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

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4
votes
2answers
155 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...
1
vote
1answer
35 views

Time scaled polynomial Brownian Motion

I want to choose constants $a$ and $b$ such that the process $$X_t = t^aP\left(\frac{B_t}{t^b}\right)$$ is a martingale, where $B_t$ is a Brownian Motion and $P(y)$ is a polynomial of degree n. Thus ...
2
votes
0answers
46 views

Lebesgue Measure of “excursions” of Brownian Motion

I know that the set $S$ where a standard Brownian motion $M:=B[\mathbb{R}]$ attains a strict local minimum is a.s. dense in $\mathbb{R}$. For every point $s \in S$, consider the interval $(s,t)$ such ...
0
votes
0answers
39 views

Expectation of a process with stochastic volatility

I would like to compute the conditional expectation of a stochastic process with stochastic volatility. The model is similar to Heston model except here the drift is not constant but an independent ...
1
vote
1answer
30 views

What are the first two moments of this stochastic process?

The setup. Consider a doubly stochastic Poisson (i.e. Cox) process, which is a Poisson arrival process $X_t$ with stochastic intensity function $\lambda_t$, i.e., a Poisson process whose rate is ...
1
vote
2answers
76 views

Most General Theory of Stochastic Integration

I've learnt continuous stochastic integration using the classical books: - Revuz & Yor, - Karatzas & Shreve and - Oksendal. Now I want to learn general stochastic integration, i.e. possibly ...
1
vote
1answer
48 views

Ito formula - How to calculate this differential?

Let $W(t)$ be a Brownian motion, how can I calculate the following differential: $$\int_t^T\int_0^t e^{uW(s)}dsdu $$ I do not know how to apply the Ito formula on this problem. Thanks in advance!
1
vote
0answers
21 views

Probability of the maximum of a reflecting Brownian motion [duplicate]

Let $\{W_t\}_{t\geq 0}$ be a standard Brownian motion (starting at $0$). For $T$ large enough, I would like to prove that $P(\max_{t\in[0,T]} |W_t| \leq c T^{1/3})$ is bigger than a negative power of ...
2
votes
0answers
63 views

Quadratic Variation and Semimartingales

It is clear that every (I am particularly interested in continuous) semimartingale has a well defined quadratic variation process. However, what can be said about processes that have a well defined ...
2
votes
1answer
29 views

4th-moment bound on continuous local martingale

I am struggling with this question: Let $X$ be a continuous local martingale with $X_0=0$, and such that $\mathbb{E} (\langle X \rangle^{p/2}_t) < \infty$, for all $t \geq 0$ and $p \geq 2$. ...
1
vote
1answer
51 views

Lower bound on the probability of the maximum of a reflecting Brownian motion

Let $\{W_t\}_{t\geq 0}$ be a standard Brownian motion (starting at $0$). For $T$ large enough, I would like to prove that $P(\max_{t\in[0,T]} |W_t| \leq c T^{1/3})$ is bigger than a negative power of ...
0
votes
1answer
41 views

$p$-variation of a continuous local martingale

Here are two interesting statements stated from a book which I do not know how to prove: Let $\{V^{n}_t \}$ be a sequence of processes defined by $$ V^{n}_t = \sum_{k=0}^{\lceil 2^n t \rceil -1} \big| ...
0
votes
0answers
25 views

Parabolic PDE with diffusion matrix of zero determinant

Consider a Fokker-Planck type PDE in $\mathbb{R}^2$: \begin{equation} \partial_t\rho=\mathrm{div}(\rho\nabla V)+ D^2:\left[\sigma\rho\right] \hspace{2cm} (*) \end{equation} where we have the ...
4
votes
2answers
74 views

Localisation in the proof of Ito's formula

I am reading Karatza's and Schreve's book "Stochastic Calculus and Brownian Motion" and I don't understand a strange thing as follows: Let $X=X_0 + A +M $ be a semimartingale, where $A$ is a ...
0
votes
0answers
13 views

How to optimizing a function that takes two different forms in two different regions

a,b,and P are non-negative constants. And $\theta$ is a random variable with distribution function $F(\theta)$ and density function $f(\theta)$. Denote $H(\theta)= {F(\theta)\over f(\theta)}$. No ...
-1
votes
1answer
34 views

Expectation of B(1) times stochastic integral? [closed]

I need to find the value of this expectation: $$\mathbb{E}\left(B(1) \int_0^1 f(t) dB(t)\right)$$ $B=(B(t))_{0\leq t\leq1}$ is a standard Brownian motion on $[0,1]$ and $f=(f(t))_{0\leq t\leq1}$ is ...
1
vote
1answer
34 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 ...
3
votes
0answers
174 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 ...
1
vote
1answer
58 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 ...
0
votes
2answers
47 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 ...
0
votes
1answer
34 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} ...
1
vote
1answer
55 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 ...
3
votes
0answers
167 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 with respect to a filtration ...
1
vote
1answer
55 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 ...
1
vote
0answers
30 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] = ...
2
votes
1answer
68 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
35 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
54 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
43 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
81 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
20 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.
0
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0answers
6 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
103 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
49 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
156 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
62 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
54 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
80 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
42 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
554 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
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?
1
vote
0answers
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 ...
1
vote
1answer
26 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
63 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
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
votes
1answer
66 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
202 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 ...