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Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

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19 views

Covariance of two Brownian motion integrals

Assuming that $W(t)$ is a Brownian motion, and considering two integrals $$ X :=\int_{0}^{T} W(t) d W,\quad\text{and}\quad Y :=\int_{0}^{T}(W(t)+t)^{2} d W $$ I'm looking for the covariance $Cov(X, ...
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1answer
20 views

Constructing Correlated Wiener Processes

Construction Hello. I'm reading the attached paper about the construction of correlated processes given a correlation matrix. But I am stuck on equation (2.23) -- surely it should say $c_{ik} . c_{...
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33 views

Proving one form of Ito Isometry using Functional Analysis

I would like to know whether it is possible to give a proof of (one form of) Ito Isometry using a tool which I like to call "the functional analysis"-way. Let me explain the settings first. What we ...
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1answer
50 views

Applying Ito's Identity

I came across an interesting result and a proof for it. I'm confused though about one of the lines. It goes like this. Using the equality: $$\int_0^sf_r\,dW_r=\int_0^\infty 1_{]0,s]}(r)f_r\,dW_r,\...
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1answer
37 views

Partial Derivative for Stochastic Integral

Good day, I am trying to apply Ito's lemma to find an integral but I am struggling with my choice of functions. $\int^ T _0 tdW(t) = T W(T)- \int^T_ 0W(t)dt$ Our version of Itos lemma states the ...
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19 views

A question about measurability on the Taylor expansion of Ito's formula.

I am reading the proof of Itō's formula and self-studying stochastic integration. Essentially, the authors proved the formula for dimension $d = 1$ and they ask the learner to do for greater ...
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72 views

How to calculate expected value of integral?

How to calculate $E \big[(\int ^{t} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^A (\int ^{t+h} _{-\infty} e^{\lambda u} d\tilde{L_\alpha}(u))^B\big]$, where \begin{align} \tilde{L}_\alpha (t) = \...
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1answer
31 views

SDE Integration

Does anyone know how to get the integration of the SDE below (Assume $\sigma \to 0$)? $$\dfrac{\mathrm dS_t}{S_t}=(r_d-r_f)\mathrm dt+\sigma(t, S_t)\mathrm dW_t$$ Thank you in advance! Image Link ...
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1answer
63 views

Expectation of solution to SDE $dX_t=-\tanh(X_t) dt + dW_t$

How do I calculate the expectation of the process given by the SDE $$dX_t=-\tanh(X_t) dt + dW_t, \qquad X_0=x_0$$ and $W_t$ a Wiener process? If I start with $$ d\left(e^{t/2}\sinh(X_t)\right) = e^{...
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1answer
81 views

Black-Scholes model with time-dependent volatility

We consider the Black-Scholes model with time-dependent volatility $\sigma(t)$: $$ dS_{1}(t)=rS_{1}(t)dt+\sigma(t)S_{1}(t)dW(t) $$ The question: what constant $\hat{\sigma}$ one needs to apply such ...
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1answer
23 views

Solution of a second order Stochastic Differential Equation

Consider the following SDE $$dx = (-ax)dt + \sigma db\\ dy = (-by+e^{-x})dt$$ where $a,b,\sigma>0$ and $b$ is a browninan motion PROBLEM: What is the solution of this system? How can I estimate ...
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1answer
22 views

How to compute ito's calculus without knowing its solution already?

