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

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2
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1answer
70 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
72 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
49 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
31 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
108 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
124 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
29 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
66 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
45 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
480 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
55 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
38 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
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0answers
41 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 ...
1
vote
2answers
61 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
51 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
85 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
105 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
87 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
130 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
124 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 ...
2
votes
1answer
274 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
42 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
42 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
17 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
88 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
20 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
112 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
35 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 ...
1
vote
1answer
203 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
53 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
28 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 ...
1
vote
1answer
56 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 $\sigma_{1}>0$. ...
1
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0answers
18 views

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 ...
-1
votes
1answer
170 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
votes
1answer
50 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 ...
1
vote
1answer
67 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 ...
1
vote
2answers
41 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
vote
1answer
81 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 ...
3
votes
1answer
78 views

Squared Bessel Process and Ito Lemma

$dX_t = \delta dt+ 2\sqrt{X_t} dW_t$, where $W_t$ is a standard Wiener process, Define $\tau =\frac{\sigma ^2}{2\nu(2 − \delta)}\left(1 − \exp \left(−\frac{2\nu t}{2−\delta}\right)\right)$ If ...
3
votes
0answers
48 views

Quadratic Variation of Increasing Process?

I am looking through my notes and I came across the following statement: Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, ...
1
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0answers
98 views

Solving an expectation related to CIR process

I encounter the following question Let $X$ satisfy the SDE $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$ for $s\geq t$ with $X_t=x$, where $k,\alpha,\sigma$ are positive constants. Find the ...
3
votes
2answers
59 views

Stochastic Differential Equations - A Few General Questions

I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I ...
1
vote
1answer
48 views

A distribution of a stopped Wiener process

Let $(W_s)_{s \geq 0}$ be a Wiener process and $\tau$ be a random variable with an exponential distribution with parameter $1$. Suppose that $W$ and $\tau$ are independent. Determine the distribution ...
0
votes
1answer
432 views

Does one necessarily need an MS in Math before taking a PhD in Math? [closed]

I finished bachelor's in mathematical finance and am nearly finished with master's in mathematical finance (I am already done with thesis), and I plan to pursue a PhD not in mathematical finance but ...
1
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0answers
60 views

A Doob-Meyer decomposition related question

First I will state the question and then I will show my answer, which I obtained by imposing an additional condition on the processes involved. I would like to get some help on how to solve the ...
4
votes
1answer
565 views

What is the difference between stochastic calculus and stochastic analysis?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
2
votes
1answer
71 views

Determine for which values of some parameters a stochastic integral is a Brownian motion

Let $W_t$ be a Brownian motion on $(\Omega, F, (F_t)_t, P)$. Find all values of $a$ and $b$ such that the stochastic integral $$X_t=\int_0^t a+\frac{bu}{t} \;dW_u$$ is a Brownian motion. 1)So I need ...
0
votes
0answers
347 views

Transition density of a Geometric Brownian-motion

The solution to SDE $$dS(t)=\sigma S(t)dW_t$$ is $$S(t)=S(0)\exp(-\frac{1}{2}\sigma^2t+\sigma W_t)$$ the transition density for this martingale is $$p(S(t),t;S(0),0)=\frac{1}{S(t)\sigma \sqrt{2\pi ...
1
vote
0answers
34 views

Convergence in finite-dimensional distributions of some integral

Let $(X^n_t)_{t \geq 0}$ be a sequence of random real-valued processes that converges in finite-dimensional distributions, i.e. for all $k \in \mathbb{N}$ and for all $0 \leq t_1 < \dots < t_k$ ...
1
vote
1answer
66 views

If $M_t$ is a martingale, prove $\Bbb E \left[ M_T\int_t^T h_s ds |F_t\right] = \Bbb E \left[ \int_t^T M_s h_s ds |F_t\right]$

If $M_t$ is a martingale, for $0<t<T$, prove $\Bbb E \left[ M_T\int_t^T h_s ds |F_t\right] = \Bbb E \left[ \int_t^T M_s h_s ds |F_t\right]$. I can think of $LHS=M_T \Bbb E \left[ \int_t^T h_s ...