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

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2
votes
1answer
63 views

Expression for quadratic variation

I read a book and don't understand the following: Let $X$ be a continuous local martingale and is uniformly bounded. Let $\langle X \rangle^{(n)}_t = \sum_{k \in \mathbb{N}} (X_{t \wedge t^n_k}- X_{t ...
1
vote
0answers
26 views

Application of Girsanov theorem

Let $f(t)$, $t \geq 0$ be a smooth function with $f(0) = 0$ and let $B(t)$, $t \geq 0$ be a brownian motion. Let $P$ and $Q$ be two measures on $C[0,1]$ corresponding to respectively, $B(t)$, $t \geq ...
0
votes
1answer
45 views

Computing quadratic variation and criteria for Brownian motion

Let $f(t)$ be a nonrandom and continuously differentiable function and $B(s)$ be the brownian motion. a) Computer the quadratic variation of : $X(t) = f(t)B(t) - \int_0^t f'(s)B(s)ds$ b ) For ...
0
votes
0answers
14 views

Cardinality of the set of zeros of the solution of an Stochastic Differential Equation

Let $\sigma(x)$ be smooth and bounded above and below from zero. i.e $0 < \alpha^{-1} \leq \sigma \leq \alpha$. Let $X(t)$ be a solution of $dX(t) = \sigma(X(t))\,dB(t)$ Let $A = \{t \in [0,1] : ...
2
votes
0answers
41 views

Find the density of the random variable X(t)(Kolmogorov Forward equation)

Let $V(x) = x^2 / 2+ W(x)$ where $W(x)$ is a smooth function with compact support. Let $f$ denote the probability density. $f(x) = \frac{e^{-V(x)}}{\int e^{-V(x)}dx}$. Consider the stochastic ...
1
vote
1answer
57 views

Probability that Brownian Motion hits $t+1$ before $t-1$

Compute the probability that a brownian motion starting at $0$ hits the line $t+1$ before the line $t-1$. Here is what I did: I figured it has to do with optional stopping theorem. The ...
0
votes
0answers
15 views

Poisson random measure analogue for discrete-time Markov chains

For continuous-time Markov processes one can associate a Poisson random measure. Is a there an analogue random measure for discrete time Markov chains? Thank you.
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0answers
4 views

Error from bias and noise in a linear operator

There's a result $S$ that depends linearly on some forcing $F$: $S=\int dt' G(t-t')F(t')$ Let's say I need to predict $S$, but can't measure $F$ exactly. I have both bias and noise in my ...
4
votes
0answers
78 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 ...
0
votes
1answer
44 views

How to solve this question with Itô lemma?

Let $$M(t) = \int_{0}^t Y (u)dB(u).$$ where $$E \left[ \int_{0}^t Y^2(u)du\right] < \infty.$$ Use Itô’s rule to find the differential $dQ$ of the process $$Q(t) = M^2(t) − \int_{0}^t Y^2(u)du$$ ...
1
vote
1answer
31 views

Yet Another Stochastic Process

I am asked to solve $$ E\left[ \int_0^\infty \exp(-rt)A\exp(S_t) dt \mid S_0 = s \right]\\ dS_t = \mu d_t + \sigma \, dW_t$$ where $E$ denotes the expectations operator, and $A$ is some constant. I ...
1
vote
1answer
82 views

Solving the Ornstein-Uhlenbeck Stochastic Differential Equation

I am asked to solve the following SDE: $$dX_t = (a-bX_t)dt + cdB_t,\ \text{ where }X(0) = x.$$ ($(B_t)_{t\ge0}$ is a brownian motion.) For constants $a$, $b$ and $c$ and $X$ is a random variable ...
1
vote
1answer
52 views

Application of Feynman-Kac

Let $u(t, x) = E_x[\int_0^t \! 1_{[-1,1]}(B(s))ds] = E[$Time spent by B(s) in $[-1, 1]$ up to time $t$ | $B(0) = x$]. write a differential equation for $u(t,x).$ Include appropriate boundary ...
1
vote
1answer
49 views

Finite Dimensional Distributions of Stochastic Process

If $X(t) = \int_0^t \! B(s)ds$ where $B(s)$ for $s > 0$ is a Brownian motion process. Part a) what are the finite dimensional distributions of $X(t)$? (not an explicit formula, you don't need to ...
4
votes
1answer
69 views

Prove identity in law for stochastic process driven by Brownian Motion

Let $B = (B_t)_{t\geq 0}$ be a standard brownian motion started at $0$. Consider the two following stochastic equations: \begin{equation} \begin{split} dX_t &=& (13 + 2X_t)\,dt + (6 + ...
0
votes
1answer
33 views

Convergence properties of the Ito integral

I am currently going through the proof of the existence of a solution of the SDE \begin{align} dX_t = bdt + \sigma dB_t \end{align} where $B_t$ is a Brownian motion wrt a filtration ...
3
votes
0answers
394 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 ...
1
vote
1answer
41 views

