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
35 views

Why is $dX_t=X_t(\mu_t dt+ \sigma_t dW_t)$ an Ito process?

In solving the SDE $dX_t=X_t(\mu_t dt+ \sigma_t dW_t)$ we pick $Y_t=ln X_t$ and then apply Ito's lemma on the twice differential function $f(x)=ln (x)$ .But then why is $X_t$ anIto's prcess given ...
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50 views

Multiple Wiener Integral by Ito

In the context of Wiener-Ito chaos expansion, I had a look at Ito's paper "Multiple Wiener Integral", 1951. I am puzzled by his last result, theorem 5.1, that a multiple Wiener integral ...
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0answers
15 views

Verify a stochastic integral has normal distribution.

A well-known result is that:if $\sigma$ is a non-random process,then $$\int_0^T\sigma_t\,dW_t\sim N(0,\int_0^T\sigma_t^2\,dt)$$ ( from Shreve's "Stochastic Calculus for Finance" thm 4.4.9) by means of ...
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1answer
31 views

Simple differential equation and Integral Ito

With stochastic differential equation dx(t) = dW (t), and knowing that all integrals occurring are integral Ito. Witch variable changes y = tx. How I can prove? integral between 0 and t[sdW(s)] = ...
4
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1answer
74 views

How can a random variable have random variance?

This seems counter-intuitive to me since variance is a difference of expectations and afaik, unconditional expectation is a real number. Apparently, $X_t$ where $dX_t = Y_t dW_t$, where $Y_t$ is an ...
2
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1answer
42 views

How to find the distribution of the following stochastic integral of a geometric Brownian motion?

$K_{\phi,\lambda}(r)=\int_{0}^{r}\exp\{(r-s)\phi+\lambda(W_r-W_s)\}dB_s$ where $W$ and $B$ are independent standard Brownian motions, and $(\phi,\lambda) \in \mathbb{R} \times \mathbb{R}_+ $ The ...
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0answers
15 views

Stochastic Integrations with respect to non-martingale

Does there exists a theory of stochastic integration with respect to processes which are not local-martingales? For example if I have a general stochastic process $X_t$, can I integrate certain other ...
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1answer
24 views

application of Holder's inequality from Oksendal's book on SDEs

I am following the proof of the existence of solutions of SDE: let $b(t,x)$ and $\sigma(t,x)$ be Lipschitz continuous and consider the following SDE $dX_t=b(t,X_t)dt + \sigma(t,X_t)dB_t$. Define ...
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0answers
22 views

Time integral of Brownian motion's running maximum

Let $\mu \geq 0$ and consider $B_{\mu}(t) := B(t) + \mu t$ a one-dimensional BM with drift $\mu,$ and let $M_t := \max_{0 \leq s \leq t} B_{\mu}(t)$ be its running maximum. My question involves two ...
2
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0answers
30 views

How to solve a nonlinear SDE analytically

I have a numerical solution for the following equation: x'(t)= x(t) - x^3(t) + n(t) where n(t) is a white gaussian noise with zero mean and unit variance. I am a bit confused on how to go about the ...
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0answers
12 views

Covariance of nonlinear sde

My problem is to compute the covariance of the following Ito process $$ dX_t=AX_t+\sum_{k=1}^{n}B_kX_tdW_k, $$ where $A,B_k$ are nonlinear operators defined on a complex separable Hilbert space $H$. ...
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0answers
64 views

Convergence in $L^2$ of the stochastic integral $\int\limits^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s$

Let $e\in \mathbb{R}^+$ and $B_t$ 1-dimensional Brownian motion. Consider $$X_t=\int^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s.$$ How to show that $X_t \to 0$ in $L^2$ as $e\to0$? Obviously the ...
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0answers
11 views

How to compute this integral using Ito isometry? [duplicate]

I am trying to evaluate the following integral: $E\Bigg[\Bigg(\int^{t}_{0} \frac{B_s}{e}1\big(B_s\in(-e,e)\big)\Bigg)^2\Bigg]$ I cannot figure out how to apply Ito isometry when the indicator ...
4
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1answer
90 views

Construction of Ito integral

This is in regards to constructing the Ito integral, specifically the first step of approximating bounded, continuous functions by elementary functions. Let $(\Omega, \mathcal{F}, P)$ be a ...
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1answer
66 views

Do we have the following?

