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

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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 ...
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1answer
9 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 ...
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1answer
28 views

Geometric Brownian Motion [on hold]

I am new there. How can I calculate following expected value: $$E[X(s)\times X(t)]$$ where $X$ is Geometric Brownian Motion, i.e. $X(t) = exp[(\mu - 0.5\cdot \sigma^2)t + \sigma\cdot W(t)]$ ...
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0answers
16 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 ...
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1answer
30 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 ...
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1answer
16 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 ...
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0answers
16 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 ...
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1answer
17 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 ...
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2answers
19 views

System of Stochastic Diff Eq

How can I solve the system of stochastic differential equation $$dX_{1}=X_{2}dt+adW_{1}$$ $$dX_{2}=-X_{1}dt+bdW_{1}$$
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1answer
17 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, ...
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1answer
24 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} ...
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23 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 ...
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0answers
16 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 ...
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0answers
9 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 ...
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0answers
15 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 ...
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0answers
23 views

Is the expectation of an integral equal to the intergral of expectation?

I have to calculate the expectation of the integral between $t_0$ and $t_1$ of a random variable $S(t)$ can we say that: $E[\int^{t_1}_{t_0} S(t) dt]= \int^{t_1}_{t_0} E[S(t)] dt$?
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2answers
44 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 ...
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31 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 $x_0$. ...
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0answers
14 views

Stochastic Calculus for Pure math

Is there any use of taking Stochastic Calculus for pure math career (not financial math)? I am undergraduate and I've already taken courses in Calculus, Analysis and Real Variables. Thanks!
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0answers
29 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 ...
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1answer
26 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. ...
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1answer
43 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 $d$ ?
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1answer
50 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 ...
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1answer
31 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 ...
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1answer
32 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 ...
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0answers
8 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 $E[Y_{i,t}]^2 < C_1 \cdot \epsilon^k$ ...
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0answers
19 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?
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1answer
26 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 ...
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0answers
14 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) ...
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1answer
33 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 - ...
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0answers
14 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 $1$.
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1answer
23 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 ...
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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 ...
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0answers
13 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 ...
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1answer
35 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 $M= X \cdot W$. ...
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1answer
42 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 ...
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1answer
23 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 ...
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0answers
21 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 ...
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0answers
26 views

Method for finding a arbitrage opportunity when market price of call is incorrect

The solution of the Black-scholes equation is the price of a European call. And the option price assumes the underlying stock is a geometric Brownian motion with volatility $\sigma_{1}>0$. ...
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two different Monte Carlo approaches

Assume that the function $f$ is integrable and maps $[0, 1]$ into $[0, 1]$. Consider estimating $\int_0^1 f(x)\,dx$ using two different Monte Carlo approaches. The standard approximation is applied in ...
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Fixed Point involving limits of Integrals

Assume $\omega$ is a random variable with a p.d.f $f(\omega)$. There is a function $\lambda(\omega):[0,1]\rightarrow[0,1]$ such that $\int_0^1\lambda(\omega)f(\omega)d\omega=\bar{\lambda}$ with ...
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0answers
20 views

Basic Stochastic Calculus

Let $B_t$ be brownian motion. Then if I need to calculate $\mathbb{E}[2(B_2-B_0)+(B_2+B_1)(B_3-B_2)]$ is this simply $0$ as independence results in: $\mathbb{E}[2(B_2-B_0)] + \mathbb{E}[B_2+B_1] ...
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0answers
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Pricing an option on a mean-reverting assets

In an universe we have two assets and a predictor: $\frac {dS_{1,t}}{S_{1,t}}=(\mu_{1,1}+\mu_{1,2}X_t)dt+\sigma_{1,1}dB_{1,t}+\sigma_{1,2}dB_{2,t} $ $\frac ...
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0answers
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How is the following solution derived to solve the SDE?

Let $Y_t$ be the Ito process given by $$dY_t = \theta_t dX_t - \frac{1}{2}\theta_t^2 dt $$ By applying Ito Lemma to $f(Y_t,t) = e^{Y_t} = Z_t$, we get the following SDE $$dZ_t = \theta_tZ_t dX_t$$ ...
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0answers
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Functions of Brownian Motion and Time

Sorry, this will be a little long. I'm currently working on a problem where I basically have an SDE logistic equation: $$dX_t = diag(x_1,\cdots, x_n)[b+Ax-\lambda \eta(t)] dt + diag(x_1,\cdots, ...
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1answer
48 views

How do I solve this SDE (stochastic differential equation)?

I am stuck in trying to solve this equation \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} Here, $b$ is a constant. I am trying to apply my usual methods for ...
2
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1answer
42 views

Stochastic integral where the integrator is zero in probability

We are given a continuous semimartingale $Y$ and a continuous process $B$ of finite variation. Hence, we know that $\langle B \rangle$, the quadratic covariation of $B$, is zero in probability. I now ...
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1answer
45 views

Application of Ito's formula to log and exponential

Let $X$ be a strictly positive continuous semimartingale with $X_0 = 1$ and define the process $Y$ by $$ Y_t = \int_0^t \frac{1}{X} dX - \frac12 \int_0^t \frac{1}{X^2} d \langle X \rangle. $$ Let the ...
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2answers
31 views

What is the distribution given by $\int^t_0 W_s^2ds$

Define $X_t=\int^t_0 W_s^2ds$, what will be the distribution of $X_t$? My approach is as follow: Let $f(s)=W_s^2s$, by Ito's lemma we have $X_t=W_t^2t-2\int^t_0W_ssdW_s-\frac{t^2}{2}$. Discretize ...
1
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1answer
33 views

Infinitesimal Random Variable

I have been very confused by the idea of infinitesimal random variables, namely letting $\{Z(\omega,t)\}_{t\in\mathbb{R}}$ be a stochastic process. What do we mean by $dZ$. Is this meant by ...