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

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4
votes
1answer
46 views

Can we derive the PDE followed by a marginal transition probability density?

A pair of correlated stochastic processes follow the SDEs \begin{align} dX_t&=a(t,X_t)\,b(t,Y_t)\,dt+c(t,X_t)\,d(t,Y_t)\,dW_t, &&X_0=\bar{x}\\ dY_t&=f(t,Y_t)\,dt+g(t,Y_t)\,dZ_t, ...
1
vote
0answers
44 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
3
votes
1answer
54 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even though its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
1
vote
1answer
18 views

Stopped brownian motion

Assume $B_t$ is a standard complex (or 2D if you wish) brownian motion and $\tau$ is a stopping time relative to $B_t$. I want to know if it is possible to construct another brownian motion $W_t$ such ...
0
votes
1answer
44 views

What is the distribution of a stochastic process?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be a random variable on ...
2
votes
1answer
64 views

Regarding “Two Singular Diffusion Problems” by William Feller

I'm currently reading the research paper, Two Singular Diffusion Problems, by William Feller (1950). However, I don't understand how Feller derived the solution $(3.5)$ given equation $(3.4)$ in his ...
3
votes
0answers
42 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 ...
1
vote
1answer
40 views

Prove that a right-continuous stochastic process is product measurable

Let $X=(X_t,t\ge 0$ be a real-valued stochastic process on a measurable space $(\Omega,\mathcal{A})$ with almost surely right-continuous paths $\mathbb{F}:=(\mathcal{F}_t,t\ge 0)$ be a filtraiton on ...
4
votes
1answer
96 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
1
vote
0answers
18 views

Passing Expectation into Series (specifically Sine)

I want to show that this is true: $${ \mathbb{E}\big[\sin X_t \big]} = \sum_{n=0}^{\infty} \frac{(-1)^{n}{ \mathbb{E}\big[ X_t^{2n+1} \big]}}{(2n+1)!}$$ ($X_t$ is a Brownian Motion). By linearity I ...
-1
votes
1answer
59 views

What is the integral of a family of diffusion processes? [closed]

Let $S$ be an infinite subset of $[0,1]$. For all $s \in S$, let W_s(t) be a standard Wiener process. Definite P(s)_t = \mu(P,s,t) dt + \sigma(P,s,t) dW^s_t Can we characterize? $$F_t= \int_S P(s)_t ...
-1
votes
2answers
51 views

Ito Differential Equation example [closed]

Could someone explain Ito through an example as following? How to use Ito differential equation to find $dy$ , where $y = e^{w(t)}$
0
votes
0answers
50 views

Expectation of squared Ito integral

Let $\omega$ be a standard Brownian motion. How do you compute the expectation involving the square of an Ito integral: $ ...
0
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0answers
17 views

Prove that an operator from $L^2(\Omega;C(s,T;\mathbb R^n ))$ into itself is well defined

I need an help proving the following estimate. First, we fix the notation. Let $L^2(\Omega;C(s,T;\mathbb R^n ))$ be the set of continuous and adapted processes $\{X_t:t\in [s,T]\}$ (valued in ...
2
votes
1answer
35 views

Function of mean square continuous process

I have been asked to prove that, if $\{X_t\}$ is a ($n$-dimensional) mean square continuous process and $f:\mathbb{R}^n \rightarrow \mathbb{R}^d$ is a Lipschitz function, the process $\{f(X_t)\}$ is ...
1
vote
1answer
44 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
2
votes
1answer
19 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
1
vote
0answers
24 views

Condition for the mgf of a stochastic integral to be finite

Fix $t>0$, let $B$ be a Brownian motion and let $\sigma$ be a previsible process such that $$\mathbb{E}\left[\text{exp}\left(\frac{1}{2}\int_0^t\sigma_s^2ds\right)\right]<\infty.$$ Then is ...
3
votes
1answer
44 views

Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
2
votes
1answer
42 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
1
vote
1answer
26 views

gaussian process convergence

if I have a series of gaussian processes : ($W_{t}^{n}$ is gaussian process for every n) and I know that for every t there exist $W_t $ s.t $ E|W_t^n-W_t|^2\to0 $as $n\to \infty$. how can I show that ...
0
votes
1answer
23 views

Expectation of a Wiener process at a Stopping Time - 2

I am working through an answer to the following question and I do not understand a statement given towards the end of the solution, specifically why $\tilde{W}(\sigma) = 1$. (This question is related ...
1
vote
2answers
38 views

System of Stochastic Diff Eq

How can I solve the system of stochastic differential equation $$dX_{1}=X_{2}dt+adW_{1}$$ $$dX_{2}=-X_{1}dt+bdW_{1}$$
1
vote
1answer
51 views

How do we apply Ito's lemma to a product of functions

In finance an optimal portfolio choice it is common to use some tools of stochastic calculus. Going through a book, I found the following statement, \begin{equation} a_t = \int_s N_t(s) P_t(s) ds ...
0
votes
1answer
29 views

Expectation of a Wiener process at a Stopping Time

I am working through an answer to the following question and do not understand an expectation which takes place at the end. $\textbf{Question:}$ Define the following stochastic process \begin{align} ...
0
votes
0answers
29 views

Application of Ito's Lemma to stochastic integrals

From my understanding, the Ito Integral is a random variable itself. Suppose we have $X_t=\int_0^t Z_udZ_u$. To find $dX_t$, I would think we can apply Ito's Lemma. However, how would the partial ...
1
vote
0answers
21 views

Proof: Sum of two independent gaussian vectors is a gaussian vector

I want to show that the sum of two independent gaussian vectors is a gaussian vector. We had, that a gaussian vector can be written as $X=A*Z+b$ where $A$ is a real matrix, $b$ is a real vector and ...
0
votes
0answers
15 views

Distribution of SDE numerically from Fokker-Planck.

I'm aware of some numerical methods related to SDEs such as Euler-Maruyama, Milstein etc. However, couldn't one also simulate the equivalent Fokker-Planck equation via finite element methods? This ...
0
votes
0answers
20 views

Variance estimation of a diffusion process

The framework of this question is a 1 dimensional diffusion process, defined ny the following equation: $dx_t=adt+bdw_t$ Where $w_t$ is a standard berownian motion and and $a$ is a constant drif ...
1
vote
2answers
54 views

Differential and Differential Equation - Difference in meaning?

I am a little confused, an exercise by a teacher has been set which says: For $X_t = 2e^{B_t}$ Where $B_t$ is brownian motion at time $t$. a) Find the stochastic differential $d(X_t)$ b) Find the ...
3
votes
0answers
39 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...
0
votes
0answers
40 views

Jump-Diffusion Process: How to calculate the expectation of integral of S(t)

Having a jump-diffuion process $S(t)$ and the transition density $f_{dS(t)}(x)$. How can I calculate the Expectation of the integral of $S(t)$ between two instants $t_0$ and $t_1$? $S(t_0)$ is ...
1
vote
2answers
40 views

SDE Modeling: Ito vs. Stratonovich

In my SDE class last semester there were some hints that sometimes an SDE model makes more sense in the Ito sense, and sometimes in the Stratonovich sense. This was explained very briefly and vaguely. ...
3
votes
1answer
60 views

How to compute stochastic integral: $\int_0^t d(B_s^2)$

Here, $B_t$ is Brownian motion at time $t$ What property is used to compute the integreal $\int_0^t d(B_s^2)$? Shouldn't there be some other variable attached with the differential $d$ ?
2
votes
1answer
85 views

Girsanov's theorem and absolutely continuous restrictions

Let $W$ be a Brownian motion on some probability space $(\Omega, \mathcal{F}, P)$. Let $\mathbb{F}^W$ be the filtration generated by $W$ and let $X$ be a process that is progressively measurable ...
1
vote
1answer
53 views

