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

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Progressively Measurable for Rigth Continuous Adapted Processes

Any adapted and right continuous process $X_t$ is progressively measurable. For the above statement, I found proof in several books. They all have similar argument as follows. For a given $t > ...
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75 views

Variance process of stochastic integral and brownian motion

Let $(W_t)$ be a Brownian motion with respect to a filtration $(\mathcal{F}_t)$. For all $t \geq 0 $ set $$X_t = \int_0^t W_s^2 \mathrm{d} W_s,\qquad Y_t = W_t^7.$$ Find the covariance process ...
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2answers
40 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
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96 views

Why is the canonical filtration of a Brownian motion left-continuous?

Let $\{W_t, t\geq 0\}$ be a Brownian motion, and has a.s. continuous sample paths. Let $\{\mathcal{F}^W_t, t\geq 0\}$ be the canonical filtration, i.e. $\mathcal{F}^W_t=\sigma(W_s, 0\leq s\leq t)$. ...
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24 views

Remove drift from exponential Weiner process

I have the following problem: let $X_t$ solve $$ dX_t = b X_t \, dt + \sigma X_t \, dW_t$$ where $W_t$ is a Weiner process. Find $s(\cdot)$ such that $Y_t = s(X_t)$ is a martingale. We can see by ...
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46 views

Stochastic process using Markov chain (thief on the run!!)

I'm given an exercise where we are to simulate a thief escaping from an officer. The thief (let's call him/her T for simplicity) and an officer (O) have four cities to be in. Let's call the cities A, ...
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71 views

Two stochastic processes with the same distribution inducing different measures

I am currently reading Strook's $\textit{Probability Theory: An Analytic View}$, and I am confused by the following statement on page 156: "I take for $D(\mathbb{R}^N)$ the measurable structure given ...
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42 views

What is the distribution of a Brownian motion evaluated at times defined by Brownian motion?

Let $X_t$ and $Z_t$ be independent, $\mathbb{R}$-valued Brownian motions. For each $t$, the process $X_{|Z_t|}$ defined as $$\omega\mapsto X_{|Z_t(\omega)|}(\omega)$$ is measurable (with respect to ...
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33 views

Is exit probability monotonic in drift and diffusion coefficient?

Let $W$ be Brownian motion. Let $b_t$ and $\sigma_t$ be adapted to $\mathcal{F}_t^W$. Consider the SDE $$dx_t=b_tdt+\sigma_tdW_t.$$ Assume that $b$, $\sigma$ are such that $x$ stays non-negative. Fix ...
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45 views

Ito formula for integral function

Let $$dS_t = \mu S_t dt + \sigma S_t dW_t$$ where $W_t$ is a Wiener process. Let $$Z_t = e^{-r(T-t)} \int_{t}^{T}{h(u,S(u))du} = g(t,S)$$ where $h$ is a known function of $t$ and $S$. How can we ...
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63 views

Rewriting Diffusion Processes: Combining Independent Wiener Processes

In stochastic calculus, a rule of thumb for computations is $(dW_t)^2 = dt$ for a Wiener process $W_t$. Say we have a diffusion process $dX_t = dW^1_t + X_t dW^2_t$, with $W_t^1, W_t^2$ independent ...
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1answer
41 views

\otimes notation question

what does this notation mean: $f_t $ is $\mathscr B_t \otimes \mathscr F_t$-measurable for every $t\in[0,T]$ and $\Bbb E \left[ \int_0^T \mid f_t \mid^2 dt \right]$ and what alternatives may be used? ...
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1answer
91 views

Feynman-Kac representation for a PDE

I have the following PDE: $$ u_t + r x u_x + \frac{\sigma^2 x^2}{2} u_{xx} + h(t,x) u_y - ru =0 $$ $$ u(x,T,y) = y $$ I wanted to check whether the following representation is correct (I used ...
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3answers
61 views

Random variable stochastic bigger than random variable

I have a exercise, which I don't know how to show. It goes like, X is a continuous random variable with support $(-\infty,\infty)$. Consider the random variable $Y=X+\Delta$, where $\Delta$ is ...
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18 views

What is the effect on the eigenvalues of reducing a column of a stochastic matrix.

