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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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Probabilistic solution of Hydrogen atom PDE?

I know that the Feynman-Kac formula gives a representation of the solutions of PDEs of the form $$ \partial_t u = Lu + Vu$$ for some differential operator $L$ and a bounded potential $V$. From the ...
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7 views

Product Rule for Expectation

Denote $f$ and $g$ two functions of $x_t$, where $x_t$ is a Markov diffusion: $$ dx_t = \mu(x_t)dt + \sigma(x_t)dZ_t $$ For $T \geq 0$, I would like to conceptually express $$\frac{\partial E[f(x_T)g(...
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22 views

Applying Ito's formula to $D^x(t)=(\partial/\partial x)S(t)$

Let $\gamma:[0,T]\times (0,\infty)\to \mathbb{R}$ be continuous, bounded above, $(\partial/\partial s [\gamma(t,s)])$ is continuous in $(s,t)$. Consider the following process $$dS_{\gamma}^x(t)=S^x_{\...
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Example of ANY stochastic process (SDE), with reversible distribution

Can anyone provide an example (as simple as they like) of a process $X_t$ on $\mathbb{R}$ solution to $dX=\sigma (X,t)dt+b(X,t)dW$. Where $W$ is a Brownian Motion, and $\sigma$ and $b$ can be any ...
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1answer
40 views

Are $\int_0^t \text{sign}(W_u) \, dW_u$ and $W_t$ independent for a Brownian motion $(W_t)_{t \geq 0}$?

Let $(W_t)_{t \geq 0}$ be a Brownian motion. Consider $$ X_t = \int^t_0 \text{sign}(W_u) \, dW_u $$ where $$\text{sign}(x) := \begin{cases} 1, & x \geq 0, \\ -1, & x>0. \end{cases}$$ ...
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1answer
25 views

Doobs Martingale inequality

I am confused on how to calculate the expectation of an integral on this question. Use the Doob martingale inequality to estimate $\mathbb{E}\sup_{0\leq s \leq t} \mid\int_{0}^{s} \cos(u)dB(u)\mid^...
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27 views
+50

Conditional expectation for brownian motion

Consider two brownian motions $(W_t)_{t\ge 0}$ with starting point $x$ and $(W'_t)_{t\ge 0}$ with starting point $y$. Define $T:=\inf\{t\ge 0:W_t=0\}$, the time when $W_t$ is equal to $0$. I want to ...
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1answer
11 views

Limit in distribution of increasing variance normal random variable

Suppose you have a sequence of normal random variables $X_n \sim N(0, n)$. Is there any random variable $X$ such that $X_n \to X$ in distribution. Here, I use the following definition: $X_n \to X$ ...
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2answers
37 views

Supremum of Brownian motion

I am trying to understand the proof in "Roger and Williams" for the Lemma Lemma: Let $B_t$ be a Brownian motion, then$$P(\sup_t B_t=+\infty,\inf_t B_t=-\infty)=1$$ Let $Z:=\sup_t B_t$, they started ...
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38 views

Fourier expansion of bounded noise

I have the following problem: Let $\theta$ be a i.i.d. random variable with distribution $U[-\pi,\pi]$, $Wt$ the standart Wiener process and $\sigma$ a real number. Define the stochastic process $$Y_t ...
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Distribution of $W_{t^2}-W_t$

Given a Brownian motion $W_t$ and a time-changed Brownian motion $W_{t^2}$, I want to find the distribution of $X_t=W_{t^2}-W_t$. From the definition, there exists a Brownian $B$ such that $\int_0^t \...
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1answer
19 views

Progressively Measurable on a Topological Space

In Baldi's book, Stochastic Calculus, Proposition 2.1 states that a right continuous adapted process, taking values in a topological space $E$ will be progressively measurable. The proof starts out ...
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2answers
50 views

Solve SDE for Brownian Bridge

Let $(B_t)$ be a one-dimensional Brownian motion and $y \in \mathbb{R}$. Show that the solution to the SDE $$dX_t^y=dB_t + \frac{y-X_t^y}{1-t}dt$$ with initial value $X_0^y = 0$ on $[0,1)$ is given by ...
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21 views

