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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82 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
62 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 ...
2
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1answer
68 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 ...
0
votes
1answer
29 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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0answers
15 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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0answers
84 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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0answers
42 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. ...
1
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1answer
29 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
57 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
votes
1answer
48 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 ...
0
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0answers
60 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 ...
0
votes
1answer
31 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 ...
2
votes
1answer
70 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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0answers
19 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
51 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
110 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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0answers
78 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 ...
1
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1answer
62 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 ...
0
votes
1answer
30 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
0
votes
1answer
50 views

What is the distribution of this random variable? [closed]

Find the distribution of this random variable: $$X_t=\exp\left(t \int_0^t sdW_s\right)$$ knowing that $W$ is a Brownian motion in the filtered space $(\Omega, \mathcal{F},P,(\mathcal{F}_t)_{t\geq0} ...
0
votes
1answer
78 views

Solution to the linear SDE $dX_t = \alpha X_t \, dt + \sqrt{2} dB_t$ using Itô calculus

So if I have the following generator and an initial condition: $$A(f)(x) = \alpha x f'(x) + f''(x) \\ X_0 = x \in \mathbb{R}^+$$ I've been asked to find $X_t$ and assume that $\alpha$ is a constant. ...
2
votes
2answers
421 views

Matlab Code to simulate trajectories of Ito process.

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
0
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0answers
51 views

Stochastic Differential equations with $\sin(x^2)$ as drift.

Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance. ...
1
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1answer
76 views

An exponential martingale [closed]

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
3
votes
0answers
71 views

Multipe Ito Integrals

Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ...
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0answers
99 views

Write the Hamilton Jacobi Bellman equation

Consider the following stochastic optimal control problem. \begin{equation} V(t,x) = \max_{u}\,\, \log \left(\mathbb{E}\left[\int_{0}^{T} u^{2}(t)dt\right]\right) \end{equation} subject to the ...
0
votes
1answer
56 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
-3
votes
2answers
92 views

How to solve the SDE $dX_t = \frac{b-X_t}{T-t} \,dt + dW_t$?

SDE: $$dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T, \qquad X_0 = a$$ Answer: Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$, then $$\begin{align*} ...
0
votes
1answer
113 views

Solve the SDE $dX_t = \frac{1}{2}\sigma(X_t)\sigma'(X_t)dt+\sigma(X_t)dW_t$

Solve this SDE: $dX_t=\frac{1}{2}\sigma(X_t)\sigma'(X_t)dt+\sigma(X_t)dW_t$ with $X_0=x_0$ My try is let $f(x)=\int_{x_0}^{x}\frac{dy}{\sigma(y)}$ and $(f^{-1})'=\sigma(x),(f^{-1})''=\sigma'(x)$ ...
1
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1answer
55 views

integral approximation (law of large numbers)

I am totally at a loss with this question and don't even know where to begin. Let $g:[0, 1]\rightarrow \mathbb{R}$ be a measurable and Lebesgue-integrable function. $U_1, U_2, \dots$ be a series of ...
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0answers
47 views

partial derivative of stochastic variable inside an integral

Very simple question, is it correct to take a partial derivative of stochastic variable inside an integral. If not, why? is$ \frac {\partial}{\partial R} \int_q^Q R(v) dv = \int_q^Q dv$ ? where R is ...
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0answers
26 views

Integral of a non-linear step function on closed interval

I need to compute the following integral for a random variable $a$ with known support and CDF: \begin{equation} \int_{a^L}^{a^H} \left( \sum_{j=1}^{N} \begin{cases} B_j a \mbox{ if } a \leq a_j^*\\ 0 ...
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0answers
32 views

Expected value of solution of SDE

Is there any way to find expectation of $X_t$ defined by the following SDE? $dX_t = -[\sin(2X(t)) + \frac{1}{4}\sin(4X(t))]dt + \sqrt{2}\cos^2 x dB(t), X(0)=1, t \in [0,\tau),$ where $\mathbb{B}$ is ...
0
votes
1answer
41 views

Question on Ito Isometry and bounds of integration

I am trying to find the variance of $\int_t^T(T-s)~dW_s$ I was wondering if this approach is correct: $$ Var~(\int_t^T(T-s)~dW_s~)=\mathbb E~[~(~\int_t^T(T-s)~dW_s~)^2~]=\mathbb ...
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2answers
72 views

Show $E[h(X)] = \int_0^{\infty} h'(t)P[X>t]dt$ and the first two moments

Let $X\geq 0$ be a real random variable and $h:\mathbb{R} \rightarrow \mathbb{R}$ a monotonously growing, continuously differentiable function with $h(0)=0$. Show: $E[h(X)] = \int_0^{\infty} ...
1
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1answer
42 views

