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Questions tagged [stochastic-calculus]

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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Why do trajectories differ when introducing a second Brownian motion in SDE Simulation?

I am working on simulating a SDE using both the direct approach and a transformed variable approach. My SDE is: \begin{equation} dX_t = \kappa X_t \, dt + \sigma \sqrt{X_t} \, dB_{1,t} - \gamma \sqrt{...
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Convergence of the martingale $M_t = \frac{1}{\sqrt{1-t}}e^{-\frac{B_t^2}{2(1-t)}}$ to zero, as $t \to 1^-$

I am self-learning applied stochastic calculus from the text: A first course in Stochastic Calculus, by L.P. Arguin. Exercise problem 5.18 in my text, asks to prove the almost sure convergence of a ...
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Running Supremum of standard Brownian motion and probability distribution [closed]

I am reading Optimal Stopping and Free-Boundary Problems by Peskir and Shiryaev and noticed a result on page 151 as follows: $\mathbb{P} (\sup_{t \geq 0} (B_t - \alpha t) \geq \beta) = \exp(-2\alpha \...
Harry Wang's user avatar
2 votes
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Solution of a square root SDE

In my research I have a family of SDEs that frequently pop up and I can't find an analytical solution for them, nor can I sample from them exactly. For $a, b \in \mathbb{R}$ the SDE is \begin{equation}...
perojov's user avatar
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Two processes with the same law, their supremum? [closed]

Suppose that there are two continuous-time processes $X = (X_t)$ and $Y = (Y_t)$ with $t \geq 0$, they have the same law under probability measure $P$. Could we say that $\sup_{0 \leq t \leq 1}(X_t)$ ...
Nightraidtown's user avatar
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Intuitive understanding of the definition of the $\sigma$-algebra of a stopping time $\tau$

I would like to better understand the basic intuition behind the definition of the $\sigma$-algebra of a stopping time $\tau$. Definition. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space ...
Quasar's user avatar
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Markov versus Martingale

My text provides an illuminating example about the fact that Markov processes are not martingales in general and martingales are not Markov processes in general. First, the standard brownian motion $(...
Quasar's user avatar
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A cadlag Feller process for $\mathcal F$ is Markov w.r.t $\mathcal F_+$ (Th. 46, Chap. 1, Stochastic Integration - Protter)

In page 35 of the book Stochastic Integration by P. Protter, he defines a Feller process as follows: Then he states the following theorem. In the proof, he used the following strategy: Next, he ...
Jeffrey Jao's user avatar
2 votes
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Tail Probability of r.v. $X_t$ from a Langevin diffusion

Backgrounds: An overdamped Langevin dynamics on $\mathbb{R}$ is defined as the solution to the following SDE: $$ dX_t=-\nabla V(X_t)\,dt+\sqrt{2\beta^{-1}}\,dB_t,\qquad X_0=x_0. $$ If $V(x)=\frac{x^2}{...
Hirofumi Shiba's user avatar
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Stochastic Integral and Ito's Isometry: Sharp-bracket process [M] or Angle-bracket process <M>?

I am learning stochastic integral, and I have noticed that the Ito's Isometry is sometimes stated using the quadratic variation process $[M]$ (e.g., pg. 47, Eq. (27.3), vol. 2 of Rogers & William),...
Mingzhou Liu's user avatar
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31 views

Distributional equality between SDEs

Let $B(t)$ be a Wiener process. We know that $X^{(1)}(t) = B(t)$ and $X^{(2)}(t) = -B(t)$ both weak solutions to the SDE \begin{equation} dX(t) = dB(t), \quad X(0) = 0. \end{equation} Now let $B(t)$ ...
perojov's user avatar
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Expectation of Solution to SDE, logarithmic extension of Vasicek

I have the solution to the following model $$dr(t)=r(t)(\eta-a\log r(t))dt+\sigma r(t)dW(t)$$ which, through the Vasicek model and a change of variable, I found to be $$r(t)=\exp[Y(0)e^{-at}+\frac{\...
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Show that $S(\omega) := \sum _{i=1}^{Y(\omega)} X_i(\omega)$ is a square-integrable random variable in $L^2(\Omega,\mathcal A, P)$

Let $n \in \Bbb N,M \geq 0$ be constants,$( \Omega,\mathcal A,P)$ be a probability space, $Y : \Omega \to \{0,1,\dots ,n\}$ a random variable and $X_1,\dots, X_n : [-M,M]$ random variables with: (i) ...
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regarding vector of random variables

In my research work I came across the following expression $P = \underbrace{A_1(\textbf{h}_2^T(n)\cdot \Psi \cdot \textbf{h}_1(n)+A_2h_0(n))}_{Part 1}$ + $w(n)$ ---(1) where $\textbf{h}_2(n)$ is a $M \...
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Can a random field of Ito diffusions spend positive time in a Lebesgue measure 0 set?

