$$ where $J_c(s) = \int_{0}^{s}e^{-c(s-r)}dB(r)$, is an OU process. I simulate the data using Matlab and the sample codes are ...
1
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1answer
49 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
2
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0answers
117 views

Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds} $$ where the ...
2
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0answers
72 views

Question on averages of Ito Integral: $E(\int_0^t X_sdB_s \int_0^t X_sds)=?$

Given some probability space, assume $X_t$ is a square integrable continuous process adapted to the filtration $\mathcal{F}_{t}$ generated by the standard Brownian process $B_t$. I denote by $(X.B)_t=\...
2
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2answers
534 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
7
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1answer
319 views

Application of the Burkholder Davis Gundy inequality

The proof of the Feynman-Kac formula uses a lemma which I need to prove, but I can not figure it out. The lemma is the folllowing: Let $X$ be a weak solution of $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t$$...
2
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1answer
185 views

Conditional expectation brownian motion

Somebody has an idea on how to tackle this quantity $$ \mathbb{E}\left[ \left. \frac{\int_{0}^{T}{{{e}^{\alpha {{W}_{t}}}}}dt}{\int_{0}^{T}{{{e}^{-\alpha {{W}_{t}}}}}dt+\int_{0}^{T}{{{e}^{\alpha {{W}_{...
2
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0answers
78 views

Question about a Bessel process

Are there any explicit path-wise solutions for a 3 dimensional Bessel process? E.g. the Ito SDE: $$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$ where $W_t$ is a standard Wiener process.
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2answers
55 views

brownian noise and stochastic differential equations

Consider the SDE $$dx=3x(t)dt+dW(t)$$ Where we're dealing with Brownian noise. Now, dW comes from $$dW(t)=\int_0^{dt}ds\ \eta (s)$$ As far as I understood, $\eta$ is the noise distribution (...
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0answers
18 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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0answers
26 views

Quadratic covariation

I am trying to solve small task from stocastic calculus. It can be shown that for stochastic processes X and Y , the quadratic covariation satisfies the polarisation formula - $[X, Y](T) = 1/2 ([X + ...
3
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1answer
53 views

Expectation value of stochastic process

For which $k>0$ process $X=(e^{kW_s^2})_{s \ge 0}$ belong to $\mathcal{L}^2_{\infty }(W)$ and for which belong to $\Lambda ^2_{\infty }(W)$. Set one localization sequence $(\tau_n)_{n \ge 0}$ for ...
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1answer
18 views

Vasicek equation [closed]

Vasicek interest rate stochastic differential equation is $$dR(t)=(\alpha-\beta R(t))dt+\sigma dW(t)$$ where $\alpha , \beta$ are positive constants. I need to use Ito-Doeblin formula to compute $...
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0answers
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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0answers
15 views

Interpretation of the stochastic integral of simple functions

A simple function $h$ (or random staircase function) is defined as $$ h(t) = \sum_0^{n} \xi_i I_{(t_i,t_{i+1}]}(t) $$ where $0<t_1< \ldots < t_{n+1}<T$ and the $\xi_i$ are random ...
0
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1answer
36 views

Using of Ito formula with martingales

We have exam test - $\alpha,\beta \in \mathbb{R}$ and $N(t)=e^{\beta t}cos(\alpha W(t)).$ It is necessary to calculate $\mathbb{E}[cos(\alpha W(t))]$. I know that $\beta$ can be chosen so that $N$ ...
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2answers
66 views

What's the variance of the following stochastic integral?

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive. I ...
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1answer
27 views

Ito integral for simple stochastic process

I need for $l=1,2......$ prove that $E[W^{2l} (t)]=$ $\frac{(2l)!}{2^l l!}$ and $E[W^{2l+1} (t)]=0$ I know that Ito integral for simple stochastic process satisfies $E[I^2 (t)]=E\int_0^t\Delta^2(u)...
5
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1answer
2k views

How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?

I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion. And its analytic ...
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1answer
41 views

Girsanov theorem calculations help

I need help to understand a couple of calculations in this Girsanov theorem related SDE problem. I have five questions as stated below. Let $X_t$ solve the Ornstein-Uhlenbeck equation $$dX_t = X_t\, ...
3
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1answer
45 views

Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
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1answer
28 views

Question about Ito integration in SDE in Stochastic optimal control

Here is my question statement. I cannot understand the last equality. Let $U=[-1,1]$. \begin{equation} \mathcal{U}[0, T] = \left\{ u:[0,T] \rightarrow U \mid u \text{ is } \{\mathcal{F}_t\}_{t\geq0}\...
3
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0answers
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The integral is the area under the curve. Is there a similar notion for stochastic integrals?

As discussed in the answers to this question, the integral is defined to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. ...
2
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0answers
52 views

A Simple Stochastic Integral Asymptotics

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth condition $...
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0answers
51 views

How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
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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&...
3
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0answers
34 views

Find (a,b) such that aX+bY is a Brownian motion

Let $$\begin{cases} dX_t = \mathrm{sin}(X_t+Y_t) dW_t \\ dY_t = \mathrm{cos}(X_t+Y_t) dV_t \\ X_0=Y_0=0 \end{cases}$$ Where $(W,V)$ is a two-dimensional Brownian motion and $(X,Y)$ be a strong ...
0
votes
1answer
32 views

Stochastic integral of $\mathrm{sin}(X_t+Y_t)\mathrm{cos}(X_t+Y_t)$ [closed]

As part of an exercise I am doing from stochastic analysis I need to compute the following integral: $$E(\int_0^x \mathrm{sin}(X_t+Y_t) \mathrm{cos}(X_t+Y_t)dt)$$ where $X_0=0$ and $Y_0=0$. Presumably ...