Show that a stochastic process is a martingale

Use Ito's formula to prove that the following stochastic process is a $\{\mathcal{F_t}\}$- martingale. a) $X_t = e^{\frac{1}{2}t}cosB_t \ \ \ \ (B_t \in \mathbb{R})$ So ...
0
votes
1answer
50 views

Stochastic differential equation with trigonometric functions

I heard that the following SDE can be solved analitically by substitution: $dX(t) = - \left[ \sin (2 X(t) ) + \frac{1}{4} \sin (4 X(t) ) \right] dt + \sqrt{2} \cos^2 X(t) dB(t),$ $X(0) = 1, \; t \in ...
0
votes
1answer
46 views

Ito's process and martingale [duplicate]

Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...
0
votes
1answer
32 views

SDE transformation using a primitive of a function?

Consider the following SDEs : (E) : $dX_t = (\alpha b(X_t) + {1\over2}b(X_t)b'(X_t))dt + b(X_t)dB_t$ (E') : $dY_t = \alpha dt + dB_t $ prove that E can be transformed to E' using : $ ...
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0answers
63 views

How to write the Hamilton Jacobi Bellman equation

We consider the following optimal control problem \begin{equation} V(t,x)=\max_{u}\mathbb{E} ( \log [\int_{0}^{T}u^{2}(t)dt + U(X(T))]) \end{equation} subject to the state process \begin{equation} ...
1
vote
1answer
117 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
3
votes
1answer
140 views

Expectation of Ito integral, part 2, and Fubini theorem

I previously asked a question (Expectation of Ito integral). I have additional questions on the same subject. Let's say that we have an Ito process such as $$ X(t)=X(0) + \int_0^t a ds + \int_0^t b ...
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1answer
88 views

Expectation of Ito integral

The expectation of an Itô stochastic integral is zero $$ E[\int_0^t X(s)dB(s)\,]=0 $$ if $$ \int_0^t E[X^2(s)]ds\,<\infty $$ It is sometimes possible to check this condition directly if the ...
0
votes
0answers
70 views

BMO martingale and exponential martingale

Consider the BSDE, $$ Y_{T}-Y_{t}=\sum_{i=1}^{n} \int_{t}^{T} Z_{s}^{i}dB_{s}^{i} - \frac{1}{2}\int_{t}^{T} \left| Z_{s}\right|^{2}ds $$ where $B$ is a standard Brownian motion on a complete ...
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2answers
72 views

Moment generating function of the stochastic integral $\int_0^t \alpha_s \, dW_s$

Question: Let: $$ Y_t=\int_0^t\alpha_s \, dW_s $$ where $\alpha_t$ is a deterministic, continuous integrand and $W_t$ is a P Brownian motion. Calculate the moment generating function of $Y$. I can ...
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0answers
39 views

SDE with no weak solution

I'm facing the followingd d-dimensional SDE: $$dY_t=\sigma(h_t)\,dB_t$$ In addition it holds, that: $h_t\in H$ and $H$ is compact (for example the simplex of $R^n$) the proces $h_t$ is progressivley ...
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1answer
33 views

I want to show $\operatorname{Cov}(X(t),X(s))=\min(s,t)- \frac{st}{T}.$

i have this Equation with Condition $X\left(0\right)=a $ and $ 0\le t \lt T$ $$dX\left(t\right)=\frac{b-X\left(t\right)}{t-T}dt+dB\left(t\right)$$ I solved and got $$X\left(t\right)= ...
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0answers
43 views

Girsanov Measure Question.

If $Z_t = exp^{\int_0^t X_s dW_s - \frac{1}{2} \int_0^t (X_s)^2 ds}$ is a martinagle then by Girsanov's theorem, the measure $P_T$ defined by $P_T(A) = E^P(AZ_T)$ is mutually absolutely continuous ...
0
votes
1answer
132 views

Expectation of product of stochastic integral and brownian motion

Find the covariance: $$ COV((\int_t^T(T-s)dW_s), W_t) $$ I used the covariance formula: COV(X,Y) = E(XY) - E(X)E(Y) = E(XY) as E(X)=E(Y)=0 But I am stuck on figuring out the expectation of the ...
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vote
1answer
48 views

Stochastic Integral Help

Let W(t) be a Brownian Motion. Show that the integral: $$ \int_t^T W(s)ds $$ can be written in terms of the stochastic integral: $$ \int_t^T (T-s)dW(S) $$ Is there an error with this question? I ...