Questions on the calculus of stochastic processes, or processes that have a random component.

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2
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1answer
26 views

What is a good book for learning Stochastic Calculus?

I am in search of a good book for learning Stochastic Calculus from a purely mathematical/statistical point of view. Almost all the books I see are based on Finance. Also, please specify the ...
1
vote
0answers
18 views

How to calculate the differential of the following stochastic integral?

Let $$Y_t=\int_t^T f(t,s)\ \mathsf dW_s$$ I want to compute $\mathsf dY_t$. This suggests me to consider how to find $\mathsf dY_t$ for $$Y_t=\int_t^T f(t,s)\ \mathsf dW_s$$ or $$Y_t=\int_t^T g(t,s)\ ...
0
votes
0answers
13 views

Stopping of quadratic covariaton

I am given two local martingales $M$ and $N$ and a stopping time $\tau$. We work on a finite time interval $[0,T]$. I want to prove $$\langle M,N\rangle^{\tau}=\langle M^\tau,N\rangle$$ using the ...
2
votes
0answers
43 views

one dimensional SDE with zero drift

I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, ...
0
votes
0answers
38 views

Algebra behind Feynman-Kac theorem?

According to many sources, The Feynman-Kac theorem in Equation (1) below is the solution to Equation (3) - if X(t) follows a diffusion such as in (2). (Most Important) - Can someone show the algebra ...
1
vote
1answer
25 views

Covariance of Ornstein-Uhlenbeck process

$U(t)=e^{-\mu t}W(\frac{\sigma^2e^{2\mu t}}{2\mu})$. The problem is to find $Cov[U(t),U(t+s)]$. I used the identity, $W(\frac{\sigma^2e^{2\mu t}}{2\mu})=W(\frac{\sigma^2e^{2\mu t}e^{2\mu s}}{2\mu ...
2
votes
0answers
33 views

What calculus material can prepare me for MCMC?

I am looking to revise calculus from scratch to move on to Monte Carlo Markov Chain Methods and Quasi Monte Carlo Methods. I studied calculus properly a couple of years ago however it was back in high ...
0
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0answers
17 views

Is there a big difference between runge kutta 4th for ODEs vs SDEs?

I was working on 2nd, 4th order runge kutta method for stochastic differential equations. I saw 2nd formula for ODEs and SDEs. There is some difference between their formulas . Unfortunately I can't ...
3
votes
0answers
19 views

What is the Skewness of a Geometric Brownian Motion?

Consider a GBM : $$S(t) = S(0)\exp\left({(\mu-\frac{1}{2}\sigma^2) t + \sigma W_t}\right)$$ $$d\log S(t) = (\mu-\frac{1}{2}\sigma^2) t + \sigma dW_t$$ $$\frac{d S(t)}{S(t)} = \mu t + \sigma ...
1
vote
1answer
58 views
+50

How can I solve this stochastic system of equation?

$(B_1(t),B_2(t))$ is a 2-dimensional standard Brownian motion. $\alpha , \beta$ are constant. The system of equations is: $$dX_1(t)=X_2(t)dt+\alpha dB_1(t)\\dX_2(t)=-X_1(t)dt+\alpha dB_2(t)$$ I tried ...
0
votes
0answers
12 views

Exact solution to nonlinear backward SDE

I have read a paper about numerical SDE. After deriving the method, it uses the method to calculate the following nonlinear cases: $$\begin{cases} dX_t=ud\tau+\sigma ...
0
votes
0answers
8 views

ODE where the derivative is a function of a stochastic process

Suppose we have a linear ODE, $\frac{dv}{dt} = 1 + xv $, where the coefficient $x$ is an Ornstein-Uhlenbeck process, $dx = (x_0 - x)dt + \sigma d\omega$. Is there a way to express this ODE as an ...
0
votes
0answers
13 views

Finding a function to use for Ito's Lemma

The original problem was to show the following stochastic process has a global solution: $$dx_i = x_i\left(b_i-\sum_{j=1}^4 a_{ij}x_j \right)dt+\sigma_ix_idW_t$$ To do so, they considered the ...
2
votes
0answers
14 views

Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
0
votes
1answer
35 views

How to find exact solution of this volterra equation?

I was working on numerical solution of this equation (by block pulse). $$x(t)=1+\int_{0}^{t}s^2x(s)ds+\int_{0}^{t}sx(s)dB(s)\\t \in[0,\frac{1}{2}]$$B(t) is standard brownian motion. Author of the ...
2
votes
0answers
27 views

What does it means of Normalization term of Gibbs distribution?

