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

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1answer
16 views

More on the Existence and Uniqueness of the solutions of an SDE Proof

An extract from the proof of the existence and uniqueness of the solution of a SDE from Oksendal. I cannot see how holders inequality and the ito isometry are applied.
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1answer
18 views

Part of Proof of the Uniqueness of the Solution of SDE's

This is an extract from Oksendal's SDE of the proof of the uniqueness of the solution of a SDE. I cannot see how the $P[|X_t-\hat{X_t}|=0 \ \ \ \text{for all t} \in \mathbb{Q} \cap [0,T]]=1$ is ...
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0answers
14 views

Product of independent continuous local martingales is local martingale

Revuz-Yor's book mentioned if $M$ and $N$ are independent continuous local martingales, then $MN$ is still local martingale. But I don't know how to prove it. Any help, thanks!
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1answer
15 views

Finding the unique Martingale Convergence Representation of a given r.v.

According to the martingale representation there exists a unique $g(t,\omega) \in \mathcal{V}(0,T)$ such that $M_t = E[M_0]+\int^{t}_{0} g(s,\omega) dB(s); \ \ \ t \in [0,T]$ Find g in the case ...
2
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1answer
27 views

More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...
2
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0answers
21 views

Preliminaries of the Martingale Representation Theorem

I cannot understand why we are taking a dense subset of $[0,T]$. Furthermore, I cannot see a result that would allow each such $g_n(B_{t_1},\ldots,B_{t_n})$ to be approximated in ...
2
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1answer
38 views

Ito formula applied to $\frac{1}{t}\int_0^t W_s ds $

I got this expression and I have to calculate its differential by the Ito formula, $W_t$ denotes the Brownian motion: $$\frac{1}{t}\int_0^t W_s ds $$ I calculate the derivative of ...
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1answer
20 views

Evaluating Stratonovich integral from definition

I am struggling to evaluate the integral $\displaystyle \int^{T}_{0} B_t \circ dB_t $ from definition. So far I have that $\begin{align} \displaystyle \sum ...
1
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1answer
27 views

Continuity problem in derivation of general ito integral

This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...
0
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1answer
16 views

Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
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1answer
18 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
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1answer
19 views

Continuity theorem in Itô integral explanation

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.
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1answer
14 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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0answers
31 views

First hitting time Geometric Brownian motion

I have the following problem: My Process underlies the SDE $ d W_t = \mu W_t dt + \sigma W_t d B_t $ with $B_t$ being a standard Brownian motion, $\mu,\sigma >0$, i.e. $W_t = S_0 \exp\Big( ...
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0answers
24 views

Matlab code for Simulation of SDE [duplicate]

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
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0answers
26 views

Product of stochastically independent random variables

Let $X, Y, Z$ be three stochastically independent random variables that are quadratic integrable (quadratintegriertbar is the German term, I didn't find a exact translation). No which statements hold ...
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0answers
9 views

solve stochastic partial differential equations where initial value function does not have compact support

In stochastic calculus, there are several techniques of solving initial value problems for partial differential equations. Kolmogorov BE and Feynman-Kac formulas (as well as others) require that the ...
1
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1answer
49 views

is $(x-6)^2$ in $C_0^2$?

My math problem involves using a theorem that requires $f(x)=(x-6)^2$ to be in $C_0^2$. I'm trying to understand what $C_0^2$ means and how to check whether a function belongs to it. The course I'm ...
2
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2answers
91 views

Matlab Code to simulate trajectories of Ito process.

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
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0answers
43 views

Stochastic Differential equations with $\sin(x^2)$ as drift.

Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance. ...
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0answers
34 views

Brownian motion starts fresh variant

It is a standard result that if $W_t$ is a Brownian Motion and $S$ is a stopping time of the standard filtration $F_t$ then we have that $B_t = W_{S+t} - W_S$ is a Brownian Motion. I quote the ...
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0answers
30 views

solving partial differentiation using finite difference method

I have been trying to solve right hand side (RHS) of the following one-dimensional partial derivative equation: $\frac {\partial p} {\partial t}=\frac {\partial} {\partial x} ({D(x)}e^{-\beta V(x)} ...
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0answers
39 views

Property of G- Stochastic Calculus

i have maybe a stupid question about an equation. It is said that \begin{equation} \inf\limits_{P \in \mathcal{P}}\mathbb{E}_{P} \left[\int_0^T \varphi_{x}(t,X_{t})X_{t}\pi^{T}_{t}\,\mathrm ...
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0answers
46 views

What is a.e. a.s

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such ...
0
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1answer
21 views

inverse function type SDE

SDE $dX_t=-a^2\sin X_t\cos^3X_tdt+a\cos^2X_tdW_t$ with $X_0=x_0$ I think this is inverse type of SDE, refer to Itô's formula and SDE. However, I can't find the inverse funcition. My try ...
3
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0answers
64 views

Multipe Ito Integrals

Im working on a Lemma 10.8 in the Book "Numerical Solution of Stochastic Differential Equations by Kloeden And Platen" I have been stuck on one point. Can somebody help me to understand how he moved ...
2
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1answer
74 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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1answer
29 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
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2answers
62 views

How to solve SDE

SDE: $dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T$ $X_0=a$ answer Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$ and ...
1
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1answer
37 views