I'm having trouble computing Itô's calculus. Take $\int_0^tB_sdB_s$ for example, can I solve it base solely on Itô-Doeblin Formula, instead of assuming $f(x)=\frac{x^2}{2}$? Please help!
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23 views

A Version of Fubini-Tonelli Theorem for Hilbert Space Valued Functions

I'm currently working on a project in which we define a new type of integral. And I'm trying to intechange the integral with expectation, something like $\mathbb{E} \left[ (\mathcal{N})\int f dW \...
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31 views

$B_t^3 - 3t B_t$ is a $L^2$ martingale ($B_t$ being a standard Brownian motion)

By Itô's formula I get that \begin{align} d(B_t^3 - 3t B_t) &= (3 B_t^2 dB_t + 3 B_t dt) - 3(B_t dt + 3 t d B_t) \\ &= (3 B_t^2 + 3t) d B_t \end{align} which seems related to martingale ...
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22 views

Second order derivatives in Ito formula for Brownian motion and local martingale

Itô's formula for a $\mathcal{C}^2$ function of two variables F reads: \begin{align} F(X_t, Y_t) &= F(X_0, Y_0) + \int_0^t \frac{\partial F}{dx}(X_s, Y_s) \, dY_s + \int_0^t \frac{\partial F}{dy}...
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17 views

Multivariable Stochastic (Stratonovich) Integration

Suppose we have two Wiener process variables, possibly correlated, $W_1$ and $W_2$. If I Stratonovich-ly integrate them, is it true that: $$\int W_1dW_2 = (1/2)W_1W_2 $$
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1answer
65 views

Itô integral and stock prices in discrete time.

Let $X_n\sim \text{Bernoulli}(\frac{1}{2})$ with state space $\{-1,1\}$ i.i.d for all $n$. Set $S_n=X_0+X_1+...+X_n$. Suppose that $X_0=2$ a.s. then $\{S_n\}$ has independent increment $\text{...
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1answer
28 views

Stochastic integrals and stopping times

Suppose $\{H(s,\omega): s\ge 0 , \omega \in \Omega \}$ is progressively measurable and $\{ B(t): t \ge 0\}$ a linear Brownian motion. Show that for any stopping time T with: $\mathbb{E} \left[ \int_{0}...
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2answers
40 views

Two Definitions for $E(X)$

Given that $\int_{\Omega}|X(\omega)|dP(\omega)<\infty$ and $\mu_X$ to be the induced probability measure of $X$ on $\mathbb{R}^n $. Why is it that \begin{equation*} \int_\Omega X(\omega)dP(\omega)...
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13 views

Why the stochastic integral is a $L^2$ convergence ? Pointwise convergence doesn't work?

We define $$\int_{0}^Tf(X_t,t)dB_t:=\lim_{n\to \infty }\sum_{k=0}^{n-1}f(X_{t_{i-1}},t_{i-1})(B_{t_{i+1}}-B_{t_i}),$$ where the limit is taken in $L^2$, $\{t_0,...,t_n\}$ is a partition of $[0,T]$ s....
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20 views

Malliavin Calculus: If $u \in \mathbb D^{1,2}(H)$, then $\delta(u) \in \mathbb D^{1,2}$

I am reading a lecture note on Malliavin caculus. "An intro to Malliavin calculus. In proposition 1.6.7, it is claimed that If $u \in \mathbb D^{1,2}(H)$, then $\delta(u) \in \mathbb D^{1,2}$. ...
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21 views

SDE: How to calculate this kind expectation of SDE solution?

For a Stochastic Differential Equation model: $$ dx = bx(1-x/F)dt + cxdB $$ It has exact solution with initial $x(t_0)$: $$ x(t) = \frac{x(t_0)\exp((b-\frac{c^2}{2})\,t + c\,B(t))}{1+b\,\frac{x(t_0)...
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0answers
14 views

Is an exponential function of a stochastic process a smooth function?

