Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

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3
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1answer
735 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 $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
2
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2answers
74 views

Itō formula as presented in “Stochastic Equations in Infinite Dimensions” by Giuseppe Da Prato

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula: Given Hilbert spaces $(U,\langle\;\cdot\;,\;\cdot\;\...
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1answer
25 views

brownian motion and process C1 (order 1 of continuity)

Here is my problem, With probability 1 (ie: a.s) the brownian motion $(B_t)_{t\in[0,T]}$ is continuous (which is define on a classic probability space $(\Omega, \mathcal{F}, (\mathcal{F}_{t})_{t>0}...
3
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1answer
63 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
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0answers
28 views

Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where     &...
2
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28 views

Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
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14 views

Gaussian filtering

I'm reading a paper and don't get how they tackle the drift of a gaussian process. We are in the setting of isonormal Gaussian processes. Let $Z$ be a Gaussian process with covariance operator $\...
2
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0answers
27 views

Derive an Itō formula for $f(t,X_t)$ where $X_t=X_0+tY+W_tZ$ and $f:[0,\infty)\times H\to\mathbb R$ and $H$ is a Hilbert space

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be Fréchet ...
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0answers
29 views

Prove that a sum of random variables converges against an Itō integral

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be ...
2
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1answer
47 views

Is this an adapted process?(deterministic integrator in Itô-process)

Assume you have a probability space with a filtration, $(\Omega,\mathcal{F},P,\{\mathcal{F}_t\})$. Assume that the stochastic process $X_t$ is adapted to this filtration, and is jointly measurable ...
4
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1answer
110 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution $X_t$...
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0answers
25 views

For Ito diffusion, what is the difference between two measures $Q^x$ and $P$?

I am confused about the difference between $Q^{s,x}$ and $P$ for the following ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad t\ge s;\quad X_s=x.$$ Followings are from most books: given the ...
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0answers
27 views

Itō isometry in Hilbert spaces

Let $U$ and $H$ be separable Hilbert spaces $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $\mathfrak L:=\mathfrak L(U,H)$ be ...
2
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1answer
94 views

Ito integral of average of the square of a Wiener signal?

How do we evaluate the average of the square of a Wiener signal? Standard case: Typically, the signal average is $S(t)=\frac{1}{T}\int_{0}^{T}s(t)dt$, where we can write the integral in Ito form $S(...
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33 views

Models for Probability Density Functions with unknown parameters and given mean and variance

The PDF $f(x)$ of a non-negative random variable $x$ has the structure $$f(x)=\exp (a-bx-cx^{2})$$ where $a$, $b$ and $c$ are any model parameters. It is assumed that $c\ge 0$ so that $f(x)$ does not ...
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12 views

Representation of the optimal filter measure as the measure of a diffusion process

In "Mitter SK, Newton NJ. A Variational Approach to Nonlinear Estimation. SIAM J Control Optim. 2003 Jan;42(5):1813–33", it is shown that the path estimation measure $P_{X|Y}(\cdot,y)$ for the ...
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0answers
35 views

Why is a discounted price process a local martingale under the Risk Neutral Measure?

I'm familiar with the fact that if the stochastic process $\left( g(t) \right)_{t \in \left[0 , T \right]}$ is almost surely square integrable, i.e. $\mathbb{P}\left( \int_0^t |g(s)|^2ds < \infty \...
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0answers
18 views

show continous local martingale

I've to proof if $f([M],M)$ is a continuos local martingale, where $M$ is a continous local martingale and $\frac{\partial f}{\partial x_1} + \frac{1}{2} \frac{\partial^2 f}{\partial x_2^2} \equiv 0$ ...
3
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1answer
66 views

Question regarding Brownian motion

Hello I have two questions regarding the construction of Ito's integral in Øksendals book from here: http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf On page 25 he lists these 3 properties ...
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0answers
92 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
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0answers
32 views

show that a function with brownian motion is a martingale

Let $B=(B^1,B^2)$ be a two-dimensional Brownian motion w.r.t. the Filtration $\mathcal{F}^B$. Show that $(M_t^2)_{t\in \mathbb{R}_{+}}:=(e^{B_t^1} \cos(B_t^2))_{t\in\mathbb{R}_{+}}$ I've tried it ...
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1answer
37 views

Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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0answers
18 views

bromnian motion and use of Lebesgue's differentiation theorem

Let $M$ be a Brownian motion with $M_0=0$ and $V\in L(M)$. Use Lebesgue's differentiation theorem to prove that there exists a predictable process $H\in L(M)$ such that $V\cdot M$ and $H\cdot M$ are ...
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1answer
35 views

Stochastic control HJB equation

I am trying to solve this optimal control problem : $ V(x,t) = inf( E[\int_{0}^{1}(x(t)^2 - \frac{1}{2}u^2(t))dt + x(1)^2])$ subject to $dx(t) = u(t)dW_t$ $x(0) = x_0 \in R $ $u(t) \in [-1,1] $ ...
2
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0answers
30 views

Integral Representation of Brownian Motion [duplicate]

B is a Brownian motion with values in $\mathbb{R}$. I have to find a process $(F_t)_{t\in[0,T]}$ such that $X=E[X]+\int_0^T F_s dB_s$, for $X=B_T$, $X=\int_0^T B_tdt$, $X=B^2_T$, $X=B^3_T$ and find a ...
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67 views

Rephrase a multiparameter SDE indexed by time and space as an infinite dimensional SDE indexed by time

Let $\mathcal V_t\subseteq\mathbb R^3$ be the bounded space occupied by a closed particle system at $t\ge 0$ and $$[0,\infty)\ni t\mapsto X_t\in\mathcal V_t\tag 1$$ be the movement of a fixed particle ...
1
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1answer
31 views

The Stratonovich Integral and its meaning as the limit in mean square of a sum?

I am studying the Stratonovich Integral and on wikipedia, Stratonovich Integral, it states that the integral, for a process $X:[0,T] \times\Omega \to \mathbb{R}$, as: $$ \int_0^T X_t \circ dW_t $$ ...
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0answers
33 views

Exact definition of time correlation function

Given a stochastic process $X(t)\in \Omega=\mathbb{R}$, markovian, ergodic and with a unique stationary distribution (e.g. a Boltzmann distribution), the time correlation function is usually defined ...
0
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1answer
17 views

Solving a simple, linear type SDE

I am a bit confused by SDE's. I am trying to solve the SDE $dX=(c-\mu X )dt+\sigma dB$, with $\mu,\sigma,c$ constants and $X_0=x_0$ deterministic. I already know the solution of $dX=fdt+gdB$ with $X(...
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0answers
15 views

showing exponential tightness.

Let $\{X_i\}$ be a sequence of iid random variable with common distribution $\mu$. Define $S_n=\frac{1}{n}\sum_{i=1}^n X_i$ and let $\mu_n$ be its distribution. I now need to show somehow, that $\{\...
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0answers
34 views

Application of Stochastic Calculus to Interest Rate Model (Ito's Formula)

Above is my question. Now, the setting is of mathematical finance, but the part that I'm stuck on isn't directly related to finance, but stochastic calculus (hence posting on this site). We have the ...
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0answers
8 views

Find Itˆo diffusions $X_t = (t-2)^2_+W_2^4W_t$ in the differential form

I have $Y_t = (t-2)^2_+W_2^4W_t$. (The notation $x_+$ means the positive part of x, i.e. max(x, 0)) I try to write $Y_t$ in the differential form, that is: $$dX_t = U_tdt + V_tdW_t$$ In order to ...
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1answer
32 views

How many types of martingale related stochastic processes are there?

Previously I had thought that the only concepts in this direction were martingales, submartingales, and supermartingales. However, at least when discussing quadratic variation and stochastic ...
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1answer
35 views

Local martingale implies martingale

Let $M$ be a right-continuous local martingale such that $M^*_t \in L^1(P)$ for all $t \in \mathbb{R}_+$. Here \begin{align*} M^*_t(\omega) = \sup_{0 \leq s \leq t} |M_s(\omega)|. \end{align*} Now, I ...
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Can we apply an Itō formula to the solution of a SPDE?

