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

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2
votes
1answer
26 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
26 views

A Feynman-Kac style derivation of a survival probability of a Compound Poisson process

Let $$R_t = u + \beta t - \sum^{N_t}_{i=1}U_i$$where $u\geq 0$, $\beta > 0$, $N_t$ is a Poisson counting process with intensity $\lambda$ and $U_i$ are jumps having a probability density $\nu(y)$ ...
0
votes
0answers
12 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
23 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
13 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
14 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
124 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
19 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 ...
1
vote
1answer
50 views

Using Feynman-Kac, compute the following:

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 ...
-1
votes
0answers
22 views

Derivative with respect to the probability density function

I want to find the derivative of ln[P(Y∣X)/P(Y)] with respect to the P(Y|X), where P(Y|X) is the conditional probability distribution function of random variable Y given the random variable X and ...
0
votes
1answer
73 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
19 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 ...
3
votes
1answer
64 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
26 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
28 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
19 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 ...
0
votes
1answer
21 views

How to apply Holder inequality to prove the following?

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
25 views

Brownian bridge sde

The SDE for the Brownian bridge is the following: $dX_t = \dfrac{b-X_t}{1-t}dt+dB_t$ with the solution $X_t = a(1-t)+bt+(1-t)\int_{0}^t \dfrac{dB_s}{1-s}$. The expectation and covariance are: ...
0
votes
0answers
17 views

Quadratic Variation of Stochastic Integral of Simple Predictable process

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Good Integrator. ...
-1
votes
0answers
22 views

How to arrive the following conclusion? [duplicate]

There is a statement as follow: $E(|X_1(t)-X_2(t)|)\leq\int_0^t \kappa[E(|X_1(t)-X_2(t)|)]ds$, where $\kappa$ is a strictly increasing concave function such that $\kappa(0)=0$ and ...
0
votes
0answers
37 views

How to make the following conclusion?

There is a statement as follow: $E(|X_1(t)-X_2(t)|)\leq\int_0^t \kappa[E(|X_1(t)-X_2(t)|)]ds$, where $\kappa$ is a strictly increasing concave function such that $\kappa(0)=0$ and ...
1
vote
1answer
22 views

Proving Ito's product rule

From Wikipedia the multidimensional Ito lemma is: If $\mathbf{X}_t = (X^1_t, X^2_t, \ldots, X^n_t)^T$ is a vector of Itō processes such that $d\mathbf{X}_t = \boldsymbol{\mu}_t\, dt + ...
0
votes
0answers
17 views

Cross variation

I have a question about the following argument. I see in my book a claim that given 2 stochastic integrals : \begin{align}X_1&:=\int_{0}^{t}f_s\mathsf dM_s\\ X_2&:=\int_{0}^{t}g_s\mathsf ...
1
vote
0answers
29 views

Stochastic Integral of Simple Predictable Process is a Martingale

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Martingale. I ...
1
vote
0answers
22 views

A Question About Probability of ratio of $\max(\cdot)$?

In My field , I reached to this problem. Assumptions: Consider $x_i,\hat{x}_i$ are iid (identical and independent) samples of a joint distribution (e.g., exponential). And also, assume we have $N$ ...
1
vote
1answer
32 views

what would be power series of $x_t = e^{\beta_t} $ if $\beta_t$ is a Brownian motion process?

In general the power series of $e^x =1+x/1!+x^2/2!+x^3/3!+...$ but because the process is random we can't apply the direct differentiation than how can i write it's power series.In the book stochastic ...
2
votes
2answers
51 views

Stochastic Exponential - Protter

I am trying to understand the proof of Theorem 37 at page 84 of the book Stochastic Integration and Differential Equations by P. Protter. In the proof there is the following statement, referred to ...
0
votes
1answer
22 views

Distribution of stochastic integral w.r. to brownian motion

Let $B=(B_t)_{t \geq 0}$ be a standard brownian motion, $T > 0$ and $f : [0,T] \rightarrow \mathbb{R}$ a continuous function. I want to determine the distribution of the following integral: ...
1
vote
1answer
48 views

Quadratic Variation of Quadratic Variation

Consider a good integrator $X$ (semi-martingale) and the relative quadratic variation process indicated by: $Y_t:=[X,X]_t$. Why is that: $$[Y,Y]_t=0 \ \ \ \ \ and \ \ \ \ \ \ [X,Y]_t=0 \ \ ?$$ ...
1
vote
0answers
88 views

How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
2
votes
0answers
58 views