Geometric Brownian Motion as the limit of Binomial Tree

I know that GBM can be discretely approximated by methods such as Euler-Maruyama, and it can be shown that Binomial tree converges to GBM at the continuous time limit. However I'm having a hard time ...
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7 views

Exact requirements for interchanging mean square derivatives with limits and expectations

I've recently took an applied course in stochastic processes, and I'm far from satisfied with how we dealt with mean square derivatives. Firstly, we defined a mean square derivative of a process $\...
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1answer
37 views

Local martingale up to a stopping time vs local martingale

Let $M$ be a stochastic process, and let $T$ be a stopping time. We call $M$ a local martingale up to $T$ if there exists a sequence $(T_n)$ of stopping times such that $T = \text{sup} T_n$ and $M^{...
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0answers
32 views

Standard Brownian motion almost surely not $0$. [duplicate]

Consider a continuous standard brownian motion $(B_t)_{t\ge 0}$. I want to show, that $$\mathcal{L}(\{t\ge 0: B_t=0\})=0$$ I also have a hint that I need to show that $B:\mathbb{R}_+\times\Omega\...
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1answer
42 views

How can I use the moment generating function to prove independence between $X_s$ and $X_t-X_s$

Consider a process $X_t$ with the dynamics $$dX_t=f(t)dW_t$$ where $W_t$ is Brownian motion and $X_0=0$. Define $\sigma^2(t)=\int_0^tf^2(u)du$ Clearly $X_s$ and $X_t-X_s$ for $t>s$ are independent ...
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1answer
32 views

Deterministic Integral of a Predictable Process is Predictable

I was reviewing a proof of existence of solutions to stochastic evolution equations which takes the form of a fixed point argument on the space of predictable processes such that $$ \sup_{t\leq T}\...
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14 views

Does a given First Passage Time Distribution imply a single possible Fokker-Planck equation?

Consider a 1-dimensional Continuous Markov Process $X(t)$ with fixed and constant absorbing boundaries, let's say at $\pm \theta$, and with starting point at $X(t=0)=0$. This setting will lead to a ...
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21 views

adapted, increasing, (locally) integrable variation process is a (local) submartingale

I read a theorem that an adapted, increasing, (locally) integrable integrable, variation process is a (local) submartingale. (here increasing includes right continuity). Definition: A process is $\...
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23 views

Show that a bivariate Ito process has normal distribution

I have to show that, if $W_t$ is a 1-d Brownian motion then $\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution. Hint: apply Ito formula to this bivariate process. Any idea or suggestion on ...
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0answers
17 views

question on semimartingales and jump processes

I am trying to understand a jump diffusion process of peter tankov. I studied economics, so i don't understand a lot about stochastic calculus. At a certain point, i have this equation: $\frac{dC_t}{...
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1answer
58 views

Prove that $(X_n)$ converges to finite limit almost surely.

Let be $(X_n)$, $(Y_n)$ and $(Z_n)$ positive sequences on filtration $(F_n)$. Assume that $E|X_n| < \infty$ and $E(X_{n+1}|F_n) \leq (1 + Z_n)X_n + Y_n$ and $\sum_n Y_n < \infty$ and $\sum_n Z_n ...
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0answers
14 views

Discrete Time “Girsanov” with sub-Gaussian noise

Context I seem to recall from a course in stochastic calculus a few years back that for a random vector $X_t$ with dynamics $$dX_t = f(x_t)dt + g_t dW_t$$ where $W$ is $\mathbb{P}$-wiener there ...
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1answer
56 views

If $X$ is an Ito process, is $\mathbb E(\int X \mathrm d X)$ convex?

