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

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15
votes
0answers
432 views

Uniqueness of Brownian motion

May be it is a dumb question, but it vexed me a little bit. I understand the construction of the Brownian motion (first use Kolmogorov extension theorem to construct value at dyadic times and then use ...
11
votes
0answers
208 views

Removing deterministic discontinuities from semi-martingales

Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le ...
9
votes
0answers
557 views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and ...
7
votes
0answers
113 views

How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
7
votes
0answers
105 views

Higher math and statistics/probability

So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ...
7
votes
0answers
83 views

Ornstein-Uhlenbeck SDE solution

I'm following this solution of $$dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t \tag1 $$ And the question is whether its solution $$X_t=\theta+e^{-\kappa(t-s)}(X_s-\theta)+\sigma\int_s^t ...
7
votes
0answers
250 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
6
votes
0answers
79 views

Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Let$^1$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace ...
5
votes
0answers
42 views

Can Stochastic Integration be Further Generalized?

Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ...
5
votes
0answers
49 views

Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
5
votes
0answers
84 views

Brownian motion on sphere proof?

proving the brownian motion on the sphere equation the stratonovich form differential equation $$\partial X=n(X)\times \partial B$$ the equation in ito's form becomes $$dX=n(X)\times ...
5
votes
0answers
398 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
5
votes
0answers
212 views

Brownian Motion and stochastic integration on the complete real line

I'm struggling to understand stochastic integration over intervals containing zero, i.e. integrals of the form $\int_{a}^{b} X_s \, d B_s$ where $-\infty \leq a < b \leq \infty$, $(X_t)_{t \in ...
5
votes
0answers
112 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
5
votes
0answers
130 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
4
votes
0answers
58 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the joint probability distribution ...
4
votes
0answers
52 views

Stochastic Integral of Particle Scattering

I have a stochastic process that describes a particle moving through a field of randomly distributed particles and undergoing scattering collisions (modeled simplistically) off of them. In its ...
4
votes
0answers
77 views

Is $X_t = tW\left(\frac{1}{t}\right)$ a Martingale?If not, how could it be a Brownian Motion?

As is proved, $X_t = tW\left(\frac{1}{t}\right)$ is a Brownian motion. For example see Theorem 4.2 in this paper http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf I'm just confused because ...
4
votes
0answers
103 views

Confusion about localization

I am a bit confused about the following result I read: Let $(\Omega,\mathfrak{A},\mathfrak{F},\mathbb{P})$ be a filtered probability space. Let $\left\{X_t\right\}_{t\in[0,T]}$ be a continuous and ...
4
votes
0answers
65 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
0answers
119 views

Why is a predictable stochastic process called *predictable*?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I$ be an index set $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be a stochastic ...
4
votes
0answers
77 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
4
votes
0answers
214 views

Using Girsanov theorem to prove density of stopping time

Let $B$ be a standard Brownian motion and for $a>0$ and $b>0$, and set $$\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$ There are at least two ways to solve the following problem (the other one ...
4
votes
0answers
917 views

Can I get a PhD in Stochastic Analysis given this limited background?

General advice on PhD apps welcome Given my limited background in stochastic analysis and other information (below), can I apply for a PhD with stochastic analysis for my dissertation topic? 1/4 I ...
4
votes
0answers
113 views

Regarding proof of converse to Girsanovs theorem

This is regarding an argument from Arbitrage Theory by Thomas Björk - Theorem 11.6, but is attempted self contained. Consider a Wiener process W on probability space ...
4
votes
0answers
200 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
4
votes
0answers
220 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
4
votes
0answers
68 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
4
votes
0answers
470 views

Spectral process for the Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process $X(t)$ is a centered, Gaussian process with covariance function $$B(s,t) = e^{-\vert t-s \vert /2}$$ The spectral measure is abs. cont. w.r.t. the Lebesgue measure ...
4
votes
0answers
231 views

Brownian motion integral

Let $(B_t)$ be a standard Brownian motion, $f$ a continuous function and $X_t = \int_0^t f(s)B_s ds$. I was able to prove that $(X_t)$ is a Gaussian process with zero mean and trying to find the ...
4
votes
0answers
210 views

Integral representation of fractional Brownian motion

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
4
votes
0answers
367 views

Can infinitesimal generator be defined by the time-inhomogeneous stochastic process?

