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.
2,284
questions
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Simplifying a Stochastic Integral to a Summation
I am writing my undergraduate dissertation on Self-Exciting Processes where the conditional intensity function is expressed as:
$$\lambda_t | \mathcal{H}_t = \lambda_0 + \int_0^t \phi(t-s)dN_s = \...
- 21
1
vote
1
answer
46
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Independence stochastic integral
I've been finding some difficulties in an exercise about the stochastic integral:
Consider the following stochastic process:
$$X_t=\int_0^t\sigma_udW_u$$
Where $\sigma_u$ is a cadlag deterministic ...
- 662
5
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0
answers
26
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Show that $\xi X \mathbb 1_{(t_1,t_2]} \in \Lambda_T^2(W)$ and $\int_{t_1}^{t^2}\xi XdW=\xi \int_{t_1}^{t_2}XdW$
Let $X \in \Lambda_T^2(W), t_1<t_2<T$ and let $\xi$ be any random variable $\mathcal F_{t_1}$ - measurable. Show that $$\xi X \mathbb 1_{(t_1,t_2]} \in \Lambda_T^2(W)$$ and $$\int_{t_1}^{t^2}\xi ...
- 137
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Using the definition of a stochastic integral directly show that $\int_0^t W_s^2 dW_s=\frac 13 W_t^3 - \int_0^t W_s ds$
Using the definition of a stochastic integral directly show that $$\int_0^t W_s^2 dW_s=\frac 13 W_t^3 - \int_0^t W_s ds$$
There are many solutions to this task on the forum, but they use the the ...
- 137
-1
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0
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25
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Ito-Formula for a Poisson-Process [closed]
I am new to Stochastic Theory and trying to understand (Prop 20.13) of this Article
https://personal.ntu.edu.sg/nprivault/MA5182/stochastic-calculus-jump-processes.pdf
(The Ito-Formula for a Poisson-...
- 91
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votes
1
answer
36
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Missing term in Ito Integral
If I define a deterministic process $V(y,t)$ then by Taylor I have:
$$
V(0+\delta y, 0+\delta t) = V(y_0, t_0) + \frac{\partial V}{\partial y}|_0\delta y + + \frac{\partial V}{\partial t}|_0\delta t + ...
- 242
1
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1
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38
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Find the stochastic differential equation that is solved by $e^{Y_t}$
Let $Y_t=\left(\int_0^t e^{t-s}dB_s\right)^2$. Find the stochastic differential equation that is solved by $e^{Y_t}$. The answer must be given in differential notation.
What I did was rewrite $Y_t$ as
...
- 981
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22
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Some Questions About Covariation Calculation
When I'm reading "An Introduction to SPDE" by John B. Walsh, I met some problems in the chapter about Stochastic Integrals.
In Page 290, the author defined the covariance measure
$$Q(A,B,(s,...
- 31
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1
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48
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Linear SDE with locally Lipschitz coefficients and without linear growth
I am seeking clarification on the existence and uniqueness of a strong solution to a stochastic differential equation (SDE) in the context of a Brownian motion. Let $\mathrm{W}$ be a $q$-dimensional ...
- 19
3
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1
answer
58
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Find a function $b \in L^p([0,1])$ for $p \in [1,2)$ such that the laws of $W$ and $W + \int^{\cdot}_0b_s ds$ are mutually singular.
I want to find a $f \in L^p([0,1])$ where $p \in [1,2)$ such that the laws of $W$ and $W + \int^{\cdot}_{0}f_s ds$ are mutually singular. I know the following result where $D[0,1] =
\lbrace F \in ...
- 1,040
4
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1
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59
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Why are stochastic integrals of not simple processes adapted?
If $X$ is a continuous adapted stochastic process, and $Y$ is a square integrable martingale, then how do we know that $I\left(t\right)\dot{=}\int_{0}^{t}X_{s}dY_{s}$ is also adapted?
I know that by ...
- 1,258
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1
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60
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When can the order of a Ito integral and a.s. limit be interchanged
When can the order of a Ito integral and a.s. limit be interchanged i.e. assume you got
$Y_t^n\rightarrow Y_t$.
When can we say
$$\int\limits_0^t Y_t^n dW_s \rightarrow \int\limits_0^t Y_t dW_s$$
I ...