Help with integral (inner product of stochastic and deterministic process)

i need to calculate an integral of the form $$ X = \int_0^T w(t) \sin (\omega t) dt $$ where $w(t)$ is a stochastic normal process (white noise), $\sin(\omega t)$ is deterministic. How do I do that? ...
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0answers
33 views

Second (centered) moment for martingales

Take the process ${x}_t$ following geometric Brownian motion (GBM) $$x_t=\mu x_t \,dt+\sigma x_t \,dW_t$$ with $x_0>0$ known. It has first moment equal to $$\text{E}[x_t]=x_0 e^{\mu t}$$ and second ...
2
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0answers
50 views

Proof of the Key Renewal Theorem

I try to prove the Key renewal theorem by using the renewal theorem. In my book, it is written that it can be done by proofing the theorem first for indicator functions, then for step functions and ...
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0answers
19 views

Expected value of solution of SDE

Is there any way to find expectation of $X_t$ defined by the following SDE? $dX_t = -[\sin(2X(t)) + \frac{1}{4}\sin(4X(t))]dt + \sqrt{2}\cos^2 x dB(t), X(0)=1, t \in [0,\tau),$ where $\mathbb{B}$ is ...
0
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1answer
18 views

Question on Ito Isometry and bounds of integration

I am trying to find the variance of $\int_t^T(T-s)~dW_s$ I was wondering if this approach is correct: $$ Var~(\int_t^T(T-s)~dW_s~)=\mathbb E~[~(~\int_t^T(T-s)~dW_s~)^2~]=\mathbb ...
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0answers
43 views

Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
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0answers
29 views

Girsanov theorem conditions

If we have an adapted function $f(t)$ such that $\int_0^t f(s)ds\,<\infty$, then the Girsanov exponent can be defined: $$ Z(t):=\exp\left( \int_0^t f(s)dW(s) - \frac{1}{2} \int_0^t ...
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0answers
7 views

Stationary distribution for OU process driven by fractional brownian motion

Consider the SDE driven by a fractional brownian motion $$ dX_t = \kappa (\omega - X_t) dt + \eta dW_t^{H} $$ where $W_t^{H}$ is a fractional brownian motion with Hurst parameter H. I am interested ...
0
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1answer
28 views

Ito's process and martingale [duplicate]

Let ${W_t}$ be 1 dim Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. My try is below. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why ...
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1answer
24 views

SDE transformation using a primitive of a function?

Consider the following SDEs : (E) : $dX_t = (\alpha b(X_t) + {1\over2}b(X_t)b'(X_t))dt + b(X_t)dB_t$ (E') : $dY_t = \alpha dt + dB_t $ prove that E can be transformed to E' using : $ ...
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1answer
47 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
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0answers
21 views

Visualization help for random Environment models

Hi im stuck on simple random environment models. Let $\Omega=P_{k}^{\mathbb{Z}^{d}}$ where for $k>0$ fixed. $P_{k}$ denotes the set of (2d)-vectors $(p(e))_{|e|=1,e\in \mathbb{Z}^{d}}$ with ...
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1answer
93 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
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1answer
54 views

SDE and Stochastic calculus

$W_t$ is 1 dimension Brownian morion. $X_t=(cosW_t,sinW_t)$ Write SDE about $X_t$ I thought that $f(t,x)=(cosx,sinx)$, but I can't how "$t$" is expressed. I heard that the hint of this question is ...
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0answers
22 views

Question regarding Notes on Strong Markov Property

I wrote the following notes from a lecture a couple of weeks ago and I don't understand a particular line. Suppose $B_t$ is a Brownian Motion Now look at $B^x_t = x + B_t$ which is a BM starting at ...
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0answers
25 views

IID implies Ergodicity

The environment space is given by $\Omega:=P^{\mathbb{Z}^{d}}$, where P contains the 2d-vectors serving as admissible transition probabilities. An Element $\omega \in \Omega$ is defined as ...
2
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0answers
47 views

Most probable path of diffusion process

Suppose we have an Ito diffusion $X_{t}$ on $\mathbb{R}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (1) \end{align*} where $W_{t}$ is a standard Brownian motion. If $B = 1$, ...
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0answers
18 views

Product of Geometric Brownian motions

Let $S,P$ be geometric BMs: $$dS_t=S_t(\mu dt + \sigma dW_t^1)$$ $$dP_t=P_t(\tau dt + \beta (\rho dW_t^1+ \sqrt{1-\rho^2}dW_t^2)$$ Where $W^1$ and $W^2$ are independent standard BM. I want to show ...
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1answer
39 views

Approach to Solving SDE

I am trying to find the solution to the SDE: $$ dX_t=aX_tdt+(b+cX_t)dW_t $$ for $t\ge0$, $X_0>0$, constants $a,b,c$ Would appreciate any hints as to how to approach this using ito's formula, I'm ...
2
votes
1answer
61 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
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2answers
37 views

Brownian Bridge conditional probability

The problem is to show that the density $P[W_{t_1} \in dx_1,...,W_{t_n}\in dx_n | W_T = b]$ is the density of a Brownian bridge from $a$ to $b$. $W$ is Brownian motion. The density of a Brownian ...