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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Computational rules for expectations of functions of wiener processes.

What are some general rules that are helpful for computation/calculation of expectations such as $$ E(X_t | \mathcal{F_s} ), $$ where $X $ is a function of Brownian motions $W_t$ and $\mathcal{F}$ is ...
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1answer
27 views

Paley Wiener stochastic integral

Sorry for the stupid question, no answers necessary anymore! let $(B_t)_{t\in [0,1]}$ be a standard Brownian motion and $F\in C[0,1]$ differentiable. Then the sequence (which is an easy version of ...
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26 views

Convergence in $L^2(\Omega\times (0,T))$

Let $$f_i=\exp(\int_0^T h_i(s)\,{\rm d}W_s-1/2\int_0^T h^2_i(s)\,{\rm d}s)$$ where $W_s$ is a brownian motion in a probability space $(\Omega,F,P) $ and $h_i\in L^2(0,T) $. Suppose $F_n\to F$ in ...
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1answer
53 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
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1answer
43 views

Integral with respect to brownian motion

Let $f$ be a continuous function on $[0,\infty)$ and $B_t$ be a standard Brownian motion. Define $X_t=\int_0^t f(s) dB(s).$ a) Show that $X_t$ is Gaussian and computer its covariance $C(X_s, X_t)$ ...
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1answer
33 views

absolute continuity - Dirac measure with respect to gaussian measure [duplicate]

Let $a \in \mathbb{R}$ and Dirac measure $\delta_a (A) = 0$ if $a \notin A$ and $\delta_a(A) = 1$ if $a \in A$, and let $\mu_1$ be the one-dimensional gaussian measure. Let $\mu$ and $\nu$ be two ...
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1answer
39 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
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0answers
11 views

Determining Bounds of a Generating Function of a Stopping Time [duplicate]

Consider the diffusion process $$DX_t=b(X_t)dt+\sigma(X_t)dW_t$$ where $\sigma\sigma*$ is positive definite and $b, \sigma$ are smooth and bounded. Given a one-dimensional domain bounded from 1 side ...
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1answer
31 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
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2answers
34 views

simple stochastic differentiate

someone can help me to differentiate $$a(t-1)+bt+(1-t)\int_{0}^{t}\dfrac{dB_s}{1-s}?$$ I've tried but I really don't know how to do with the last part.. Thank you somuch for your help
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1answer
20 views

Kunita Watanabe Identity

I am looking for a proof of the following version of Kunita Watanabe Identity: "Let $M,N \in M_{c,loc}$ and $H$ be a locally bounded previsible process. Then $[H \cdot M, N ] = H \cdot [M,N]$" I ...
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2answers
57 views

Stratonovich integral

I'm having some troubles to calculate the Stratonovich integral $I(sin)(t)=\int_{0}^{t}\sin{B_{s}}dB_{s}$. I've tried with the limit of ...
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1answer
55 views

Show independence of stochastic integral and stochastic process

Let $ M_t $ and $ N_t$ be two continuous local martingales with respect to a filtration $ \mathcal{F}_t $. Suppose that $ M_t $ and $ N_t$ are independent and set $X_t = \int_0^t M_s^4 \mathrm{d} M_s ...
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1answer
37 views

Why can $\int_0^t f''(X_s) \, d\langle X \rangle_s$ not be a local martingale?

We know from Itos formula, if $X$ is a continuous local martingale and $f$ has two continuous derivatives, we can write $f(X_t)$ as $$ f(X_t) = \int_0^t f'(X_s) dX_s + \frac{1}{2} \int_0^t ...
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0answers
21 views

basic Stochastic differential equation

I'm sorry but I'm having some troubles to find a solution of this simple stochastic differential equation, $dX_{t}=2\sqrt{X_{t}}dB_{t}+2dt$ where $B_{t}$ is a Brownian motion, please can you help ...
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1answer
124 views

Integral of Wiener Process and Central Limit Theorem

I am trying to solve the following exercise: (1) Given $W$ is a Wiener process, find a constant $M$ such that $\lim\limits_{t\to\infty} \frac{1}{t}\int_{0}^{t}\sin^2W_s ds=M$ (2) Then show ...
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1answer
72 views

How to solve a linear stochastic differential equation?