Suppose the following integrals \begin{equation} \int_t^T X_s \, ds\ \text{ and }\ \int_t^T Y_s \, ds \end{equation} are well-defined, where $X_s$ and $Y_s$ are continuous stochastic process. Do we ...
2
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0answers
61 views

Integral with respect to Brownian motion, Variance

Good day. Imagene we have a martingale $M(t)=\int_0^t f(s)dB(s)$ which satisfies Dambis-Dubins-Schwarz Theorem. At the same time $M(t)^2 - <M>(t)$ is a Martingale starting in $0$ as well. If i ...
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0answers
23 views

How to calculate the differential of the following stochastic integral?

Let $$Y_t=\int_t^T f(t,s)\ \mathsf dW_s$$ I want to compute $\mathsf dY_t$. This suggests me to consider how to find $\mathsf dY_t$ for $$Y_t=\int_t^T f(t,s)\ \mathsf dW_s$$ or $$Y_t=\int_t^T g(t,s)\ ...
2
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0answers
61 views

one dimensional SDE with zero drift

I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, ...
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0answers
22 views

Ito's representation for $L^1$ random variable

Given $(\Omega,\mathbb{F},P)$ where $\mathbb{F}$ is the $P$-complete filtration generated by Brownian motion $W$. Ito's representation says for $X\in L^2(\mathcal{F}_\infty,P)$ with zero mean,there ...
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0answers
46 views

Pathwise measurability of Ito integral under supremum norm

I'm doing my first research project on Stochastic Analysis and in order to prove something which is crucial, I need to prove the following claim: LEMMA: Denote by ...
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1answer
49 views

How to find exact solution of this volterra equation?

I was working on numerical solution of this equation (by block pulse). $$x(t)=1+\int_{0}^{t}s^2x(s)ds+\int_{0}^{t}sx(s)dB(s)\\t \in[0,\frac{1}{2}]$$B(t) is standard brownian motion. Author of the ...
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2answers
25 views

Show the stationary distribution of $\partial_tp=\partial_x(bp)+(1/2)\sigma^2\partial_{xx}p$ (forward Kolmogorov) is $p=Ce^{-2\int b/\sigma^2dx}$

I am trying to understand this proof so that I can do the exercises without having to actually memorise the formula and plug in numbers, like a lot of people do. Thanks a lot in advance! So if we ...
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0answers
16 views

Stochastic integral with respect to a poisson process

Let $N_t$ be a Poisson process with parameter $\lambda > 0$ and let $S_t = f(t, N_t)$ be a function which fullfilles the necessary conditions that we can define $\int_0^t S_{u_{-}} dN_u$. Is ...
2
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1answer
48 views

Is there a way to estimate moments of strong solution to SDE

Suppose the SDE $$\mathsf dX_t =b(t,X_t)\mathsf dt + \sigma(t,X_t)\mathsf dW_t,\; X_0 = x$$ where $t\in[0,T]$ has a strong solution. I know in general we can't find an explicit formula for the ...
1
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1answer
35 views

Burkholder's inequality for elementary stochastic integral

An elementary Burkholder's inequality for simple stochastic integral says that given nonnegative martingale $M$ and simple bounded predictable process $H$, it holds that for all $c>0$, the tail ...
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0answers
43 views

Differentiating Stochastic Integral

I was wondering how to write the following integral in differential form: $$\int^t_0 f(s,t)dW_s$$ where $W_s$ is a standard Brownian Motion. In my understanding, if $f(s,t)$ can be written as ...
2
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1answer
45 views

Compute $\mathbb{E}[\tilde{X}_t]$, where $\tilde{X}_t=X_t=(1-t)\int_0^t\frac{1}{1-s}dW_s$ for $0\le t<1$ and $\tilde{X}_t=0$ for $t=1$

I have the following exercise and I don't really understand the answer. I am going to write my professor's answer first, then a question about what I don't understand about my professor's answer and ...
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1answer
42 views

Application of Ito's isometry in deduction of Wiener Ito Chaos expansion

I am trying to learn about the Wiener Ito Chaos expansion and starting reading Oksendal's notes on Malliavin calculus where it is treated in Chapter 1. For a link to the notes, please see ...
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0answers
18 views

Questions on drifts and Girsanov transforms.

I wish to prove the following the statement: "If $\mu$ and $\gamma$ are probability measures on $C([0,\infty), \mathbb{R}^d)$, with $\gamma$ being the standard Weiner measure, $W_t$ being standard ...
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44 views

Just use the expected value for the random coefficient in a differential equation

We often encounter differential equations with some coefficients that are random variables. One way to solve these problems is to replace the random coefficient with its expected value (EV). Then we ...
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0answers
28 views

Is the space of all adapted processes with Càdlàg paths a Banach space?