Let's say I have an Ito process $X_t$, and another process $Y_t=e^{\int_t^{t+\delta} X_s ds}$. I want to know that quadratic variation of $Y_t$ and another process. I know that the quadratic ...
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0answers
30 views

Dominated Convergence Theorem in Stochastic Calculus (Infinite Dimensional Spaces)

Is there an established Dominated Convergence Theorem version for the stochastic setting? Particularly in infinite dimensional spaces. I have a hard time finding such books. Cite some resources please....
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14 views

Derivative of the fractional Brownian motion

From Jost (2008), we can see $$W_t^{H} = C_H \left\{ \int_{-\infty}^t \frac{dW_s}{(t-s)^{1/2-H}} - \int_{-\infty}^0 \frac{dW_s}{(-s)^{1/2-H}}\right\},$$ with $C_H = \sqrt{\frac{2H\times \Gamma(3/2- ...
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32 views

Markov property of SDE's solution

Considering the SDE $dX_t=b(t,X_t)dt+\sigma (t,X_t)dW_t$ ($W$ is Brownian motion)  If there exists weak solution $(X,W),(\Omega ,\mathscr{F} ,P),\{\mathscr{F}_t\}$, is $X$ Markov process? I know ...
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1answer
37 views

Stochastic Process, Brownian Motion, Ito's Formula

Assume $X,Y$ are stochastic processes satisfying $$X(t) = X(0) + \int_0^tF_X(s)ds + \int_0^tG_X(s)dW(s) $$ $$Y(t) = Y(0) + \int_0^tF_Y(s)ds + \int_0^tG_Y(s)dW(s) $$ for all $0 \leq t \leq T $. Use ...
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38 views

Integral of dirac-delta times the log of dirac-detla

$$ \int_x \delta(x)~\ln(\delta(d))~dx = 0 ? $$ Where $\delta(x)$ denotes the Dirac-Delta function, $ln(\cdot)$ is a logarithm, and $dx$ is simply the diferential of $x$ for the integral. I'm ...
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22 views

non-trivial weak solution of SDE

Let consider the SDE $dX_t=|X_t|^adW_t$ for $0<a<\frac{1}{2}$  I know that this SDE has non-zero weak solution by Engelbrt & Schmidt's theorem. But I cannot find weak solution $(X,W)$ with $...
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1answer
67 views

Why do we need absolute continuity of $\langle M \rangle_t(\omega)$ with repect to the Lebesgue measure?

I am trying to understand the proof of proposition 3.2.6 in Stochastic Calculus and Brownian Motion by Karatzas and Shreve. For $X$ bounded they use Lemma 3.2.4 in the same book and eventually claim(...
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25 views

Expectation of Ito Process

Suppose $X$ is an Ito process defined by $ dX = adt + b_tdB$ and $X_0 = 0$ where $B$ is a Brownian motion. Ito process consists of two parts: one is deterministic and the other is random. Is the ...
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11 views

Brownian motion absorbed at the boundaries

Why the solution of the following SDE $$dX_t=\sqrt{X_t(X_t-1)}dB_t$$ is considered a Brownian motion which is absorbed at the boundaries?
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1answer
57 views

Solving $dS^x(t)=S^x(t)(r(t)dt+\gamma(t,S^x(t))dW(t))$

Consider the stochastic process $S^x(t)$ with $S^x(0)=x$ $$dS^x(t)=S^x(t)(r(t)dt+\gamma(t,S^x(t))dW(t))$$and define $D^x(t)=(\partial/\partial x)S^x(t)$. Then $$dD^x(t)=D^x(t)\Big[r(t)dt+\frac{\...
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36 views

What represent ito integral ? What does means that $\int_0^t B_sdB_s=\frac{B_t^2}{2}-\frac{t}{2}$

I still have difficulties to understand Itô integral, and unfortunately, I don't really understand what represent the Itô integral. Just to inform, I asked, but it has been erased without ...
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29 views

Integral as: martingale or local martingale

I wonder when a stochastic integral is a martingale or a local martingale. Let's assume that we have a process: $X_t = X_0 + \int_{0}^{t} a_s ds + \int_0^t b_s dW_s$ Is this kind of integral a ...
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1answer
37 views

Convergence of stochastic integral to Brownian motion

Let $a \in \mathbb R$, $W(t)$ a standard Brownian motion, and $$ V(t) = a \int_0^{t} e^{-a(t-s)} d W_s. $$ Is it true that $$ \int_0^t V(u) \, du = W(t) - W(0) \quad \text{as} \quad a \to \infty $$ ...
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16 views

What is this Itô integral ? What does it represent?