- 740
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1
answer
33
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Cameron-Martin theorem for $C([0,\infty))$.
Let B be a $1$-dimensional Brownian motion. I want to show that $B_t$ and $B_t + t$ are mutually singular on $C([0,\infty))$. I know that for $T > 0$ and
\begin{equation}
D[0,1] = \Bigl \lbrace F \...
- 1,040
2
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1
answer
71
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Non-continuity of Levy area
I am trying to get familiar with Rough Path Theory by reading 'Rough Path Theory and Stochastic Calculus' by Yuzuru Inahama. He argues in favour of Rough Paths as opposed to Ito integrals as the ...
- 429
2
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1
answer
53
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How can I find a stochastic process $A_t$ s.t. $\frac{1}{S_t}=\mathcal{E}(A)_t$?
Let $\mu, \sigma$ be constants and $(B_t)_t$ a brownian motion and define the process $S_t=S_0e^{\left(\mu-\frac{\sigma^2}{2}\right)t+\sigma B_t}$. Then define $U_t=\frac{1}{S_t}$. I want to find a ...
- 2,669
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How to use Ito’s formula calculate a norm of complex function?
Suppose $u(t)\in H^{1}(0,T)$ is a complex value function that $\forall t\in[0,T]$, satisfies:
$du(x,t)=f(u(x,t))dt+g(u(x,t),t)dW$, where $W$ is a Brownian motion.
Calculate $d\Vert u(t)\Vert^p$
where $...
- 63
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1
answer
71
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Proof that the stochastic exponential is a local martingale
I'm struggling to understand the proof as to why the stochastic exponential is a continuous non-negative local martingale. My notes say the following:
where 3.6 is $Z_t = 1 + \int^t_0 Z_s dX_s$.
I ...
- 431
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2
answers
57
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Convergence of approximating integral sum in case of stochastic integrals
Is it true, that if the integrand in the
$$\int_{0}^{T}X\left(t\right)dY\left(t\right)$$
integral is deterministic (so it is the same “function” for each $\omega\in\Omega$) and continuous, then it ...
- 1,258
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1
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227
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Stochastic integration by parts of geometric brownian motion
My objective is to calculate the integral of a geometric brownian motion $Y=e^{\alpha t+\beta W_t}$, i.e. $$\int_{0}^T e^{\alpha s+\beta W_s}ds$$
and to characterize the moments of resulting random ...
- 73
2
votes
1
answer
54
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How to express the the stochastic integral into an Ito process
I am facing problem while trying to express the following stochastic integral into an Ito process.
$$X_{t} = e^{-t}\int_{0}^{t}e^{u}dW_{u}$$
I would like to express the above as
$dX_{t} = \Theta dt + \...
- 21
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57
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Showing more explicitly a proof of regularity from Karatzas and Shreve p245
Assume that $d \geq 2$ and fix any $a \in D$. We would like to prove the following from theorem 2.21 Karatzas and Shreve p245 :
Theorem:
$$\lim_{{x \to a, x \in D}} E^x \{ f(W_{\tau_D}) \} = f(a)$$ ...
- 103
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29
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Integral of birth-death process up to stopping time
We are trying to find the average sum of branches on a tree representing a birth-death process, with fixed birth and death parameters.
Let $X_t$ be the number of individuals at time $t\geq 0$, with $...
- 121
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39
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Expectation of the Ito integral of a progressively measurable process
I consider $\phi\in H_{2}^{2}$ the set of progressively measurable process such that
$$
\lVert \phi\rVert_{H_{2}^{2}} = \mathbb{E}\left(\int_{0}^{\infty}\phi_{s}^{2}ds\right) <\infty
$$
I would ...
- 1,130
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1
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solving ODE contain matrix
I am currently researching the specific image generation problem in this paper 'Score_Based Generative modeling through stochastic differential equation' At the end of page 14, the authors are using ...
- 11
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0
answers
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Asymptotic behavior of the integrand in a stochastic integral
I am dealing with a stochastic integral (expected value) as follows: Suppose $c>0$, $X_t\ge0$ is an increasing process and $f\ge0$ is decreasing in $(-\infty,\infty)$. If
$$
E\left(\int_{0}^{\infty}...
