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

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1answer
31 views

How to integrate over stochastic paths in stochastic calculus?

Suppose $X$ is a stochastic process with a certain probability distribution that is not time-dependent. $X$'s value is assumed to be a real number. Now we want to take the average of $X$ over every ...
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147 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
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1answer
49 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it ...
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2answers
36 views

Predictable Processes in Brownian Setting

Maybe it's a silly question. I've been reading Protter's book on stochastic integration. And all the integrands are required to be predictable. But from what I can recall, in the traditional ...
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1answer
29 views

Unique solution in differential equation

Given a functions g(t,T) and Q(t,T) such that $g(t,T) = - \frac{\partial}{\partial T} \ln Q(t,T)$, $Q(T,T) = 1 = Q(t,t)$, T>0 and $t \in [0,T]$ Does it follow that $Q(t,T) = exp(-\int_{t}^{T} ...
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1answer
45 views

Proving a statement in quadratic variation that ${\langle X \rangle}^{\tau} = \langle X^{\tau} \rangle$

Let $\tau$ be a stopping time and $X$ be a continuous local martingale. Let $\langle \cdot \rangle$ denote the quadratic variation. We want to show that $${\langle X \rangle}^{\tau} = \langle ...
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56 views

Existence and uniqueness of strong solution of stochastic differential equation.

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 ...
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1answer
32 views

Analytic solutions to a Stochastic Differential Equation

I want to solve an SDE as follows: $$ dX_t = \alpha(\beta - X_t)dt + dB_t,\quad X_0 = x_0 $$ where $\alpha$, $\beta$ are positive constants and B_t is a Browian motion independent to $X_t$.Is there a ...
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28 views

How to show $t \mapsto E[Z|\mathscr{F}_t]$ is a.s. borel measurable.

I'm going through Revuz and Yor and am stuck at a technicality. Suppose $Z$ is bounded and $A$ is bounded increasing continuous with $A_0 =0$. The goal of the problem is to show $E[ZA_\infty] = ...
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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 ...
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24 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 ...
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1answer
37 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 ...
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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] : ...
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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 ...
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1answer
46 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 ...
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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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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 ...
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0answers
48 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 ...
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1answer
41 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 $$Q(t) = M^2(t) − \int_{0}^t Y^2(u)du$$ ...
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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 ...
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1answer
49 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 ...
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1answer
42 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 ...
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1answer
44 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 ...
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1answer
61 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 + ...
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1answer
29 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 ...
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0answers
353 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 ...
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1answer
39 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
34 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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38 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
23 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
37 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
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 ...
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1answer
53 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
102 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
19 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
55 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 ...
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60 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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46 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
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 ...
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28 views

How to interpret a special construction of random variables

Let $\left( \Omega, \mathcal{F}, \mathbb{P}\right)$ be a probability space and $\left(f_n\right)_{n=1}^{\infty}$ a sequence of independent and identically distributed random variables with ...
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1answer
45 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 ...
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0answers
34 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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38 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
67 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
52 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 ...
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1answer
43 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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39 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
34 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