Consider the functional $F$, which is defined for each Ito process $$X(t) = \int_0^t \mu(s) \mathrm d s + \int_0^t \sigma(s) \mathrm d W(s)$$ as $$F(X) := \mathbb E\bigg(\int_0^T X(s) \mathrm dX(s)\...
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1answer
31 views

Standard Brownian motion and stopping time

Let be $B$ standard Brownian motion and let $S \leq T$ two stopping times with $E(T) < \infty $ and $E(S) < \infty$. Prove that hold $$ E[(B_T - B_S)^2] = E[B_T^2 - B_S^2] = E(T-S).$$ Please ...
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34 views

Using Itō's formula to solve a differential equation

Hi I am struggling on a past question in a previous stochastic calculus exam paper where we are considering the process $Z=(Z_{t})$ defined by $Z_{t}=\sqrt{\sqrt{2t}e^{-\sqrt{2t}}}\times B_{e^{\sqrt{...
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26 views

Compute $E(\int_0^t(1+e^{B_s})dB_s)$

Asked to calculate $E(\int_0^t(1+e^{B_s})dB_s)$ for $t\geq0$ where $(B_t)$ is standard Brownian Motion. I asked a similar question here: Compute $E\left((B_t−1)^2\int ^t_0(B_s+1)^2 dB_s\right)$, where ...
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Good material on reflecting boundaries for stochastic processes

restricting attention to continuous time, continuous state space ( say $\mathbb{R}$) stochastic processes. Can someone point me in the right direction of how to one imposes reflecting boundary ...
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1answer
23 views

Convergence of $X_n(t_n)$ when $X_n(t)$ is weakly convergent to a continuous process

Suppose a càdlàg univariate process $X_n(t)$, $t\in\mathbb{R}_+$ converges weakly to an a.s. continuous process $X(t)$ as $n\uparrow\infty$, where $X(t)$ can be described by a SDE. Suppose that $X(t)$ ...
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18 views

Apply the usual version of Itō formula to get an expression of the stochastic process {$|X_t-x|,t∈[0,t]$} as an Itō process?

Consider an Itō process $X_t=x+\int_0^tφ(s)dBs+\int_0^tφ(s)ds$. Could you apply the usual version of Itō formula to get an expression of the stochastic process {$|X_t-x|,t∈[0,t]$} as an Itō process? ...
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1answer
29 views

Hermite polynomial with brownian motion is martingale

Let $(B_t)_{t\ge 0}$ be a standard brownian motion. I want to show that $(H_n(B_t,t))_{t\ge 0}$ is a martingale, where $$H_n(x,t)=\frac{d^n}{du^n}e^{ux-\frac{u^2}{2}t}\Big|_{u=0},$$ such that the ...
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0answers
18 views

How would one solve for a process where a stochastic random variable is divided by a deterministic random variable?

I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its ...
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0answers
13 views

How to convert a higher order SDE to a system of first order SDE's? [duplicate]

Consider the following stochastic differential equation: $$\ddot{X_t}+\omega_0^2X_t=dW_t.$$ I thought the way to do this would be to let: $$dX_t=\dot{X}_tdt,$$ $$d\dot{X}_t=-\omega_0^2X_tdt+dW_t,$$ ...
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0answers
35 views

How to improve the convergence of a stochastic differential equation?

I have a stochastic differential equation, i.e, $$ d\rho_t= \hat{A} \rho_s dt + \hat{B} \rho_s \nu dt + \hat{C}\rho_s\omega_{1t} dt + \hat{D}\rho_s \omega_{2t}dt \quad , \quad t>s $$ Here A, ...
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Prove that $\lim\limits_{n \to \infty} P(\Lambda_n | F_n) = 1_{\Lambda}.$

Let be $(F)_{n}$ filtration and $ A_{n} \in F_{n}$ for every $n \geq 0$. Let be $$ \Lambda_{n} = \bigcup_{m \geq n} A_m $$ and $$\Lambda = \bigcap_n A_n. $$ Prove that $\lim\limits_{n \to \infty} P(\...
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1answer
35 views

Compute $E\left((B_t−1)^2\int ^t_0(B_s+1)^2 dB_s\right)$, where $(B_t)$ is a standard Brownian motion

Compute $E((B_t−1)^2\int ^t_0(B_s+1)^2 dB_s)$ for $t≥0$ given that $(B_t)_{t≥0}$ is a Standard Brownian Motion. Presume we will need to compute $E((B_t+B_s)-(B_s-1))^2$ to get some independent terms ...
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0answers
17 views