I don't know how to find a solution of this stochastic differential equation: $dX_{t}=(1+\delta \mu X_{t})dt+\delta X_{t}dB_{t}$ Where $B_{t}$ is a standard Brownian motion and $\mu$ and $\delta$ ...
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1answer
33 views

Deriving Black Scholes using CAPM

I am referring to http://www.frouah.com/finance%20notes/Black%20Scholes%20PDE.pdf Section 3, which is a bit more detailed version of the original derivation from ...
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1answer
101 views

Show the following definition does not give a $\sigma$-addtive measure pathwisely

Given the space of all square-integral functions over $[0,1]$: $L^2([0,1], \mathcal{B}([0,1]), m)$ and a Brownian motion $W_t$ defined on the probability space $(\Omega, \mathcal{F}, P)$, we define ...
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28 views

Ito formula for integral function

Let $$dS_t = \mu S_t dt + \sigma S_t dW_t$$ where $W_t$ is a Wiener process. Let $$Z_t = e^{-r(T-t)} \int_{t}^{T}{h(u,S(u))du} = g(t,S)$$ where $h$ is a known function of $t$ and $S$. How can we ...
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1answer
18 views

Ito Isometry - Definitions

Three different lecturers have provided three different definitions of Ito Isometry. These are: Lecturer A \begin{align*} \mathbb{E}\left[ \left(\int_{0}^{\infty} h_{s}\,dW_{s}\right)^{2} \right] = ...
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29 views

Stochastic integral in Tanaka formula

Tanaka's formula is the following result $$|B_t| = \int_0^t \text{sgn}(B_s)\, dB_s + L_t$$ I can see how to show that the stochastic integral $$M_t = \int_0^t \text{sgn}(B_s)\, dB_s$$ is a martingale ...
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1answer
33 views

Stochastic integral in closed form

Let $(W_t)_{t\geq 0}$ be a Brownian motion and $\alpha>0$ be a constant. Consider the following quantity: $$\mathbb{E}\Big(\int_0^tsdW_s{\bf 1}_{\{t^{-\frac{1}{2}}W_t>\alpha\}}\Big).$$ Can a ...
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10 views

Convergence of normalized stochastic integral

I am wondering about some results about the convergence of processes like that : $$ \frac{1}{T} \int_{0}^{T} H_{s}dM_{s} $$ with $M_{s}$ a semi-martingale when T goes to $ +\infty$ Thanks a lot :-) ...
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2answers
98 views

Martingality Theorem: Solving expectation of a stochastic integral

I am trying to prove that: $$ \Bbb E\left[\int_s^t\sigma e^{-k(t-u)}\sqrt{V_u}dW_u\right] =0$$ Where: $$ dV_t=k~(\theta-V_t)~dt+\sigma\sqrt{V_t}dW_t $$ I have attempted to use Ito's formula on the ...
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1answer
38 views

Ito with the function containing stochastic integral

Statement of problem From Oksendal SDEs question 5.18: The geometric mean reversion process is a solution to: $$ dX_t = k (a - \log X_t) X_t dt + \sigma X_t dB_t $$ In showing that solution is ...
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26 views

Ito's integral from the definition

I am doing Oksendal's book exercises one by one. I got stuck in 3.2. I need to prove, from the definition that $$\int_{0}^{t}B_s^2\text{d}B_s=\frac{B_s^3}{3}-\int_{0}^{t}B_s\text{d}s,$$ where ...
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0answers
10 views

Holder continuity, brwonian motion [duplicate]

Let $B$ stand for a brownian motion on a finite interval $[0,1]$. If i am not wrong, i think that there exists a positive constant $c$, such that almost surely, for h small enough , for all $0< t ...
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1answer
43 views

Problem with understading “mixed” integration

Using standard notation: $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \:\:X_0=x \tag{1}$$ Now in my script it is said that if we integrate both sides, we get: $$X_t=x+\int_0^t ...
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1answer
53 views

Positivity of a stochastic process

I want to simulate the paths of a stochastic process $$ dS_t = r S_t dt + \sigma S_t dW_t$$ Using the Forward Euler method, we can write: $$ S_{n+1} = (1 + r \Delta t_n + \sigma \Delta W_{n}) S_n $$ ...
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11 views