Properties of Stochastic Differential Equations

Suppose I have an SDE of the form: $$dx_i = x_if(x_1,\cdots,x_n) + \sigma_ix_idW_t $$ By defining $y_i = \log x_i$, I can change the SDE to: $$dy_i = y_i g(y_1,\cdots,y_n) + \sigma_idW_t $$ Both ...
0
votes
1answer
40 views

Show that $\lim_{|P|\to 0}\sum_{k=0}^{n-1}\frac{W(t_{k+1})+W(t_k)}{2}\left[W(t_{k+1})-W(t_k)\right]=\frac{W^2(T)}{2}$

I have this problem which I am stuck in because it seems very obvious to me that the result is correct, but I don't know how $|P|\to 0$ can be used in the proof. Thanks a lot! QUESTION: Show that ...
2
votes
1answer
29 views

limit of gaussian process

If I have a sequence of gaussian random process $X_{t}^{n}$ which converge in $L^2$ norm to a process $X_t$ for every $t$. can I say that $X_t$ is also gaussian process? thank you
1
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0answers
23 views

Show $\mathbb{E}Xf(X)=m\mathbb{E}f(X)+\sigma^2\mathbb{E}f'(X)$, for any function $f$, where $X$ is a Gaussian random variable$

I have the following problem which I am struggling to solve. I have the solution, but I think I am using the formula wrong. Any help would be really appreciated, thanks a lot in advance! QUESTION: ...
-1
votes
1answer
74 views

Proof of Itō's lemma for the Brownian motion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
3
votes
1answer
43 views

Expectation of a product involving Brownian motion

I would need to verify if this solution is fine. Let $W_t$ be a Brownian motion and $\lambda > 0, \text{ } \lambda \in \mathbb{R}$. Calculate $\mathbb{E} \left[W_t e^{(\lambda W_t)}\right]$. ...
2
votes
0answers
21 views

Considering the Black and Scholes model, check that $\ln(S_T)=2W_T$ in a particular case

I have the following problem with its solution, but I keep on getting it wrong. I would be really grateful if someone could please explain to me what I am doing wrong. Thanks! It is part C that I ...
2
votes
1answer
32 views

Harmonic functions and Brownian motion

How can I prove that harmonic functions have the mean-value property using Brownian motion ${B_t}$? I know that I need to use the fact that $B_{t\wedge\tau}$ is a martingale where $\tau$ is a ...
0
votes
0answers
26 views

The Itō integral $\sum_{i=1}^nH_{t_{i-1}}\left(B_{t_i}-B_{t_{i-1}}\right)$ of an simple process $H$ is independent of the choice of $(t_0,\ldots,t_n)$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
0
votes
0answers
14 views

limit of quadartic variation [duplicate]

I am trying to understand why : $[\int_{0}^{t}a_s dB_s]=\int_{0}^{t}a_s^2 ds$ [] is the 2-variation process, $B$ is brownian motion in the proof I have seen they used Riemman-sums to get an ...
0
votes
0answers
26 views

harmonic functions and ito formula

I am trying to prove the mean-value property for harmonic functions in $R^k$ by ito calculus. given $G$ bounded domain and $u$ harmonic function on $G$ then $u(a)=\int_{\partial B_r} u(y)ds(y)$ ...
3
votes
0answers
28 views

Computing an Ito Integral using the Definition

Let $B_t$ be a brownian motion adapted to $\mathcal F_t$. For general $\mathcal F_t$-adapted processes $X_t$ the Ito-integral could be defined as $$ \int_0^t X_s dB_s = \lim_{n\to \infty} \int_0^t ...
0
votes
1answer
35 views

Quadratic variation of the Brownian motion and Itō's lemma

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $B=(B_t)_{t\ge 0}$ be a Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $\mathcal P$ be a sequence of countable subsets ...
3
votes
1answer
36 views

Stochastic process $\exp(W_t - t/2)$ approaches zero for large $t$, but it is a martingale?

The stochastic process $$ S_t = \exp\left( W_t - \frac{1}{2} t \right) $$ is a martingale (for example this could be seen by noting that it solves the SDE $dS_t = S_t dB_t$, which has no drift). But ...
3
votes
1answer
42 views

Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on ...
3
votes
1answer
22 views

Notation in stochastic integrals

There are some notation I don't understand: Given $W_t$, $n$-dimensional Brownian motion, and a smooth function $u:R^n\to R$ my book asserts: $$E^x\left[u(W_0)\right]=u(x)$$ What is the notation ...
1
vote
1answer
54 views

Itô integral with respect to a diffusion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
3
votes
0answers
50 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
4
votes
1answer
28 views

Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...