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

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2
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0answers
41 views

Calculating a stochastic differential

Let $f$ be a real-valued function with bounded continuous second derivative $f''$, and $w(t)$ be a Wiener process. Let $$ V(t,w(t)) = f(w(t)) - \frac{1}{2} \int_a^t f''(w(s))ds. $$ I want to apply I ...
1
vote
0answers
38 views

An application of Ito-Doob differential formula

I want to apply the following formula $$dV(t,w(t)) = \left(\frac{\partial}{\partial t}V(t,w(t)) + \frac{1}{2} \frac{\partial^2}{\partial x^2}V(t,w(t))\right)dt + \frac{\partial}{\partial ...
0
votes
0answers
33 views

Stratonovich SDE and generator in divergence form

Let $a:\mathbb{R}^d\rightarrow\mathcal{S}_{d\times d}$ be a smooth map that takes its values in the space of $d\times d$ symmetric matrices and suppose there exists a $d\times d$ matrix valued ...
1
vote
1answer
52 views

Prove that the quadratic covariation is a bilinear form

If we take $X,Y,Z$ to be square integrable martingales starting at zero, we want to show that for any $\alpha\in\mathbb{R}$ we have $\langle X + Y , Z \rangle = \langle X,Z\rangle + \langle Y, Z ...
1
vote
2answers
80 views

Is this stochastic process a martingale?

I have the following process: $X_t=tB_t-\int^{t}_{0}B_s \ ds$ where $B_t $ is a Brownian motion. Is this a Gauß-process and/or a martingale? Can someone help me with this? And how can I calculate ...
1
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0answers
25 views

Preservation of the Markov Property for SDEs

Let $X$ be a continuous Markov process on $\mathbb{R}^d$ that is also a semimartingale. Let $V=(V_1,...,V_d)$ be a collection of suitably nice vector fields on $\mathbb{R}^d$ such that there exists a ...
0
votes
0answers
23 views

Expectation of Compound Poisson Process

$\mathbb{E}[e^{(\sigma-\lambda)X_t } \mathbb{1}\{X_t \geq X^*\}] $ I am not too sure how to compute the expectation of a compound Poisson process multiplied with a indicator function. The Question ...
5
votes
1answer
42 views

Compute the distribution of $\int_0^1 B_t dt$

I need an help with the following: let $(B_t)_t$ a Brownian motion. Compute the distribution of $X:=\int_0^1 B_t dt$. Integrating by parts we have that: $$\int_0^1 B_t dt=B_1-\int_0^1 t dB_t.$$ Now, ...
1
vote
0answers
17 views

Is $X_t = e^{\int_0^t B_sdB_s - \frac{1}{2}\int_0^t B_s^2 dB_s}$ a martingale?

Question: Is $X_t = e^{\int_0^t B_sdB_s - \frac{1}{2}\int_0^t B_s^2 dB_s}$ a martingale with respect to the filtration generated by $B_t$? In order to determine whether the above expression is a ...
2
votes
1answer
42 views

What is the stochastic integral of $\frac{dW_t}{W_t}$

Does anyone know the solution to the Ito integral with the scaling factor on $dW_t$ being $\frac{1}{w_t}$? In other words what is: $\int \frac{dW_t}{W_t}$ ? It looks dangerously close to what ...
0
votes
0answers
47 views

Solving infinitesimal operator in stochastic process

I am trying to understand a notion in a paper (p. 4) about identities in stochastic processes. The author uses the following infinitesimal generator of a diffusion $Y_{t}$, $t \geq 0$: $$ ...
1
vote
1answer
51 views

Integration by parts formula for Wiener integral

Hi I need an help understanding "integration by parts" in Wiener integral. I've defined this integral as in the following: let $T=[0,t]\subset \mathbb R$ we want to define $\int_T f(s) dB_s$ where ...
-1
votes
1answer
41 views

what is the answer of this stochastic integral? [closed]

as we know "ito integral "$$\int_{0}^{t}B_sdB_s=\frac{1}{2}B_s^2-\frac{1}{2}t$$now, I am searching for the solution for this one :$$\int_{0}^{t}B_s^2dB_s$$or$$\int_{0}^{t}B_s^4dB_s$$ $B_t$ is standard ...
2
votes
1answer
88 views

Question related to Kolmogorov equations

Let $d X_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$ be an Ito diffusion. If we choose a continuously twice twice differentiable function $f$ with compact support and define $u(t,x) = E( f(X_t) | X_0 = x)$ ...
4
votes
2answers
143 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
32 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
43 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
30 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
23 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
1answer
47 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
43 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
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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
43 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 ...
0
votes
0answers
14 views

Ito's formula w

Let $f(x,t)$ be a function. Suppose $f$ is $C^2$ in $x$ and $C^1$ in t, except at a finitely many places, we have an Ito's formula, which involves local time at places where $f$ is not $C^2$. My ...
2
votes
1answer
27 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
40 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
26 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
21 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
70 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
12 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
32 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
33 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
155 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
51 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
42 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
31 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 ...
2
votes
0answers
86 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 ...
1
vote
1answer
38 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
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
43 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
56 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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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
73 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 $$Q(t) = M^2(t) − \int_{0}^t Y^2(u)du$$ ...