Random variable which is convergent to $0$ but with mean $\infty$

I have problems with understanding the following example: Suppose $\left( \Omega, \mathcal{F}, \mathbb{P}\right)=\left([0,1], \mathcal{B}([0,1]) , \lambda|_{[0,1]}\right)$ and the sequence of random ...
0
votes
1answer
38 views

The Itō Integral

In stochastic calculus and specifically for mathematical finance Ito's lemma is used for time varying processes I need to know intuitively why the Ito Integral is stochastic?
1
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0answers
40 views

Ito's lemma applied to functions involving stopping times

Recently, I come across an exercise in my book that asks us to apply Ito's formula to $$Y_t = e^{rt} \mathbf{1}_{ \{ \tau \leq t \} },$$ where $\tau$ is a stopping time. However, this is an inherent ...
1
vote
1answer
24 views

Solve linear stochastic differential equation

I have to solve $dX_t=5\,dt+3X_t\,dW_t$ Let $$Y_t:=X_t\exp(-3W_t+\frac{9}{2}t)=X_t\cdot Z_t$$ Calculating differential of $Y_t$ we have ...
0
votes
0answers
47 views

Covariance matrix of a Brownian motion

Suppose that $Y$ is a d-dimentional brownian motion under a setting $(\Omega, \mathbb{F}, P)$ adapted to a filtration ${F_t}$. Then is the covariance matrix of $Y$ always diagonal? In other words is ...
1
vote
1answer
20 views

Conditional expectation of integral

$$E\Big(\int_0^2 t^2W_t^3 \, dt \mid F_1\Big)=\int_0^1 t^2W_t^3 \, dt +\int_1^2 E(t^2W_t^3 \mid F_1) \, dt=$$ $E(W_t^3\mid F_1)=E((W_t-W_1+W_1)^3\mid F_1)=E((W_t-W_1)^3\mid ...
2
votes
1answer
57 views

Show that a process is gaussian

I need an help with the following exercise. I would like to know if what I've done is correct. Let $(X_t)_{t\geq 0}$ be the process define as $$X_t=e^{\lambda t} X_0-\sigma \Big(\lambda \int_0^t ...
1
vote
2answers
122 views

How to compute the quadratic variation of a compound poisson process?

The jump diffusion model is defined as $$dS_t = \mu S_t dt + \sigma S_t dW_t + S_t d \left(\sum^{N_t}_{i=1}(V_i - 1)\right)\;\;\;\;\;\;\;(1)$$ , where ${V_i}$ is a sequence of iid non-negative random ...
0
votes
1answer
21 views

Derive the 2-D analogue of the Laplace Dispersal Kernel using RDE

I found an interesting problem. I'm looking at the Laplace Dispersal Kernel for 1 dimensional dispersal behavior. And I wonder what happens in two dimensional world? I managed to find the limiting ...
2
votes
1answer
58 views

Brownian motion and posterior distribution

I am a bit stuck on this question: Suppose that $X_t = W_t + \alpha t$, where $W$ is a standard Brownian motion, and let $\mathcal{F}_t = \sigma ( X_u: 0 \leq u \leq t)$. The drift is constant in ...
2
votes
0answers
78 views

Higher order expectation of Lévy process using Teugels martingales

I am new about stochastic calculus but I would like to know if the following procedure for computing $E\left(L^{2}_{t}\right)$ and $E\left(L^{3}_{t}\right)$ if $L_{t}$ is a Lévy pure jump process is ...
1
vote
0answers
32 views

Ito formula of a product

I would like to calculate stochastic differential of: $$X_t=\left(\int_0^t(s^3+B_s) \,dB_s \right)(2t+tB_t)=Y_tZ_t$$ I would like to use: $d(Y_tZ_t)=Z_t \, dY_t +Y_t \, dZ_t+dY_t \, dZ_t\tag{$*$}$ ...
0
votes
0answers
47 views

Ito formula - some doubts

How once can calculate stochastic differential of a process: $$Y_t=e^{t^2+\int_0^ts \, dW_s}$$ There are two approaches, which one is correct (or both?). 1) $Z_t=t^2+\int_0^ts \, dW_s$ is an Ito ...
0
votes
1answer
16 views

Conditional expectations one more time

Please someone verifies my results: 1) $E \Big( \int_0^3W_t^2dt|F_1\Big)=$(editing in progress) 2) $E \Big( \int_0^2 (tW_t+t^2)dt|F_1\Big)=E \Big( \int_0^2 tW_tdt|F_1\Big)+E \Big( \int_0^2 ...
0
votes
1answer
50 views

Change of measure method

Let $f(t)=t^2+1$, Using change of measure method calculate $$\mathbb{E}\Big(\exp \Big(\int_0^1f(t)dW_t \Big)\mathbb{1}_{\{\int_0^1f(t)dW_t\ge2\}}\Big)$$ Do you have any idea how to tackle this? I ...
2
votes
0answers
37 views