I am studying about Gibbs distribution concept and I am confusing about the term" normalization ". According to the Hammersley–Clifford theorem, an random $x$ can equivalently be characterized by a ...
2
votes
0answers
21 views

Stochastic exponentials

Let $X$ be a good integrator with $X_0=0$, then the process \begin{equation*} Z_t=\exp(X_t-\frac{1}{2}[X,X]_t)\prod_{0\leq s \leq t}(1+\Delta X_s) \exp(-\Delta X_s + \frac{1}{2}(\Delta X_s)^2) ...
1
vote
2answers
30 views

Estimate mean and variance for a truncated sample set

Assume there is a normally distributed random variable $X \tilde{} N(\mu, \sigma)$ I want to estimate $\mu$ and $\sigma$. So far the standard setting. Assume I am given a sample $(X_i)_{i=1}^N$ of ...
1
vote
0answers
26 views

Show $Y(t)=X^{(1)}(t)-X^{(2)}(t)$ and $\lim_{t\to\infty} \mathbb{E}Y^2(t)=0$ , for $dX^{(i)}=\mu X^{(i)}dt+\sigma X^{(i)}dW$

I am trying to solve this exercise which my professor has "solved" (he says what the result but not how he gets it). This is in a problem sheet which is about the Euler-Maruyama scheme. What I get ...
0
votes
1answer
23 views

interchanging spatial integral and time integral in the Brownian context

The problem is the following My attempt is inspired in the following: Consider $$F_n(x) = \int_{-\infty}^\infty h(a) u_n(x - a)\,da $$ By Itô's formula: \begin{align} &F_n(W_t) = F_n(W_0) + ...
0
votes
1answer
20 views

Doob Meyer decomposition for Super-martingales

Let $Z$ be a super-martingale with usual Doob-Meyer decomposition: $Z=M-A$. Is it true that : $A\leq M$ and therefore: $\mathbb{E}[A^2]\leq \mathbb{E}[M^2]$ ?
0
votes
0answers
48 views

Ito's Lemma / Expected Value / Variance - Mathematical Finance

Assume an asset price $S_t$ follows the geometric Brownian motion $$\Bbb dS_t = \mu S_t\Bbb dt + \sigma S_t\Bbb dWt,$$ where $\mu$ and $\sigma$ are constants and $r$ is the risk-free rate. ...
1
vote
1answer
51 views

Calculation with Ito processes, what is $ds \, dt$, $dW_t \, ds$ and $dW_s \, dW_t$?

I am working on an exercise and I am not sure how to deal with these 3 cases... For example, is $ds \, dt=0$? I know $(dt)^2=0$, but I am not sure when it is 2 different variables. And what about ...
2
votes
1answer
51 views

Gradient descent method with random perturbation

Suppose there is a function $f:\mathbb R^n \to \mathbb R$. One way to find a stationary value is to solve the ODE $\dot x = - \nabla f(x)$, and look at $\lim_{t\to\infty} x(t)$. However I want to ...
1
vote
2answers
39 views

Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$

I tried to shove this question in another one of my posts as a follow-up question, but I deleted my comment and will post it instead. I would really appreciate if someone could help me spot out and ...
1
vote
1answer
21 views

Burkholder's inequality for elementary stochastic integral

An elementary Burkholder's inequality for simple stochastic integral says that given nonnegative martingale $M$ and simple bounded predictable process $H$, it holds that for all $c>0$, the tail ...
0
votes
1answer
16 views

Show that $X_n\in\mathcal{H}$, where $\mathcal{H}:=\{h(t):h(t)\text{ is an adapted process, }\mathbb{E}[\int_0^{\infty}h^2(t)dt]<\infty\}$

I am not sure if I got this exercise right... I have 2 questions: Have I obtained the final result correctly? If so, I used Wolfram Alpha to obtain the value of the series, but how else can I obtain ...
1
vote
0answers
28 views

Differentiating Stochastic Integral

I was wondering how to write the following integral in differential form: $$\int^t_0 f(s,t)dW_s$$ where $W_s$ is a standard Brownian Motion. In my understanding, if $f(s,t)$ can be written as ...
2
votes
0answers
25 views

Generator of a stochastic process

I have a question about the generator of a stochastic process. $T>0$: fix Let $b: \mathbb{R} \to \mathbb{R}$ be a bounded measurable function. Let $\left( (X_{t})_{t \in [0,T]}, \left(P_{x} ...
1
vote
0answers
17 views

Novikov condition, martingale

I have a question about Novikov condition and martingale. $T>0$: fix. Let $(\Omega, \mathcal{F}, \left(\mathcal{F}_{t}\right)_{t \in [0,T]}, P)$ be a filtered probability space and $(B_{t})_{t ...
0
votes
1answer
16 views

Application of Ito's isometry in deduction of Wiener Ito Chaos expansion

I am trying to learn about the Wiener Ito Chaos expansion and starting reading Oksendal's notes on Malliavin calculus where it is treated in Chapter 1. For a link to the notes, please see ...
1
vote
1answer
29 views