Consider first the definition of a stochastic integral for simple predictable processes. $$I:\mathbb{S}\rightarrow\mathbb{D},\ H\mapsto I_X(H):=H_0X_0+\sum_{i=1}^nH_i(X_{T_{i+1}}-X_{T_i})$$ The ...
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1answer
49 views

Itō Integral multiplied by Riemann Integral

I was wondering whats the result of an Itō integral multiplied by a Riemann Integral. For example, what is $$\left(\int_0^T f(u)\ \mathsf dW_u\right)\left(\int_0^T g(v)\ \mathsf dv\right)$$ where $W$ ...
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1answer
50 views

Existence stochastic integral

I am trying to understand the prove of the existence of the stochastic integral for a local martinglale null at $0$ and continuous, $M\in \mathcal{M}^c_{0,\text{loc}}$, a predictable process $H\in ...
5
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1answer
81 views

Using Ito theory to decide whether $M^f$ is martingale or a local martingale

I came across the following while reading Ikeda & Watanabe book Stochastic differential equations and Diffusion processes, in page 163-164 At first the sentence $$f(X_t)- f(X_0) - ...
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0answers
26 views

Martingale and quadratic variation inequality

I have the following inequality $$\mathbb{E}(\mid[M^{\Pi^m},M^{\Pi^m}]_T^{1/2}-[M^{\Pi^n},M^{\Pi^n}]_T^{1/2}\mid^p)\leq \mathbb{E}([M^{\Pi^m}-M^{\Pi^n},M^{\Pi^m}-M^{\Pi^n}]_T^{p/2}),$$ where $M$ is a ...
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0answers
105 views

Burkholder-Davis-Gundy inequalities

I want to prove these inequalities, i.e.: For $p\geq 1\ \exists 0<c_p\leq C_p$ such that for any martingale $M$ we have the following inequality: $$c_p\mathbb{E}[[M,M]^{p/2}_\infty]\leq ...
3
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0answers
38 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
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0answers
35 views

Quadratic Variation of Stochastic Integral of Simple Predictable process

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Good Integrator. ...
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26 views

Cross variation

I have a question about the following argument. I see in my book a claim that given 2 stochastic integrals : \begin{align}X_1&:=\int_{0}^{t}f_s\mathsf dM_s\\ X_2&:=\int_{0}^{t}g_s\mathsf ...
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0answers
51 views

Stochastic Integral of Simple Predictable Process is a Martingale

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Martingale. I ...
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0answers
17 views

Weakening mean integral requirements of stochastic integrals.

Consider the Ito integral. It is well known that adapted and measurable processes $f(s,\omega)$ that satisfy \begin{align} E \Big[ \int_0^T \big| f(s,\cdot) \big|^2 ds \Big] < \infty \end{align} ...
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2answers
67 views

Stochastic Exponential - Protter

I am trying to understand the proof of Theorem 37 at page 84 of the book Stochastic Integration and Differential Equations by P. Protter. In the proof there is the following statement, referred to ...
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0answers
23 views

Empirical Quantilfunction as Integral Bound

This is my first post, so please be nice ;) I'll try to outline my problem correctly and whilst keep it as short as possible! I have to deal (for my master thesis) with the integral ...
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1answer
90 views

Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
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33 views

The Itō integral $\sum_{i=1}^nH_{t_{i-1}}\left(B_{t_i}-B_{t_{i-1}}\right)$ of an simple process $H$ is independent of the choice of $(t_0,\ldots,t_n)$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
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0answers
14 views

limit of quadartic variation [duplicate]

I am trying to understand why : $[\int_{0}^{t}a_s dB_s]=\int_{0}^{t}a_s^2 ds$ [] is the 2-variation process, $B$ is brownian motion in the proof I have seen they used Riemman-sums to get an ...
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0answers
37 views

harmonic functions and ito formula

I am trying to prove the mean-value property for harmonic functions in $R^k$ by ito calculus. given $G$ bounded domain and $u$ harmonic function on $G$ then $u(a)=\int_{\partial B_r} u(y)ds(y)$ ...
5
votes
1answer
65 views

Computing an Ito Integral using the Definition

Let $B_t$ be a brownian motion adapted to $\mathcal F_t$. For general $\mathcal F_t$-adapted processes $X_t$ the Ito-integral could be defined as $$ \int_0^t X_s dB_s = \lim_{n\to \infty} \int_0^t ...
0
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1answer
51 views

Quadratic variation of the Brownian motion and Itō's lemma

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
3
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1answer
70 views

Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on ...