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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5
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93 views

Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
0
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0answers
30 views

Generalization of the Ito formula

I have a question concerning Ito’s formula for semimartingales with jumps. I am familiar with Ito’s formula in the following setting: Let $X_t=X_0+M_t+A_t$ be an $\mathbb{R}^d$-valued continuous ...
0
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0answers
29 views

Covariance of two stochastic integrals

Consider the stochastic integral $\int_{0}^{1}J(r)M(r,\lambda) dr$ where $J(r)$ is a demeaned Ornstein-Uhlenbeck process and $M(r,\lambda)=W(r,\lambda)-\lambda W(r,1)$ a Brownian Sheet, independent of ...
2
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0answers
40 views

Why isn't this stochastic integral trivial?

I have a stopping time $\tau$ and a stochastic process $f$. Then the following equation is true: \begin{equation} \int^{t\wedge\tau}_{0}f(s)dW(s)=\int^{t}_{0}f(s)\chi_{[0,\tau]}(s)dW(s) ...
0
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0answers
22 views

How can I formally arrive at solution of “deterministic SDE”

Let $dX_t=\mu X_t dt+\sigma X_t dW_t$. We know that this is a shorthand for integral equation: $X_t=X_0+\int_0^t\mu X_s ds + \int_0^t\sigma X_s dW_s$ Now: what if our equation looks like this ...
0
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0answers
18 views

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$ and $E\{\mid \int_{\alpha}^{\beta}f(t)d\omega (t)\mid^2|\mathscr{F}_\alpha ...
0
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0answers
26 views

Solving a Stochastic PDE with two variables in time

I am trying to work on exercise 5.13 in the book Arbitrage Theory in Continuous time by Thomas Bjork. The equation to solve is; \begin{eqnarray*} \frac{\partial F}{\partial t} (t,x,y) + \frac{1}{2} ...
1
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1answer
31 views

Prove that: $E[\int^{\tau}_{0} f(t)d\omega(t)]=0$ and $E\mid \int^{\tau}_{0} f(t)d\omega(t)\mid^2=E[\int^{\tau}_{0} f^2(t)dt]$.

Suppose $f \in L^{2}_{\omega} [0, \infty]$, and $\tau$ is a stopping time such that $E[\int^{\tau}_{0} f^2(t)dt]<\infty$. Prove that: $E[\int^{\tau}_{0} f(t)d\omega(t)]=0$ and $E\mid ...
1
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1answer
37 views

Why writing $[X,Y]_t$ as $dX_t dY_t$ is so called “abuse of notation”

Why writing $d[X,Y]_t$ as $dX_t dY_t$ or $[B]_t$ as $\int_0^tdt$ is so called "abuse of notation"? Is it because $[B]_t \rightarrow \int_0^tdt$ a.s. but they are not equal?
0
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0answers
20 views

Moments of the integrated Bessel process

I am trying to compute the moments of the integrated and the integrated-inverse Bessel process. For simplicity, if $X_t$ is a BES$(d)$ assuming $d>2$, I am trying then to compute $$\mathbb E ...
0
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0answers
12 views

Which point is the correct one when integrating w.r.t a discrete martingale?

first post here so I will try to explain as well as possible. In Durrett's book of Brownian Motion and Martingales, he uses the following example: $$ X_t = \begin{cases} 0, t<T \\ \xi, t\geq T ...
1
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2answers
52 views

Why is a stochastic integral w.r.t a martingale always a local martingale?

In my course on stochastic calculus, the professor mentioned that stochastic integral w.r.t a martingale always a local martingale? How can I rigorously show this? I know that when integrating wrt to ...
9
votes
2answers
332 views

Itô's formula: Differential form

I've started a course on financial mathematics and I'm currently being introduced to stochastical analysis, spesifically Itô's formula. From the book: It is sometimes useful to use the following ...
0
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0answers
42 views

Strong Markov property of Ito Diffusion - why must the stopping time be a.s. finite ? (Oksendal 6th edition p117 )

I am reading the proof of the Strong Markov property for Ito diffusions In Oksendal 6th edition p117 Theorem 7.2.4, and I do not understand where the fact that the stopping time has to be almost ...
0
votes
2answers
32 views

Ito's Lemma Simple Application

I need help applying Ito's Lemma to show a given result. $B_t$ is standard Brownian motion $dS_t = 0.4 S_tdt + 0.5 S_tdB_t$ I need to find $dlog(S_t)$ I am told it is $(0.4-1/8)dt + 0.5 dB_t$ ...
0
votes
1answer
41 views

what's the usage of purely discontinuous martingale in stochastic integral?

Recently I'm reading Jacod's Limit Theorems for Stochastic Process ,chapter 1 and I'm confused with the general stochastic integral for semimartingales. $H$ is locally bounded predictable process. ...
4
votes
1answer
82 views

Motivation behind Ito integral

Today my professor introduced the Ito integral as a way to make sense of $$\int \sigma(u) \cdot "noise"du$$ where noise is modeled as Brownian motion. He then said: With Riemann integrals you ...
0
votes
1answer
25 views

Verifying the identity $E\left( \int^t_0 X_s ds \right)^2 = \int^t_0 \int^t_0 E(X_s X_u)\,ds\, du$

I am doing the following exercise: The thing I am struggling with is the identity given in the hint: $$ E\left( \int^t_0 X_s ds \right)^2 = \int^t_0 \int^t_0 E(X_s X_u)\,ds\, du $$ I am unable to ...
1
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0answers
20 views

Large deviation for Brownian path on $[0,\infty)$

It seems strange to me that all we can find about Schilder's theorem in the literature is on a finite interval of Brownian path. If we equip the space of continuous function starting from $0$, ...
0
votes
1answer
37 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 ...
0
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0answers
62 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 ...
0
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0answers
16 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 ...
-1
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1answer
34 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
votes
1answer
78 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
votes
1answer
48 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 ...
0
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0answers
17 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 ...
0
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1answer
25 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 ...
0
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0answers
34 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
34 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 ...
2
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0answers
14 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 $H$. ...
0
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0answers
69 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 ...
0
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0answers
13 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
votes
1answer
114 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 ...
-1
votes
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
25 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
69 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, ...
0
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0answers
23 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
57 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 ...
0
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1answer
52 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 ...
1
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1answer
27 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 ...
0
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0answers
19 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
votes
1answer
57 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
vote
1answer
39 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
54 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
votes
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 ...
0
votes
1answer
51 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 ...
1
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0answers
19 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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0answers
54 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 ...
0
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0answers
29 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 ...