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

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1answer
39 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
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0answers
493 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
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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 ...
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1answer
39 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?
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44 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 ...
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1answer
25 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 ...
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0answers
59 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 ...
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1answer
21 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
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1answer
65 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 ...
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2answers
178 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 ...
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1answer
26 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
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1answer
60 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
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0answers
88 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 ...
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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: $d(Y_tZ_t)=Z_t \, dY_t +Y_t \, dZ_t+dY_t \, dZ_t\tag{$*$}$ ...
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0answers
53 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 ...
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1answer
17 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 ...
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1answer
52 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
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0answers
38 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 ...
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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: $\displaystyle\int_0^1W_t \, dt + \frac{5}{2} +W_1$ ...
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1answer
71 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 ...
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1answer
117 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
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1answer
61 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?
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49 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
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0answers
47 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, \ ...
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0answers
22 views

variance of total residence time in up state

Hello; I really appreciate it if someone help me about this problem
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2answers
44 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.
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20 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 ...
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2answers
115 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
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2answers
116 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 $0$. ...
5
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1answer
75 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 ...
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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
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0answers
74 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
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1answer
76 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
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1answer
96 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 ...
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0answers
117 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 ...
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0answers
127 views

The Derivation of the Ito-Wentzell Formula

Is there a good derivation of the Ito-Wentzell Formula which is a generalization of the Ito's Lemma? Here are some unsatisfactory references to the Ito-Wentzell Formula: ...
2
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1answer
77 views

How to combine two conditional exponential CDF's?

Suppose one has two machines (machine A and machine B) in sequence with time to machine break down exponentially distributed with rate parameters $\lambda_A$ and $\lambda_B$. Machine A and B have a ...
2
votes
1answer
82 views

Conditional expectation with disjoint $\sigma$-algebras

Let $(B^1,B^2)$ be independent Brownian motions with corresponding filtration $\mathcal{F}_t$. Let $\mathcal{F}^2_t$ be the filtration generated by $B^2$. How does one prove that for any $s<t$ and ...
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1answer
33 views

Applying Ito's formula to a complicated expression

I am faced with some (predictable) process $(r_t)$ and let $0 \leq t \leq T$. I am baffled with the issue of applying Ito's formula to the process $$ \bigg\{ \int_{t}^{T} G(s-t, r_t) \,ds \bigg\}_{t ...
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0answers
24 views

Solve this problem involving Geometric Brownian Process

The price of a stock follows a geometric Brownian process with annual expected return rate of 20% and volatility 50%. The initial stock price is 10 euros. Compute the probability that the stock price ...
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0answers
33 views

How can I solve $E[B^4_t B^3_t]$?

How can I solve the following expected value: $$ E[B^4_t B^3_t] $$ where $ B_t $ is a standard Brownian Motion.
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0answers
32 views

Find pdf for solution of Stochastic DE

I have some troubles learning with Stochastic DE. There is a problem. Find the probability density function f(x,t), of $X_t$ where {$X_t$} is a solution of SDE: $dX_t = mdt + \sigma dW_t, X_0 = 0$ I ...
0
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1answer
107 views

Fokker-Planck equation - find probability density function

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great. Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic ...
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1answer
56 views

probability of two successive random numbers has the same starting number

Question/problem(subtask b): What is the probability of two successive random numbers has the same starting number? What we do know is that a random number generator randomizes numbers of 6-digits ...
5
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1answer
110 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
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1answer
58 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
3
votes
1answer
106 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
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0answers
28 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
0
votes
1answer
55 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...
1
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1answer
92 views

Expected Value of the exponential of a stochastic integral

What is the expected value of the following process: $$ e^{\int_0^t B_u\, du} $$ Thanks.