- 421
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Ito formula and confusion with the operator $d$
Thanks for visiting my question.
Im am currently working on this paper (https://arxiv.org/abs/2305.02523) and I am stuck at page 21 (Theorem 14 proof).
First these SDE's were defined:
\begin{align*}
...
- 23
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1
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39
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Is the reciprocal of a Wiener-process is well defined?
Is the reciprocal of a Wiener-process is well defined?
More generally, does Stochastic Calculus work with such „reciprocal processess” of the form $\left(\frac{1}{X}\right)_{t}$, where the denominator ...
- 1,258
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How to treat the last term in equation $X(t) = \int_0^t ds \hspace{0.1 cm} a(s)X(s) + \int_0^t K(t, s) dW(s) + \frac{d W(t)}{dt}$
I would like to solve an stochastic equation $X(t) = \int_0^t ds \hspace{0.1 cm} a(s)X(s) + \int_0^t K(t, s) dW(s) + \frac{d W(t)}{dt}$ numerically.
where:
$X(t)$ is the dependent variable, which ...
- 65
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What is the distribution of $Y_t = {W_t | W_T = x}$, constructed from Wiener process?
$W_t$ is a Wiener process, $W_0 = 0$. Let $Y_t \,\,\text{be}\,\, {W_t | W_T = x}$. What is the distribution of $Y_t$?
I am not so sure, I do get that $W_T - W_t = x - W_t$ would depend on $(T-t)$ and ...
- 1
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0
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64
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About Martingale Stopping Theorem
Prove that:
A right-continuous process $X$ adapted to the filtration $\{\mathcal{F}_t\}$ is a martingale if and only if for any bounded stopping time $T$, $X_T \in L^1$ and $E[X_T] = E[X_0]$.
I know ...
- 31
1
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1
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51
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Stochastic Integral Evaluation
We want to evaluate:
$$
\int_{z=0}^T \int_{s=0}^z dW(s) dW(z)
$$
But I don't really understand how to approach the problem. Is this just zero? Why?
- 37
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answers
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Ito integral of $\int_0^t B_{s/2} d B_s$
Are the more or less closed formula for Ito integral
$$
\int_0^t B_{s/2} d B_s
$$
where $B_s$ is standard 1-dimensional Brownian motion?
For example, $\int_0^t B_s d B_s = (B_t^2 - t) / 2$, are there ...
- 149
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How to solve SDE with Ito formula?
thank you very much for clicking on my question.
I'm working on this paper (https://www.duo.uio.no/bitstream/handle/10852/10566/pm12-05.pdf?sequence=1) (Page 3) and want to solve the following SDE:
$...
- 23
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Why is $X_{t+}$ measurable w.r.t $\mathcal{F}_{t_+}$?
I encountered a trouble when I read theorem 3.17 in the book named "Brownian Motion, Martingales, and Stochastic Calculus". In this book, it first state that $X_{t+}:= \lim_{s \in D \atop s \...
- 135
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1
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Stochastic multiple Ito integrals in higher order Milstein schemes (>1)
In the series expansion to obtain the form of higher order Milstein schemes you need to evaluate these two sister integrals (where W_t is the Wiener process):
$$
\int_{t_i}^{t_{i+1}}dW_u\int_{t_i}^{u}...
- 11
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1
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Are the integral and differential definitions of Ito process equivalent? [closed]
I think an ito process $X_t$ can be defined as
$$X_t := X_0 + \int_0^t\sigma_s dB_s + \int_0^t\mu_s ds.$$
(Is this an Ito drift-difussion process?)
(Why use the subscript $s$? Eg. why is it $\sigma_s$ ...
- 2,058
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Levy-Khintchine formula using stochastic integral
looking through Nicolas Privault's Introduction to Stochastic Finance with Market Examples, I ran into the following version of Levy-Khintchine formula:
$$
\mathbb E\left[\exp\left(\int_0^T f(t) dY_t\...
- 1,884
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65
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About Ito's isometry application
I've been re-reading one passage from the book of Bernt Øksendal for a whole week and still can't understand one moment in proof of that theorem:
Lemma 6.2.7.: $\hat{X_t} = E[X_t] +
\int_{0}^{t}\frac{...