Approximation of the ito SDE using backward Euler approximation

I have the stochastic SDE $ dX_{t}=a X_{t} dt+ b X_{t} dW_{t}$ I succeeded to formulate a forward Euler approximation to approximate it but I have some problems to derive the right backward Euler ...
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23 views

Conditional Expected Value of Occurrence Time in Stochastic Process

I have a stochastic process defined by the intensity function $\lambda(t:F_t)$ where $t$ is time and $F_t$ is the filtration process. The stochastic process is self-exciting and models the occurrence ...
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1answer
23 views

Solve parameter from stochastic integral

how can I solve $\rho$ from the following: $\int_0^T dV_t = \int_0^T \kappa (\theta - V_t) dt + \int_0^T \sigma \rho \sqrt{V_t} dW_t + \int_0^T \sigma \sqrt{1-\rho^2} \sqrt{V_t} dZ_t$, where $W_t$ ...
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2answers
58 views

Covariance of Ornstein - Uhlenbeck Process

I'm considering the Ornstein - Uhlenbeck process $ X(t)=x_{\infty}+e^{-at}(x_{0}-x_{\infty})+b \int_{0}^{t} e^{-a(t-s)} dW(s)$ where $a, b > 0 $ are given constants. I used the Itô Isometry to ...
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43 views

Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
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0answers
20 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
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1answer
86 views

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
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1answer
47 views

Stochastic integration by parts formula to prove identity between iterated integrals

if $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...
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1answer
38 views

Mean value theorem inside the Expectation

Consider a stochastic process $X_t$ with continuous paths. I'd like to apply the mean value theorem inside the expectation, i.e. write something like $$ \operatorname{E} \left[ \int_0^t X_s \, ...
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20 views

Change of variable in stochastic integral

Let $B$ be a standard Bronwian motion. Can we do a change of variable in the sense $s=\theta+h$ $$\int_{0}^{t+h}X_sdB_s=\int_{-h}^{t}X_{\theta+h}dY_\theta.$$ In this case what is the process ...
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0answers
59 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
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0answers
58 views

Ito formula proof for bounded functions using stopping time

I'm self studying with the Oksendal book "Stochastic differential equations" and trying to do some exercises by myself. P.57 the exercise asks for the following (a screenshot will save us typing ...
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1answer
50 views

Stochastic integral wrt the compensated Poisson random measure

I am solving the exercises in a book I have about Lévy processes ("Lévy Processes and Stochastic Calculus", Applebaum, 2003), and I cannot get my head around an exercise that seems rather simple. I ...
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1answer
22 views

Integration with respect to two different Brownian motions

Let $B$ be the standard Brownian motion. The process $W_s=B_{s+a}-B_a$ is also a Brownian motion. I just want an example of a process $X_s$ such that $$E\int_0^tX_sdB_s\neq E\int_0^tX_sdW_s.$$
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11 views

Time homogeneous asset dynamics model

I'm studying asset process. As i know, Black scholes model and CEV model is time homogeneous diffusion model. Are there time homogeneous model ???
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41 views

Stochastic Leibniz rule Ito integral

Assume that $W$ is a Brownian motion and $f=f(t,u)$ is a function of 2 variables such that for all $t$, $f(t,\cdot)$ is adapted to the natural filtration of the Brownian motion and the Ito integral ...
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26 views

stochastic integration with respect to quadratic variation

I have been studying stochastic integral with respect to Brownian motion. At some point my professor generalized our approach such that we are able to integrate with respect to general Martingales. ...
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1answer
28 views

Comparison between these Ito Lemma versions

According to wikipedia : I found another version : Please explain the difference for me.
2
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1answer
45 views

Integrated Ornstein-Uhlenbeck

Suppose we have an OU process given by the stochastic differential equation $dr_t = \kappa(\theta-r_t)dt + \sigma dW_t$. I think that the distribution of $D(t,T) := \int_t^T r_s\;ds$ is normal (I ...
2
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1answer
45 views

upper bound for Ito integral of deterministic integrand

It is well known that Ito integrals with respect to a Brownian motion cannot be defined pathwise because the Brownian motion has infinite 1st order variation. These integrals are defined as limits of ...