Pointwise convergence for predictable processes implies ucp convergence

my task is to proof the following: Let $H^n$ be a sequence of uniformly bounded predictable processes such that $$\lim_{n\to\infty}H^n_t= 0$$ almost surely holds for all $t\geq 0$. Then we have $$H^n\...
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2answers
49 views

Antiderivative of $g(x)dg(x)$

I am reading a book by Shreve "Stochastic Calculus for Finance II" and after computing a stochastic integral $\int_{0}^{T}W(t)dW(t)$ where $W(t)$ is a Brownian motion he compares it to the integral $$\...
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0answers
19 views

Approximating left continuous process $(L_t)_{0 \leq t \leq T}$ uniformly on $[0,T]$ by step functions on the Dyadics

The question is closely related to optional quadratic variation. If I have a left-continuous adapted stochastic process $(L_t)_{0 \leq t \leq T}$ with $L_0 = 0$ on the probability space $(\Omega, \...
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1answer
91 views

Show a Continuous Local Martingale is a Martingale

Let $B = (B_t)_{t≥0}$ be a standard Brownian motion started at zero, let $X=(X_t)_{t≥0}$ be a nonnegative stochastic process solving $$dX_t = 3 \, dt + 2\sqrt{X_t} \, dB_t \qquad(X_0 = 0)$$ and let $$...
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1answer
37 views

What is $E[W_t ^2 e^{(\mu W_t - \frac{\sigma^2}{2}t)}]$? [closed]

What is the expected value: $E[W_t ^2 e^{(\mu W_t - \frac{\sigma^2}{2}t)}]$ where $W_t$ is a standard Brownian Motion and $\mu, \sigma >0$ One possible hint is: take $d/d \mu$ twice. I don't ...
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0answers
31 views

What is a stationary solution to a SPDE?

I'm reading Hairer's notes on SPDEs: http://www.hairer.org/notes/SPDEs.pdf He says on page 6 that "the stationary solution to the stochastic heat equation is Gaussian free field". He never defines ...
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0answers
23 views

Question about Square Integrable Semimartingale

Let $B = (B_t)_{t \in [0,T]}$ be a Brownian motion and $\alpha = (\alpha_t)_{t \in [0,T]} $ be progressively measurable. Let $$ X = \int_0^\cdot \alpha_t dt + \int_0^\cdot \alpha_tdB_t. $$ If $\alpha$ ...
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0answers
27 views

How to obtain the relationship of beta function [closed]

Define $B ( \mu , v ) = \int _ { 0 } ^ { 1 } x ^ { \mu - 1 } ( 1 - x ) ^ { \nu - 1 } \mathrm { d } x = \frac { \Gamma ( \mu ) \Gamma ( \nu ) } { \Gamma ( \mu + \nu ) }$. For $u,v>0$ and $c>1$, ...
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0answers
30 views

What is an Itô integral, what does it represent concretely?

I'm really in truble to understand Itô integral. I can work with it without any problem, but I really don't understand what is it. And why is it an integral ? How can we interpret $$I=\int_0^T f(s,B_s)...
2
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1answer
61 views

Nonlinear term in the KPZ equation

I'm reading up on the KPZ equation through the article by Bertini and Giacomin from 1997 and some lecture notes by Jeremy Quastel, the equation in 1+1 dimensions is stated as (for $h_t$ the height of ...
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0answers
16 views

Definition of the conditional expectation operator $E^Q_{t,z}$?

I'm studying the book "Arbitrage Theory in Continuous Time" by Bjork, and the authore uses a lot the notation $E^Q_{t,z}$, where $Q$ is a probability measure and $z=Z_t$ a stochastic process, but he ...
2
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1answer
28 views

If $ S_t $ follows a log-normal Brownian motion, what SDE does the square of $ S_t $ follow?

If $ S_t $ follows a log-normal Brownian motion, what SDE does the square of $ S_t $ follow? I have found two possibles ways of solving it. But, they diverge with respect to the drift. First ...