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

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Stochastic calculus with normal distribution

For $l=1,2......$ prove that $E[W^{2l+1} (t)]=0$ I am trying to find the ways of solving the task from Stochastic calculus, but it seems to be very difficult to start with. I am really appreciate ...
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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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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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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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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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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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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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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 ...
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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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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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1answer
33 views

Is the stochastic integral of the jumps process equal to zero for a continuous integrator?

Let $X$ be a continuous semimartingale and $H$ a progressively measurable process in $L(X)$. Assume $H$ has left limits almost surely. I claim that the jumps process of $H$, denoted by $\Delta H = H - ...
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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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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] $ ...
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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
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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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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31 views

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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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 ...
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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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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?
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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 ...
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16 views

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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107 views

Good book that contains stochastic integration, martingales and Lévy-processes?

Does anyone know about any good and easy interoductory books which contins information about martingales, sotchastic integration and Lévy-processes? I have tried reading: http://www.cambridge.org/us/...
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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 $...
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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
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{...
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1answer
47 views

Hitting times for Brownian Motion (2)

In this post there is shown that for a standard Brownian motion $\mathbb{E}[\tau^p]<\infty$ for all $p \geq 1$, where \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\...
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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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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^{...
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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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1answer
26 views

definition of stochastic process on countable space

I found the following definition of stochastic process: A stochastic process on a countable state space V is a sequence of random variables $(S_n)_{n \in \mathbb{N}}$ taking values in V and defined on ...
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2answers
46 views

Measurability From countable to Uncountable sequences family of functions

I am studying Protter Stochastic Integration and Differential Equations. In a theorem on hitting times Protter has the following theorem. Let $X$ be an adapted cadlag stochastic process, and $\Lambda$...
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31 views

Geometric Brownian motion with random drift and diffusion

One of my finance professors claims that the following is a meaningful SDE. $$dX_t = \delta_t\mu X_tdt + \delta_t\sigma X_tdW_t$$ Here $W$ is BM and $\mu$ and $\sigma$ are positive real constants. $(...
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1answer
23 views

Is a compact set on a Polish space closed?

Proposition: If an exponentially tight familiy $P_\epsilon$ satisfies the LDP (large deviation principle) lower bound for open sets for a rate function I, then I is a good rate function. The proof ...
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1answer
33 views

Why is the Covariance operator continuous?

Claim: Covariance operator $C_\mu : B^*\to B^{**}$ is continuous. $B$ is a Banach space and $B^*$ its dual space. For clarification the original def. of the covariance operator is: $C_\mu: B^* \...
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13 views

Geometrical Brownian motion Passage time

Recently I have been self-studying stochastic analysis. One of the exercises was to calculate the probabilty of Brownian motion reaching certain level before time T given that W(t)=x. This wasn't that ...
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10 views

what's the relation between characteristics of $(X_1,X_2)$ and characteristics of $X_1$ and $X_2$

I am not clear how to write down of the characteristics of two-dimensional Levy process $(X_1,X_2)$ when the characteristics of $X_1$ and $X_2$ are known. More precisely, let's say $$ X_k(t) = \...
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1answer
36 views

Clarification on Stochastic Exponential

Consider a $d$-dimensional Brownian motion $B=\left(B_1,...,B_d\right)$ whose components are independent and let $A$ be a $d\times d$ squared matrix such that $\sum_{i=1}^dA_{ii}^2=1$. Define the $W=...
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Intuition behind a market uncertainty represented by a filtered complete probability space?

What is the intuition behind a market uncertainty represented by a filtered complete probability space $(\Omega, F, P, {F_t})$, on which an m-dimensional standard Brownian Motion $W(t) = (W_1 (t), W_2 ...
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1answer
35 views

Change of variable in $\varphi(s) = t$, effect in $\mathbb{d} W_t$

I'm a little confused here. If I have the stochastic integral $$ \int_0^T f(t)\,\mathbb{d} W_t $$ and perform the change of variables $t = \varphi(s)$, how will $\mathbb{d} W_t$ transform (where the ...
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51 views

Is knowledge of PDE useful for SDE?

I am a stochastic analysis student and am particularly interested in stochastic differential equations. What always struck me as odd is how little PDE (or even ODE for that matter) seems to have ...
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1answer
122 views

Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
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27 views

Showing the Haar wavelet is a complete and orthonormal sequence within $L_2[0,1]$

I define the mother Haar wavelet to be: \begin{align} \phi(t) = \begin{cases} 1 &\mbox{if } 0 \leq t < 1/2 \\ -1 & \mbox{if } 1/2 \leq t \leq 1 \\ 0 &\mbox{otherwise}. \end{...
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53 views

Variance of Brownian Integral when the end point is specified

Consider the Brownian $W_u$. Suppose you are only considering realizations of this brownian that verify both $W_0=0$ and, for a specific (given) $t$, $W_t=a$. Under these specific conditions, what is ...
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9 views

Decouple system of Ito SDEs

Consider $\begin{split} \newcommand{\d}{\mathrm d} \d x &= -yx \d t + x^2 \d B\\ \d y &= -2 y^2 \d t + 2xy \d B. \end{split}$ This is a system of two SDEs driven by the same standard ...