Suppose I have a family of Itô diffusions governed by the following SDEs: $$dX_t(x) = b(t,x, X_t(x))dt + \sigma(t,x,X_t(x))dW_t $$ with $X_0(x) = h(x)$. Suppose $x \in \mathbb{R}^d$, $X_t(x)$ is $\...
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Why does a Mathematica simulation of first hitting times of a stochastic process not match the theoretical result?

Let $ \theta, \sigma, \mu $ be positive real numbers. We consider the following mean reverting stochastic process $d X_t = \theta \cdot (\mu- X_t) dt + \sigma d B_t $ where $X_0 = x$ . Here $B_t$ is ...
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Understanding the derivation of the expression

in my research work related to wireless communication, I have the following expression: $$\tag{1} r_1 = \sqrt{P}H_1H_2(\textbf{h}_2^T\Phi \textbf{h}_1)s+\sqrt{P}H_0h_0s+w$$ where, $r_1$ is received ...
Heretolearn's user avatar
1 vote
1 answer
148 views

"Heuristic" vs. Rigorous Ito's Lemma

Assuming $X_t$ is a standard Brownian motion and $t$ is the time variable, I have learned to derive Ito's lemma for a function $F(X_t, t)$ using the following results (below, $X_t=X(t)$, and I ...
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Langevin SDE, gradient with respect to intial condition, and an associated quadratic form

I'm considering the following Langevin SDE, for $X_t \in \mathbb{R}^d$: $$dX_t^y=-\nabla V(X_t^y) dt + \sqrt{2} dB_t, \quad X_0^y = y \in \mathbb{R}^d.$$ Here, the superscript of $y$ makes explicit ...
Alan Chung's user avatar
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2 votes
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38 views

Expectation of bessel process conditioned on starting point

I want to calculate the expectation of the Bessel process for $n=3$ $$\begin{align} dX_t = dW_t + \frac{1}{X_t} dt \end{align} $$ given that we start at initial point $X_0=1$. My attempt was the ...
black's user avatar
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4 votes
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57 views

Best book for stochastic analysis on manifolds

I know there is a book by Emery on Stochastic calculus on manifolds but is there any other book/online notes with an easier exposition of the subject ? Any reference will be highly appreciated.
Soumya Ganguly's user avatar
1 vote
1 answer
47 views

Prove that solution to geometric brownian motion is correct (plug into SDE)

As many previous questions/answers point out, the solution to the geometric brownian motion stochastic differential equation (GBM SDE) $$ \left\{ \begin{array}{ll} dX_t &=& \mu X_t \,dt + \...
SebaGM's user avatar
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Extending Law of Large Number for Square integrable Martingale

Suppose we have a square integrable martingale $\{M_n\}$ with $\lim_{n\to\infty}M_t=\langle M\rangle_n=\infty$ a.s. Now, if we have a non-decreasing function $g:[0,\infty)\to[0,\infty)$, and also $$\...
Arbitor Lunae's user avatar
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Show this process is Brownian motion

Let $U(t)=W(t)-\int_0^t\frac{W(s)}{s}ds$. Show that $U(t)$ is a Brownian motion. I am struggling to prove that $\text{cov}(U(t),U(v))=\min\{t,v\}$. If anyone can help it would be greatly appreciated. ...
Cyno Benette's user avatar
3 votes
1 answer
76 views

Reverse engineering a property for Geometric Brownian Motion

During a lecture on financial mathematics we were given a side (non-homework) question to hone the SDE solving skills. Setting $\left(\Omega, \mathcal{F}, \mathbb{P},\left(\mathcal{F}_t\right)_{t \in[...
markovian's user avatar
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1 answer
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Find $P(B_2 > 0, B_8 > 0)$

We need to compute $P(B_2 > 0, B_8 > 0)$ where $B_t$ is brownian motion Now, $P(B_2 > 0, B_8 > 0) = P(B_2 > 0, B_8 - B_2 > -B_2) = P(Z_1 > 0, Z_2 > -Z_1)$ where $Z_1 \sim N(0, ...
Harsh's user avatar
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1 answer
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Showing that $M^T(N-N^T)$ is a continuous local martingale for all stopping times $T$ and continuous local martingales $M,N$