The following is the definition of infinitesimal generator from Oksendal. Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ ...
4
votes
0answers
209 views

Integrating the inverse of a squared bessel process - integrability

Let $X_t$ be a 4-dimension Squared Bessel Process (BESQ-4). Let $M_t$ be a continuous true martingale. Question: Does $\int_0^t \frac{1}{X_s}dH_s$ exist? If so, is it only a local or a true ...
3
votes
0answers
40 views

Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
3
votes
0answers
40 views

Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
3
votes
0answers
69 views

stochastic exponential uniformly integrable martingale

$N$ is a continuous local martingale and $T_c:=\inf\{t>0:[N]_t>c\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable martingale if and only ...
3
votes
0answers
44 views

Question about “Stochastic Analysis on Manifolds”

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ ...
3
votes
0answers
26 views

A stochastic volatility model

An example of stochastic volatility model: $$\begin{cases} \frac{dX_t}{X_t} &= g_t dW_t \\ dg_t &= - k g_t dt + \sigma dZ_t \end{cases} $$ where $Z_t$ and $W_t$ are Brownian motions and ...
3
votes
0answers
27 views

Stochastic process is brownian motion by Levy's characterization

I would like to know if $B_t=W_t-\int_0^t \frac{W_u}{u}du$ is a brownian motion. I know that $W_t$ is a brownian motion. For that i would like to use Levy's characterization, so I have to show that ...
3
votes
0answers
37 views

costruction of brownian motion on sphere?

i am trying to construct a brownian motion on the sphere using the method given in Price and williams paper.$\partial$ represents the SDE of stratonovich type which is converted to ito form in last ...
3
votes
0answers
25 views

Finding the mean of $X_t = \int_0^t sW_sdW_s$

For the stochastic integral, where $W_t$ is a Wiener process, I am trying to find the mean of $X_t = \int_0^t sW_sdW_s$. I have read before that any stochastic integral with $dWt$ has mean zero, but I ...
3
votes
0answers
36 views

Generating a list of numbers

A set of numbers is generated starting from $0$ in the following way: Add the current number to the resultset In a chance of 50:50, do Either add $2$ to the current number Or subtract $1$ from the ...
3
votes
0answers
43 views

The limit of the ratio of two stochastic integrals

I am just wondering how to calculate the limit of stochastic integrals. Here is one example: $$ \lim\limits_{N \rightarrow \infty}\dfrac{\int_{0}^{N}B(s)dB(s)}{\int_{0}^{N}B^2(s)ds}$$ where $B(s)$ is ...
3
votes
0answers
46 views

conversion from stratonovich SDE to Ito's form?

conversion of stratonovich SDE to Ito SDE (Where $\partial$ is differential in the stratonovich form and $d$ is in ito's form): $$\partial X_t=\sigma(X_t,t)\partial B_t+b(t,X_t)\partial t$$. ...
3
votes
0answers
31 views

Stock Price Dynamics correlated with Bond market returns

I am currently working on to derive the following form of the stock price dynamics: $$dS_t = S_t[(r_t + \psi\sigma_S)dt + \rho \sigma_S dz_{1t} + \sqrt{1-\rho^2}\sigma_S dz_{2t}$$ where the ...
3
votes
0answers
18 views

What kind of decomposition is $X_{t \wedge L}=\tilde{X}_t+\int_0^{t \wedge L} \frac{d \langle X, M^L \rangle_s}{Z^L_{s^-}}$?

In one of the papers I was reading for my masters thesis I came across a theorem with no references. Theorem: If $(X_t)$ is an $(\mathcal{F}_t)$ martingale then there exists a $(\mathcal{F}^L_t)$ ...
3
votes
0answers
17 views

Calculate expectation of stochastic integrals

I am trying to calculate $$\mathbb{E}\left[\int^t_0 e^{\lambda s}dB_s \int^{t+h}_0 e^{\lambda s}dB_s \right], $$ where $(B_t)_{t\geq 0}$ is a brownian motion, $h>0$ and $\lambda > 0$ is some ...
3
votes
0answers
31 views

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$

Why does there exist a right continuous version of the supermartingale $\{P(L >u \vert F_u),u \geq 0)\}$ where $L$ is a measurable random variable Its is clear that not all supermartingales have ...
3
votes
0answers
31 views

How do theorems like the optional stopping theorem generalize to Bochner integrable processes with values in a separable Banach spaces?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space ...
3
votes
0answers
41 views

Numerical method for SDEs

I'm using a 4th order Adams predictor-corrector method to numerically solve a regular differential equation. Now I would be interested to be able to include a noisy term to the equation -as in the ...