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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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
15 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
18 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
29 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 ...
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1answer
25 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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0answers
12 views

Does an integrable IID continuous time stochastic process exist?

Let $t\in[0,T)$ where $0 < T \leq \infty$, and assume a stochastic process exists $Z_t$. The question is: does there exist an IID stochastic process for $Z_t$ such that $Z_t \perp Z_{\tau}$ for ...
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0answers
43 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
19 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
40 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
18 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
49 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
69 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
33 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
30 views

Evaluating Stratonovich integral from definition

I am struggling to evaluate the integral $\displaystyle \int^{T}_{0} B_t \circ dB_t $ from definition. So far I have that $\begin{align} \displaystyle \sum ...
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1answer
19 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 ...
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0answers
26 views

Matlab code for Simulation of SDE [duplicate]

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 ...
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1answer
47 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} ...
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1answer
33 views

Solution to SDE 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. ...
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2answers
111 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 ...
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0answers
43 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. ...
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0answers
40 views

An exponential martingale

Let $H_{t}$ be a bounded continuous and $\textbf{F}^{B}_{t}$ an adapted process. $B$ Brownian motion. Show that $M_{t}= \exp\left(-\int^{t}_{0}H_{s}dB_{s} -\frac{1}{2}\int^{t}_{0}H^{2}_{s}ds\right)$ ...
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0answers
67 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
76 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 ...
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1answer
32 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?
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2answers
65 views

How to solve SDE

SDE: $dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T$ $X_0=a$ answer Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$ and ...
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1answer
92 views

Itô's formula and SDE

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 $(f^{-1})'=\sigma(x),(f^{-1})''=\sigma'(x)$ ...
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1answer
42 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
32 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
22 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
20 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 ...
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1answer
20 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
62 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} ...
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1answer
28 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 ...
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1answer
35 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 ...
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1answer
29 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 ...
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1answer
25 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
48 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} ...
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1answer
95 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
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1answer
46 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
42 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 ...
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0answers
36 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
54 views

MGF of stochastic integral

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 BM. Calculate the moment generating function of Y. I can solve this ...
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0answers
36 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
28 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
33 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 ...
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1answer
49 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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1answer
42 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 ...
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1answer
34 views

Solutions of SDE do not explode when drift term is zero.

Suppose we have $dX_t = \sigma(X_t) dW_t$ where $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is Borel and $W_t$ is a standard one-dimensional Brownian motion. I am trying to show that $X_t$ cannot ...
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1answer
43 views

Random variables independent

We said that two random variables $X,Y$ are independent iff we have that for $Z = X+Y$: $$P_Z(B)=\int_{\mathbb{R}}P_X(B-s)dP_Y(s) = \int_{\mathbb{R}}P_Y(B-s)dP_X(s).$$ But I still don't get this ...
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2answers
40 views

Stratonovich integral of $\sin(W^2)$

I have to solve the following Stratonovich integral: $$\int_{0}^{t}\sin(W^2_s)\circ{dW_s}$$ First of all I use the conversion from Stratonovich to Ito, obtaining ...