Let $V\subset H\subset V^\ast$ be a Gelfand triple $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(W_t)_{t\ge 0}$ be a ...
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0answers
23 views

Stopped process of maximum stopping times

Suppose $X$ is an adapted process and $\tau_1, \ldots , \tau_k$ are stopping times such that $X^{\tau_1}, \ldots , X^{\tau_k}$ are all martingales. I want to show that then $X^{\tau_1 \vee \ldots \vee ...
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1answer
43 views

Prove Wald's identities for Brownian motion using stochastic integrals

The question is as follows: Let $W$ be Brownian motion and $T$ a stopping time with $\mathbb{E} T < ∞$. Show (use stochastic integrals) that $\mathbb{E}W_T = 0$ and $\mathbb{E} W^2_T = \mathbb{E} ...
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23 views

Martingale w.r.t. the filtration $\{\mathcal{F}_{t \wedge \tau}\}$

I want to show that any martingale w.r.t. the filtration $\{\mathcal{F}_{t \wedge \tau}\}$ is also a martingale w.r.t. the filtration $\{\mathcal{F}_{t}\}$. So, suppose $(X_n)_{n \geq 0}$ is a ...
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0answers
24 views

Ito Formula for increments of Ito Processes

Let $X_{t}=X_{0}+\int_{0}^{t}a_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}$, $W_{t}$ is a standard BM. How can I apply Ito formula to $(X_{t}-X_{s})^{2}$? Should I use a multidimensional version?
2
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1answer
55 views

Definition/Construction of Wiener Measure

I want to make sure I understand this rigorously: Assume we already know that Brownian motion $B_t$ on $[0,\infty)$ exists/how to construct it. Every $\sigma$-field considered is implicitly assumed ...
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26 views

Compound Process and its compensator

I have always implicitly thought that for a counting process $N_t$, defining the compound process $$\sum_{i=1}^{N_t} X_i,$$ where $X_i$ are i.i.d, was pretty much equivalent to constructing a ...
2
votes
1answer
139 views

How to prove convergence of process and stopping time

We consider the random function $X^n=(X^n_t)_{t\geq 0}$ with values in the Skorokhod space $\mathcal{D}$ of càdlàg paths, and suppose that it weakly converges (i.e in distribution) to $X=(X_t)_{t\geq0}...
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Correlation and covariance matrices

I have a sample cross correlation defined as, $\hat R^N_{eu}(\tau)=\frac{1}{N}\sum_{t=1}^{N-\tau}e(t+\tau)u(t)$ where $e(t), u(t)$ are independent, then I can use the central limit theorem to show ...
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0answers
17 views

Two ways of understanding $\sigma\{Y_t, 0\leq t \leq T\}$

Given a probability space $(\Omega, \mathcal{F}, P)$ and a stochastic process $Y_t$ with continuous path, are there two ways of understanding $$ \sigma\{Y_s, 0\leq s \leq t\}?$$ First is looking at $...
3
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0answers
28 views

Conditional expectation on Function space

This question is from a notation in section 13.4 of the book "Linear and Nonlinear Filtering for Scientists and Engineers" By N U Ahmed In this section, the author is deriving the Zakai ...
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19 views

What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
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1answer
50 views

Show the existence of a right-continuous modification

Suppose ($X_t$)$_{t \geq 0}$ is a stochastic process with independent increments and the function $t \rightarrow \mathbb{E}X_t$ is continuous. Prove that $(X_t)$ has a right-continuous modification. ...
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vote
2answers
96 views

If one stochastic process is a modification of another, then they have the same finite probability distribution.

On page 2 in Karatzas and Shreve: Brownian Motion and Stochastic Calculus it is said that a stochastic process $Y$ is a modification of $X$ if for all $t$: $P(X_t=Y_t)=1$. If both are stochastic ...
2
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1answer
32 views

Indicator function minus probability on an event is a martingale

Define \begin{align} \epsilon_j = \mathbb{1}_{A_j} - \mathbb{P}(A_j) = \begin{cases} 1- \mathbb{P}(A_j) \qquad &\text{if } \omega \in A_j\\ - \mathbb{P}(A_j) \qquad &\text{otherwise }, \end{...
3
votes
1answer
51 views

$X_t=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{n}})dB(u)$ is a Brownian motion for suitable non-zero constants $a_0,\ldots,a_n$

Let $B(t)$ be brownian motion. Show that for any integer $n \geq 1$, there exist nonzero constants $a_{0},\ldots,a_{n}$ such that $X_{t}=\int_{0}^{t}(a_{0}+a_{1}\frac{u}{t}+\ldots+a_{n}\frac{u^{n}}{t^{...