Given is that a cadlag adapted process $X=(X_t)_{t\geq 0}$ is a martingale if and only if $\mathbb{E}X_T=\mathbb{E}X_0$ and $X_T\in L^1$ for every bounded stopping time $T$. Now let $M,N$ be two ...
Daan's user avatar
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Proof that excessive function are also regular (super-harmonic)

In page 117 of Shiryaev's book optimal stopping rules, he claims that the excessive functions are also regular under some condition and state the proof is analogous to another proof, but I fail to ...
Stocavista's user avatar
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1 answer
19 views

What is $L^2(\mathcal{P}, \nu)$ if $\mathcal{P}$ is the previsible $\sigma$-algebra?

Let $(\Omega, \mathcal{F}, \nu)$ be a finite measure space, and let $\mathcal{P}$ be the previsible $\sigma$-algebra on $\Omega \times [0, \infty)$. The notes I am reading refer to $L^2(\mathcal{P}, \...
J. S.'s user avatar
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2 votes
0 answers
33 views

Brownian motion runs through a circle

Let $(B_t)$ be standard Brownian motion starting from 0, and $\mathbb{R} \to \mathbb{R}/2\pi\mathbb{Z} = S^1$ be a canonical projection. I want to calculate the expectation of the stopping time $T$, ...
nessy's user avatar
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3 votes
1 answer
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Distribution of a stochastic integral $\int_0^t W_1(s) dW_2(s)$ with independent $W_1$ and $W_2$.

Consider two independent 1-d Wiener processes $W_1(t)$ and $W_2(t)$ with $W_1(0) = W_2(0) = 0$. I would like to know the distribution of the process $$Y(t) = \int_0^t W_1(s)dW_2(s). \tag{1}$$ I reason ...
MonteNero's user avatar
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1 answer
48 views

Proving an inner product equality related to Ito process

I'm taking a course in stochastic differential equations and I'm having a trouble proving a fact that was stated as a middle step in discussing Ito processes without a proof in one of the lectures. ...
Jordie Vincent's user avatar
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26 views

Example of a function with infinite (total) variation and zero quadratic variation

In the first chapter of "Introduction to Stochastic Calculus with applications" by Klebaner, there is a very brief mention of the existence of functions with zero quadratic variation but ...
jdp's user avatar
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4 votes
1 answer
102 views

Quadratic variation of the square of Brownian motion

Let $B_t$ be the Brownian Motion. Find the quadratic variation of a martingale $ M_t = B_t^2-t$. My solution: By Ito's formula for $f(t, x) = x^2-t$, we know $$d(B_t^2-t) = 2B_t dB_t$$ thus $\langle M ...
nessy's user avatar
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5 votes
1 answer
63 views

Showing bounds of Stochastic Process

Suppose that we have the SDE: $$ dZ_t = 2Z_t(1-Z_t)dt + 4Z_t(1-Z_t)dB_t $$ With $Z_0 = \frac{1}{3}$. How can I show that $0 \leq Z_t \leq 1$. I have tried solving the 'alalogous' differential ...
Lehmann's user avatar
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0 answers
29 views

Limit of expectation of function of stopping time

Let $\tau_{n}$ be a stopping time such that $P(\tau_{n} \rightarrow \infty) = 1$, and consider the following limit: $$\lim_{n\rightarrow \infty} E(e^{t\wedge \tau_{n}}f(X_{t\wedge \tau_{n}}))$$ Can I ...
Ethan Davitt's user avatar
4 votes
1 answer
66 views

Expressing a continuous local martingale as an integral against a Brownian motion

I'm interested in the following problem. Suppose $X$, $X_0=0$ is a continuous local martingale with quadratic variation $$ [X]_t = \int_0^t A_s\mathrm{d}s $$ for a non-negative previsible process $(...
raj's user avatar
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2 votes
0 answers
202 views

Support of the Fractional Stationary Ornstein Uhlenbeck process (first kind)

I am interested in the support of the fractional stationary Ornstein Uhlenbeck process (first kind). In particular, I want to know whether smooth functions (or even only smooth functions starting at 0)...
Chris's user avatar
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2 votes
0 answers
28 views

Doob's inequality for Wiener Fields

Let $W^{(n)}$ a $n-$fold Wiener field, i.e. a Gaussian separable real-valued field on $\mathbf{R}_{+}^N=\left\{t=\left(t_1, \ldots, t_N\right): t_i \geq 0\right\}$ with zero mean and covariance ...
BabaUtah's user avatar
2 votes
0 answers
48 views

Can every $C^2$ function defined on a closed set in $\mathbb{R}^d$ be extended to $C^2(\mathbb{R}^d)$?