Solve stochastic differential equation

I have to solve: $dX_t=(4t-3X_t)dt+2tX_tdW_t=4tdt-X_t(3dt+2tdW_t)$ Let $$Y_t:=X_t \exp\Big(-3t-\int_0^t2sdW_s+\frac{2 t^3}{3}\Big)$$ $dY_t=X_td\Big[\exp \Big(-3t-\int_0^t2sdW_s+\frac{2 ...
2
votes
0answers
39 views

Calculate conditional expectation and variance

I have to caluclate the following expressions, can sb verify my results, please? $$E\left(\int_0^2W_t \, dt \mid F_1\right)$$ My result: $\displaystyle\int_0^1W_t \, dt + \frac{5}{2} +W_1$ ...
1
vote
1answer
70 views

Integral on interval $[-\infty,W_t]$, $W_t$ is Brownian motion

Basicaly I have an expectation of an integral on the interval which contains Brownian motion and it look like this. $$ E\left[e^{W_t}\cdot\int_{-\infty}^{W_t} e^{-z^2}dz\right] $$ $W_t$ is Brownian ...
0
votes
1answer
74 views

What is the quadratic variation of compensated poisson process? [closed]

I want to know what is the quadratic variation of a compensated poisson process. $$[N-\lambda t, N - \lambda t]_t = \sum_{0 \leq s \leq t} (\Delta (N_s - \lambda s))^2 = ? $$ This is as far as I ...
2
votes
1answer
47 views

A good book on Brownian motion

Can you suggest me a good book on Brownian motion, where it is introduced as a limit of measures on polish spaces like $C[0,1]$ and subsequently stochastic calculus is discussed?
2
votes
0answers
40 views

Ito's formula applied to a stochastic function

The Ito's formula stated in my book is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and $(X_t)_{t \geq0}$ is a ...
2
votes
0answers
40 views

Sufficient condition for martingale property

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space and $M=(M_t)_{t\geq 0}$ an $\mathcal{F}_t$-adapted stochastic process. If $$ \forall t<s, \ ...
0
votes
0answers
22 views

variance of total residence time in up state

Hello; I really appreciate it if someone help me about this problem
1
vote
2answers
43 views

How can I solve this expected value?

Good evening, how can I solve this expected value? $$ E \Bigl[ B_1 \int_0^{x} B_u du\ \Bigr] $$ where $B_t$ is a standard Brownian Motion and x > 0.
0
votes
0answers
19 views

Stratonovich integral of Wienere process [duplicate]

I need an help with the following exercise. Let $(W_t)_{t\geq 0}$ a Wiener process on $(\Omega, \mathcal E, \mathbb P)$ and let $I=[0,T]$ be an interval. We want to prove that the Stratonovich ...
1
vote
2answers
105 views

Stochastic Calculus - Ito decomposition

I have got one question about Ito decomposition. Suppose $W_t$ is a Brownian Motion: $X_t = W_t^2 + \int_0^t(W_t^3-1)du$ How to get $dX_t$? I am quited comfused by the integral. Should we calculate ...
2
votes
2answers
85 views

Product of stochastic integral and brownian motion

I am trying to compute the following expectation: $$ M_T = \mathbb E\left[W_T\int_0^T\,t\,d W_t \right] $$ where $0<t<T$ and $W = (W_t)_{t\geq 0}$ is a standard Brownian Motion started at $0$. ...
4
votes
1answer
61 views

Show that $E[X_t^2]<\infty$

Show that $E[X_t^2]<\infty$, where $$ X_t=e^{3W_t-\frac{3t}{2}}-3e^{W_t-\frac{t}{2}}\underbrace{\int_0^te^{2W_s-s}ds}_{A_t},\quad. t\geq0, $$ where $t$ is a fixed number and $W_t$ is Brownian ...
1
vote
0answers
48 views

Proof that limit exists in $L^2$ sence

Proof that exists $L^2$ limit $$ \lim_{\varepsilon\downarrow 0} L(t,\varepsilon)=\lim_{\varepsilon\downarrow 0}\frac{1}{\varepsilon}\int_0^t\mathbf{1}\left(W_s\in(-\varepsilon,\varepsilon)\right)ds, ...
3
votes
0answers
66 views

Second derivative of a convex function in the Itō–Tanaka formula

This is the form of the Itō–Tanaka formula I have (Revuz and Yor): For $f$ a convex function and $X$ a continuous semimartingale, $$f(X_t)=f(X_0) ...
4
votes
1answer
75 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
1
vote
1answer
94 views

Using Markov Property in solving PDE/SDE

I am solving the PDE I used Feynman-Kac and eventually arrived at $F(t,x)$ $ = E[X_T^2|X_t = x]$ $ = E[(X_t \pm \sigma (W_T -W_t))^2|X_t = x]$ (iirc) So, I try to evaluate $E[(X_t \pm \sigma ...
1
vote
0answers
89 views

Girsanov theorem for Ito diffusion process

I am getting confused about some important point of Girsanov theorem used for diffusion process. Starting with the diffusion $$dX_t=a(X_t)dt+b(X_t)dW_t$$ where $W_t$ is a P-Brownian motion. One can ...