I know that $\int_0^T g(W_t,t)dW_t$ is a continuous martingale, but I can't get what it represent exactly and can't find any justification anywhere. It's written every where that it's fundamental in ...
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1answer
29 views

Solution to stochastic differential equation $dY = YdW$, where $Y(0)=1$?

How to find the solution to stochastic differential equation (SDE): $dY = YdW$, where $W(t)$ is Brownian motion and $Y(0)=1$? The solution to the corresponding ordinary differential equation is easy ...
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1answer
30 views

Stochastic integration rules

I stumbled upon a stochastic integration result in a book and I'm not sure how it is derived. It goes as follows: Suppose the process $Y^{(t,y)}$ has dynamics under the equivalent martingale measure $$...
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1answer
38 views

Deterministic Integral of a Predictable Process is Predictable

I was reviewing a proof of existence of solutions to stochastic evolution equations which takes the form of a fixed point argument on the space of predictable processes such that $$ \sup_{t\leq T}\...
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22 views

Application of Gram-Charlier expansion for Swaption pricing with drift extension

I am doing a project for university and I'm stuck at some point. The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension. I already found out how ...
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1answer
45 views

Stochastic differential equation with respect to general stochastic process

I am a beginner with stochastic processes and I want to clarify some ideas that I just learnt. Say I have a stochastic process: $$X_t = X_0+ \int_0^t \sigma_s dW_s $$ where $W_s$ is a standard ...
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36 views

Show that stochastic integral is Gaussian by independency of increments

Let $h \in L^2[(0,1)]$ and consider the process $(X_t)_{t \in [0,1]} = \big( \int_0^t h(s) \text{d}B_s \big)_{t \in [0,1]}$, where $B_s$ is Brownian motion. By construction of the integral I know ...
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20 views

Apply the usual version of Itō formula to get an expression of the stochastic process {$|X_t-x|,t∈[0,t]$} as an Itō process?

Consider an Itō process $X_t=x+\int_0^tφ(s)dBs+\int_0^tφ(s)ds$. Could you apply the usual version of Itō formula to get an expression of the stochastic process {$|X_t-x|,t∈[0,t]$} as an Itō process? ...
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0answers
22 views

How would one solve for a process where a stochastic random variable is divided by a deterministic random variable?

I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its ...
3
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1answer
111 views

Show a Continuous Local Martingale is a Martingale

Let $B = (B_t)_{t≥0}$ be a standard Brownian motion started at zero, let $X=(X_t)_{t≥0}$ be a nonnegative stochastic process solving $$dX_t = 3 \, dt + 2\sqrt{X_t} \, dB_t \qquad(X_0 = 0)$$ and let $$...
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0answers
32 views

Covariation of two processes

Consider the processin $[0,T]$, $Y_t=\int_0^t f(s,\omega)dW_s$, where $\int_0^t E[f(s,\omega)^2]ds<\infty$ and $W$ is a BM. Consider also, $X_t = \int_0^t M(s,\omega)ds$, where $M$ is some ...
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0answers
25 views

Integral representation of a stochastic differential equation

If I have a stochastic differential equation gives as $$ dX_t = aW_t^2dt +bW_t^3dW_t$$ where $w_t$ is a wiener process and $a,b$ are real numbers. How can I reach the integral representation of $X_t$? ...
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1answer
64 views

Maximal inequality for the Itō integral

Let $W$ be a Brownian motion and $X$ be a predictable process with $$\operatorname E\left[\int_0^t|X_s|^2\:{\rm d}s\right]<\infty\;\;\;\text{for all }t\ge0.$$ Now, let $p\ge2$. How can we show ...
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70 views

Is the strong solution of a SDE adapted to the filtration generated by the driving Brownian motion?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $\xi$ be an $\...