When reading Page 147 of the book "Continuous Martingales and Brownian Motion" by Daniel Revuz & Marc Yor, I am confused with the Remark $3^\circ$ of (3.3) Theorem (Ito's formula).In ...
Jesen's user avatar
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3 votes
0 answers
105 views

Understanding the reverse SDE

When reading papers/blogs about diffusion models, it is stated that if $X_t$ is the unique solution to an SDE $$dX_t=f dt+g dB_t,$$ where $B_t$ is the standard Brownian motion (starting from $B_0=0$), ...
learner with 's user avatar
2 votes
1 answer
78 views

Find a PDE for $f$ satisfying $f(t,Y_t) = \exp(- \frac{\gamma^2}{2} t + \gamma W_t) E[\exp(\frac{\gamma^2}{2} T - \gamma W_T) F(Y_T) | \mathcal{F}_t]$

I am studying a course on Stochastic Calculus for Finance and am struggling with the following question: Given $dY_t = b(t,Y_t) \, dt + \sigma(t, Y_t) \, dW_t$ where $\gamma \neq 0$, and $$f(t,Y_t) = ...
FD_bfa's user avatar
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1 vote
0 answers
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For $A_t= B_t^l Y_t \mathbb{E}(\frac{A_T}{B_T^l Y_T} \mid \mathcal{F}_t)$, find the values of $l$ to replicate $A_t$ by a self financing portfolio $X$

Background: In attempting to resolve the below problem, I have arrived at an answer that appears to counter intuition (and therefore, I suspect that it is wrong). I would appreciate assistance in ...
FD_bfa's user avatar
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0 votes
1 answer
24 views

Proving the Conditional Expectation of a Uniformly Distributed Random Variable Given the Sum of Two i.i.d. Uniform Random Variables [closed]

Consider two independent and identically distributed (i.i.d.) random variables (X) and (Y), each having a uniform distribution over the interval ([0, 1]). Define a new random variable (Z) as the sum ...
AshCode002x's user avatar
1 vote
1 answer
56 views

May the sum of Wiener processes be a Wiener process?

May $X_t = W_t + \tilde{W}_t$ be a Wiener process, if $W_t, \tilde{W}_t$ are Wiener processes? I know that: $X_t$ may be not a Wiener process, e.g. in case $W_t = \tilde{W}_t$. If $W_t$ and $\tilde{...
Sergei Nikolaev's user avatar
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0 answers
15 views

Prove $(g_i ◦ X_i) , i ∈ \{1, . . . , n\}$, are independet random variables (Check my proof please)

Let $(Ω, \mathcal A, \Bbb P)$ a probability space, $n ∈ \Bbb N$ and let $(R_i, \mathcal R_i), (S_i, \mathcal S_i), i ∈ \{1, . . . , n\}$,measuring spaces. Let $X_i: Ω → R_i, i ∈ \{1, . . . , n\}$, ...
gagamaga's user avatar
1 vote
0 answers
42 views

Existence of Fokker-Planck equation under Cauchy boundary condition.

A 1D Fokker-Planck equation within a constrain region is uniquely characterized by three functions and a boundary conditon: A drifting term $\mu(x,t)$. A diffusion term $\sigma(x,t)$. An initial ...
陈进泽's user avatar
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1 vote
0 answers
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Change of variable in functional Fokker-Planck equation

Starting from the following (steady-state) functional Fokker-Planck equation for the probability of a path $\{\phi(t)\}_{t=0}^T$ \begin{equation} 0 = \frac{\delta}{\delta{\phi}} \left\{ \frac{\...
Ludens's user avatar
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1 vote
0 answers
44 views

Understanding Uniqueness of SDEs

I am a little confused about SDEs and their unique solutions. Let's say I write an SDE $dX_t = A_t dt + B_t dW_t$. Is it not the case that I can write the solution $X_t = X_0 + \int_0^t A_s ds + \...
I_cosine_this's user avatar
2 votes
1 answer
101 views

Understanding the use of Indicator function

in my research work I came across following expression: $A = \mathbb{P}[X\geq \tau_s, D\leq \frac{c}{2}(T_c-\frac{cT_c}{2B_s}), Y\geq \tau_c]$---(1) where $\mathbb{P}$ denotes probability, $X,Y$ are ...
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