Calculate $\mathbb{E}[M_{\alpha}^{p}(t)]$ for all $p>0$ and $t>0$, where $M_{\alpha}(t):=e^{\alpha W_t-\frac{\alpha^2}{2}t}$, $t\ge 0$

I am going through this solved problem but I don't understand some steps. My professor is notorious for making errors very often so don't hold back if you think he's wrong... Or if I am wrong. I am ...
3
votes
1answer
27 views

Evaluate $\mathbb{E}\left(\left[W\left(\frac{k}{n}\right)-W(t)\right]^2\right)$ for all $t\in\left(\frac{k}{n},\frac{k+1}{n}\right]$

I am trying to do a past exam paper to practice, but I don't know if I have answered this question properly... I would really appreciate it if someone could double check it. Thanks a lot! QUESTION: ...
2
votes
0answers
22 views

Probabilistic interpretation for Fokker-Planck equation

It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated ...
-1
votes
0answers
39 views

polar coordinate transformation

If we have an equation $\mathcal{L_I}=\prod \mathrm{exp}\bigg(-\lambda_j \displaystyle\sum\limits_{m=1}^{\Psi_{j}}\binom{\Psi_j}{m} ...
2
votes
1answer
28 views

Absolute continuity counterexample of a stochastic process

This example is from Stochastic Modelling and Applied Probability by Sören Asmussen (2010) p.358. The setup is the following: Let $\{Z_{t}\}$ be stochastic process on a Skorokhod space $D$ and a ...
0
votes
0answers
18 views

Is the space of all adapted processes with Càdlàg paths a Banach space?

Consider first the definition of a stochastic integral for simple predictable processes. $$I:\mathbb{S}\rightarrow\mathbb{D},\ H\mapsto I_X(H):=H_0X_0+\sum_{i=1}^nH_i(X_{T_{i+1}}-X_{T_i})$$ The ...
1
vote
0answers
29 views

Itō Integral multiplied by Riemann Integral

I was wondering whats the result of an Itō integral multiplied by a Riemann Integral. For example, what is $$\left(\int_0^T f(u)\ \mathsf dW_u\right)\left(\int_0^T g(v)\ \mathsf dv\right)$$ where $W$ ...
0
votes
0answers
17 views

Impose initial condition on partial differential equation

After solving a Fokker-Planck equation (using expansion in eigenfunctions) I have obtained the following, general solution for the probability density: \begin{equation} p(x,t) = \int_0^\infty dk~ ...
1
vote
0answers
31 views

Fundamental theorem for Malliavin derivative and Lebesgue integral

I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics. Let ...
3
votes
5answers
144 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
1
vote
0answers
21 views

Simple Stratonovich product for physical system

I was reading a physical textbook and they used the "Stratonovich product" referred to $v_1 \circ dW_1 = \frac{1}{2}[v_1 + (v_1+dv_1)]dW_1$. I think this product is from the Stochastic process, thus ...
0
votes
1answer
52 views

Using Feynman-Kac, compute the following: [closed]

Let $B(t)$ be Brownian Motion and let $\alpha$ be a constant and $T>0$. Compute $\mathbb{E}_{B_{0} = x}\left[\exp\left(-\alpha \int_0^T B(s)^2 ds\right)\right]$. I'm just having a hard time with ...
0
votes
1answer
78 views

Analytic solution to stochastic differential equations

I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. ...
0
votes
1answer
17 views

Change from stochastic exponential to exponential of Lévy process - Applebaum

In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process. I am not sure about ...
2
votes
1answer
25 views

Stationary distribution for Kolmogorov Forward Equation

Given $X_t$ which satisfies the following SDE, $$ dX_t = f'(X_t)dt + \sigma dW_t $$ where f is an infinitely differentiable function, and $f'$ above is the first derivative of $f$. We know that ...
4
votes
1answer
75 views

How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$ where $a\neq 0$ and $b\geq 1/2$, is ...
3
votes
0answers
28 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
1
vote
0answers
29 views

Is the following counter example correct?

In the book "Malliavin Calculus and related topics", the author states that $\|F\|_{k,p}=((E(|F|^p)+\sum_{n=1}^k E(\|D^n F\|^p_{H^k}))^{\frac{1}{p}}$ has monotonicity property, i.e. $\|F\|_{k,p}\leq ...
0
votes
0answers
21 views

PDEs, Monte-Carlo methods and hyperbolic problems

I often hear that Monte-Carlo methods provide good solutions to elliptic and parabolic type PDE problems. The main apparent reason being that the Feynman-Kac formulae, modernly derived from the Ito ...