Exponential Martingales - Properties

This question relates to the exponential martingale, \begin{align} Y(t) = \exp\left(-\int_{0}^{t} \lambda(s)\,dW(s) - \tfrac{1}{2} \int_{0}^{t} \lambda^2(s)\,ds \right) \end{align} and specifically ...
1
vote
1answer
62 views

How to understand the definition of weak convergence of stochastic processes

I have some problems with the definition of $\textit{weak convergence of stochastic processes}$. To ask my question, we start with two well-known definitions corresponding to measures and random ...
0
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0answers
9 views

prove $\sum\limits_{t=m+2}^n \sum\limits_{k=m+1}^{t-1} a_k \cdot X_{1,t-k} \cdot X_{2,t} = O_p(n^{1-\nu}) $ for $n \longrightarrow \infty$

Here are the preconditions required for the Lemma I have to prove: Let $X_{i,t}$ and $Y_{i,t}$ be random variables such that $E[X_{i,t}]^2 < \infty$ and $E[Y_{i,t}]^2 < C_1 \cdot \epsilon^k$ ...
0
votes
0answers
27 views

Wiener measure of smooth function in space of continuous function.

How do we show that the Wiener measure of class of smooth functions in $C[0, \infty)$ is 1?
1
vote
1answer
30 views

proving independence of stochastic integrals

Does anyone know how to show that the stochastic integrals \begin{equation} \bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}} \end{equation} are ...
1
vote
0answers
15 views

Confusion with indexes in this Stochastic D.E

I need to solve for $dS_n = 2S_ndt + 3S_ndB_t$ with $S_0 = 2$ If I were to substitute Ito's formula, would it appear in this form:? $d \ln S_n = f'(S_n)dS_n + \frac{1}{2} \sigma ^2 (S_n) ...
2
votes
1answer
46 views

Exponential Martingales

This is a two-part question concerning exponential martingales. It is stated that an application of Ito's lemma to \begin{align} \rho_t = \exp\left[-\int_{0}^{t} \lambda_s\,dW_s - ...
0
votes
0answers
18 views

Integral of Constant Parameter Martingale

What is the $\int_{1}^{t}W_1W_sdW_s$. This is the question solved by Kuo in his paper an extension of the Ito's Integral (2008) but there limit runs from $0$ instead of $1$.
1
vote
1answer
35 views

Local martingale being true martingale

I am doing a question Let $X$ be a continuous local martingale and suppose $\mathbb{E}\left[\sup\limits_{0\leq s\leq t} |X_s|\right]<\infty$ for each $t\geq 0$. Then $X$ is a true martingale. In ...
0
votes
0answers
25 views

Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
0
votes
0answers
15 views

Explicit computation of a simple expectation

Let $N_t=P_t-\lambda t$ be the compensated Poission process. Has anyone seen either of the following expected values $$E\Big[\Big(\int_0^tf_s\,dN_s\Big)^k\Big]\quad \text{or}\quad ...
1
vote
1answer
54 views

Applying the martingale representation theorem

I'm having trouble applying the martingale representation theorem to examples of Brownian martingales $M$ and contruct a process $X$ such that if we have a Brownian motion $W$ then $M= X \cdot W$. ...
0
votes
1answer
46 views

Show that a certain functional of Brownian motion is a martingale

Question: Show that $(W^2_{t}-t)^2 - 4 \int_{0}^{t} W^2_{u} du$ is a martingale. I understand how to show that $(W^2_{t}-t)$ is a martingale, and I know that $4 \int_{0}^{t} W^2_{u} du$ is the ...
0
votes
1answer
25 views

Conditional (Truncated) Expectations > Unconditional Expectations

$x$ is a continuous random variable with $pdf$ given by $f$ in the interval $[0,1]$. There is a continuous function $\lambda(x):[0,1]\rightarrow[0,1]$ with $\lambda'(x)>0$ such that its ...
0
votes
0answers
25 views

Finding solution to this stochastic differential equation

Let $W, Z$ be two correlated Brownian motions with $dW\,dZ=\rho\, dt$. We also have the following three processes: \begin{align} dD_t &= rD_t \,dt & & (D_T=1, r>0)\\ dS_t &= rS ...