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
62 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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30 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
392 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
98 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
65 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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2answers
422 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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1answer
90 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
41 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
76 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
106 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
68 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
43 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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1answer
197 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
111 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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2answers
131 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
111 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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1answer
52 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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0answers
39 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
53 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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2answers
142 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
85 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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0answers
44 views

Ito's integral from the definition [duplicate]

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
12 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
53 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
57 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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43 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
28 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
200 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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0answers
56 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
54 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 ...
2
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1answer
96 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 ...
2
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1answer
78 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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134 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
86 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
133 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
34 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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19 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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180 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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97 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
31 views

Comparison between these Ito Lemma versions

According to wikipedia : I found another version : Please explain the difference for me.
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1answer
73 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
92 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 ...
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82 views

Expectation of e^(cX) if X is a geometric Brownian motion

(Edit:) The short version: Calculate $$E[e^{cY}]$$ if $c < 0$ and $Y$ is lognormally distributed, i.e. $\log(Y) \sim N(\tilde\mu, \tilde\sigma^2)$. The long version: I want to calculate ...
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1answer
34 views

Ito integrals and the Euler scheme

I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ...
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1answer
90 views

A question about Malliavin calculus

An application of Malliavin calculus is to calculate the sensitivity of financial Greeks. However, as in the theory of Malliavin calculus, to take the derivative of a random variable, we need to ...
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21 views

Milestein Scheme

Im struggling in the following schemes. I cant understand how the first scheme is equivalent to the second one. Can somebody help me? Thanks in advance. Moreover there is a typo error in the ...
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0answers
52 views

Why the Ito isometry implies this equality? [duplicate]

If $${\rm Cov}[dW_t,dB_t]=\rho \, dt$$ then why $\mathbb{Cov} \left( \int_0^t \sigma_{1}(s) \mathrm{d} W_s, \int_0^t \sigma_{2}(u) \mathrm{d} B_u \right)$ $\stackrel{\text{Ito isometry}}{=} ...
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0answers
153 views

Expected value of correlated stochastic integrals

I do not understand the following result: Suppose $dz_\chi$ and $ dz_\xi$ are correlated increments of standard Brownian motion with $dz_\chi dz_\xi=\rho dt$ you have the following expectation ...
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125 views

The completed natural filtration of brownian motion is right-continuous, proof?

I have a question concerning a claim in J.F. LeGall's book Mouvement brownien, martingales et calcul stochastique. Let $(\mathcal{F_{t}})$ be the canonical completed filtration on $\Omega$ of a real ...
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1answer
106 views

Evaluating Stratonovich integral from definition

$\bf 3.9.$ Suppose $f\in\mathcal V(0,T)$ and that $t\to f(t,\omega)$ is continuous for a.a. $\omega$. Then we have shown that $$\int\limits_0^T f(t,\omega)dB_t(\omega)=\lim_{\Delta ...