The following is for any 2 right stochastic matrices $A_x$ & $A_y$ of equal size $n$x$n$ with known eigenvalues $\lambda_{x1}-\lambda_{xn}$ and $\lambda_{y1}-\lambda_{yn}$ respectively. Also given ...
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53 views

Problem with understading “mixed” integration

Using standard notation: $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \:\:X_0=x \tag{1}$$ Now in my script it is said that if we integrate both sides, we get: $$X_t=x+\int_0^t ...
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1answer
57 views

Positivity of a stochastic process

I want to simulate the paths of a stochastic process $$ dS_t = r S_t dt + \sigma S_t dW_t$$ Using the Forward Euler method, we can write: $$ S_{n+1} = (1 + r \Delta t_n + \sigma \Delta W_{n}) S_n $$ ...
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1answer
40 views

Intuition underlying stopped martingales

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is ...
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110 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
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53 views

Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
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1answer
55 views

Specify the distribution of two discrete independent variables

So I'm preparing for a reexamination for an introductory statistics course, last time I had trouble finding a way to specify the distribution of variables/vectors. So my question is, maybe a more ...
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296 views

The Lévy-Khintchine formula and integrability conditions of a random measure

I am trying to see the connection between the Lévy-Khintchine and the integrability conditions of a Lévy measure. The literature seems to always connect both, but I cannot make sense of this relation ...
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153 views

Applying Ito lemma to multi dimensional semimartingales

Let $X_t=(X^1,\dots,X^d$) be a d-dimensional semimartingale. Then the formula for Ito's lemma I have found in several places, including wikipedia, is: $f(X_T)-f(X_0)=\sum_{i=1}^d\int_0^T ...
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1answer
84 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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51 views

Global optimization

Assume that I want to find the global minimum of a non-linear, non-convex, multidimensional function subject to several restrictions. Could you recommend me any deterministic strategy which can ...
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1answer
156 views

Stochastic inequality, true?

Consider two stochastic processes $X$ and $Y$ satisfying the following SDEs (with the same drift!): $$X_t = x + \int_0^t b(X_s)ds + B_t$$ $$Y_t = y + \int_0^t b(Y_s)ds + B_t.$$ If $0<x<y$, is ...
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50 views

Continuity problem in derivation of general ito integral

This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...
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31 views

Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
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31 views

Continuity theorem in Itô integral explanation

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.
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28 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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14 views

What is the steady state of the objective function in the following equation?

If we assume that $u$ in the time interval $\Delta t$ follows $N(\mu\Delta t, \sigma^2\Delta t)$ in the following equation : $$ R_{t} + max_{u} (\mu - u) \frac{\partial R}{\partial V} + (\sigma^2/2) ...
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1answer
234 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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102 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
63 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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47 views

Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
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70 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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72 views

Most probable path of diffusion process

Suppose we have an Ito diffusion $X_{t}$ on $\mathbb{R}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (1) \end{align*} where $W_{t}$ is a standard Brownian motion. If $B = 1$, ...
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2answers
29 views

Let Y be a random variable with $0\le Y\le 1.$ [duplicate]

Let Y be a random variable with $$0\le Y\le 1.$$Show that $$var(Y)\le 1/4 $$ and that $$var(Y)= 1/4 $$ if and only if P(0)=1/2=P(1).
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96 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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96 views

Power spectral density of convolution of stochastic processes

I was wondering what it is the result of convolving two WSS processes in terms of power spectral densities. I know that, the output $Y(t)$ of a generic linear time invariant system with impulse ...
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39 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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66 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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54 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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38 views

convergence of Ito integral

Suppose there is a deterministic process $\phi$ in $L^2(R)$. Need to prove that $\int_0^n \phi_u dW_u$ converges in $L^2(P)$ to some $X\in L^2(P)$ as $n\rightarrow\infty$. Also need to show that ...
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1answer
31 views

Integrability and exponential integrability

I'm working on a paper, and I don't know if there is some kind of typo or if I just don't get what seems obvious to the author. Note : I'll be working with probabilities, but I guess this would be ...
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25 views

About Ito's martingale representation theorem.

I encounter the following problem when i read a paper. then wen can define a filtration and a martingale as follows: My question is : Does the martingale representation theorem still hold true ...
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34 views

Strong versus weak solutions of SDEs

I am wondering if someone could provide me with both an intuitive and a mathematical explanation of the difference between strong and weak solutions of stochastic differential equations. Thanks...
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84 views

Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
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120 views

Sample continuity of Brownian motion

I wanted to know if the Brownian motion and the fractional Brownian motion are almost surely sample continuous or not? Many thanks.
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51 views

Is this a Brownian motion

I am learning SDE, and here is some basic things I have trouble with, Let $B(t)$ be a Brownian motion, and $F \in \mathcal L^2$ is any stochastic process and I know $\int_0^tF(s)dB(t)$ is Ito process ...