- 153
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0
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19
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Extending Schilling's proof of Ito process approximation by simple processes for one-dimensional case to multivariate case
Below is the proof of Lemma 18.5 from Rene Schilling's Brownian motion which states that an Ito process can be approximated uniformly in probability by a simple Ito process.
Now it is stated in the ...
- 5,233
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0
answers
85
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Laplace transform of a stochastic process
Let $R := (R_1, R_2)$ be a two dimensional diffusion process defined by the following SDE:
$$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t}\mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \mathrm{d}W_t$$
$$\...
- 21
1
vote
0
answers
39
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Multi-dimensional Itô-isometry
I am looking to prove the following equation: $$\left(\int_s^t \sigma(X_r)dW_r\right)\left(\int_s^t \sigma(X_r)dW_r\right)^\top = \int_s^t \sigma(X_r)\sigma(X_r)^\top dr$$
for a function $\sigma(x)\in ...
- 23
1
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0
answers
33
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Reverse engineer Ito's Lemma to find $X_t$ for $dX_t = [\nu - \gamma\left(X_t - \nu\,t \right)]dt + \sigma\,dW_t$ [duplicate]
Find an integral expression for $X_t$, where $X$ is an Ornstein-Uhlenbeck-type process
governed by the stochastic differential equation:
$dX_t = [\nu - \gamma\left(X_t - \nu\,t \right)]dt + \sigma\,...
- 481
2
votes
0
answers
70
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Expectation Value of the Product of a Time integral and a Ito Integral
Consider a stochastic process $X_t$
\begin{equation}
dX_t = a(X_t)dt + \sigma dW_t
\end{equation}
where $W_t$ is a Wiener Process.
I know the expectation value of the product of two stochastic ...
- 21
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votes
1
answer
86
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How to compute the second derivative of a function of a stochastic process?
Background
Suppose we have a state $\mathbf{x}(t) \in \mathbb{R}^{d \times 1}$ evolving according to
\begin{equation}
\mathrm{d}\mathbf{x}(t) = \mathbf{F}(\mathbf{x}(t)) \mathrm{d}t + \mathbf{G}(...
- 129
1
vote
1
answer
72
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Differentiating Stochastic Integrals (Ito Integrals)
If I wanted to differentiate a stochastic integral, is this logic correct? $$X_t = \int_0^t B^2_s dB_s\\ dX_t = d \int_0^t B^2_s dB_s \\ dX_t = B^2_tdB_t - B^2_0dB_0 \\
dX_t = B^2_tdB_t \quad (B_0 = ...
1
vote
0
answers
74
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How is Leibniz integration rule applied to stochastic integrals? [duplicate]
I am working through the mathematics behind the Hull-White short rate model and am currently stuck on how to take the partial derivative of and evaluate a stochastic integral when looking at how bond ...
1
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0
answers
23
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Doubt in representation of multiple stochastic integral, Kallenberg Th. 11.25
This is a question about Kallenberg's textbook, Foundations of Modern Probability, theorem 11.25. Let $I_n$ denote the Weiner-Ito stochastic integral on $L^2([0,1]^n,\lambda^n)$, with $\eta$ an ...
- 653
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0
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29
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Deriving stochastic process trajectory integral variance
Suppose the process $x_t=u_t+v_t\int\limits_{0}^tg_t\,dW_t$ is given. Here $W_t$ is the standard Wiener process, $u,v,g$ are some deterministic funcions, and $g$ is such that the integral is well ...
- 13
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0
answers
42
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First order stochastic dominance with binary interaction
I have two variables $X_1$ and $X_2$ $\in [0,1]$, where $X_2$ (strictly) first order stochastically dominate (FOSD) $X_1$, i.e., $F_{X_2}(x) \leq F_{X_1}(x)$ for all $x$.
Then I have, $Y$ and $Z$, ...
- 109
-1
votes
1
answer
117
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Approximation Lemma in the construction of Ito Integral [closed]
I am self-learning basic stochastic calculus. In my book, the author first defines the Ito integral for simple step adapted processes and then extends it to a larger class $\mathcal{L}_{c}^{2}(T)$ of ...
- 5,320