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

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11
votes
1answer
442 views

Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
2
votes
2answers
85 views

Distribution of $\int_0^t e^s dB(s)$

Consider $$\left(\int_0^t e^s dB(s)\right)_{t\in [0,1]}$$ where $B$ is a Brownian motion. What is the distribution of this process? Since $f(s) = e^s$ is continuously differentiable, $B$ is process ...
0
votes
2answers
68 views

Incorrect use of the scaling relation for Brownian motion?

I want to calculate $$CoV\left(B_1,\int_0^1 B_t dt\right) = \int_0^1 CoV\left(B_t,B_1\right) dt= \int_0^1 \min(t,1)dt = 1/2$$ On the other hand, one could also use the scaling relation for Brownian ...
1
vote
1answer
509 views

Implication of Lévy-Khintchine theorem/representation

I have trouble understanding the use/implication of the Lévy-Khintchine theorem. One possible way to state it is the following: The characteristic function $\varphi$ is infinitely divisible if and ...
2
votes
1answer
1k views

Different versions of Girsanov theorems?

I am reading two different versions of Girsanov theorem regarding change of measure to preserve Brownian motion. Wikipedia has the following Girsanov theorem: If $X$ is a continuous process and ...
0
votes
1answer
51 views

What does an unaugmented sigma field mean?

What does an unaugmented sigma field mean in Wikipedia's Girsanov's theorem? Then for each $t$ the measure $Q$ restricted to the unaugmented sigma fields $\mathcal{F}^W_t$ is equivalent to $P$ ...
2
votes
1answer
314 views

Ito integral almost sure and $L^2$ limit

why does one define the Ito integral as the $L^2$ limit, although it can be shown by Doob's martingale inequality and Borel-Cantelli lemma that there exists a t continuous version, which is ...
1
vote
1answer
122 views

Different meanings of $\int_0^T X(t) dt$, and its meaning in Ito isometry?

Given a stochastic process $X: [0,T] \times \Omega \to \mathbb R$, I realized there are different meanings of $\int_0^T X(t) dt$. $\int_0^T X(t, \omega) dt$, $\forall \omega \in \Omega$ or a.e., ...
3
votes
0answers
299 views

Integrability in Ito isometry

Itō isometry from Wikipedia: Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to ...
1
vote
1answer
114 views

What are the norms in Ito isometry?

Itō isometry from Wikipedia: Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to ...
1
vote
1answer
509 views

Finding the distribution of an Ito integral. $\int_0^t sB_s \, \mathrm{d}s$

I'm a little baffled by this, I'm supposed to find the distribution of $X_t$ where, $X_t=\int_0^t sB_s \, \mathrm{d}s$. What I can think of is to consider the process $$\begin{align} Y_s &= ...
2
votes
0answers
193 views

Looking for a proof of a dominated convergence theorem for Lebesgue-Stieltjes integrals

From what I've read there exists a similar theorem to the dominated convergence theorem for Lebesgue integrals, which is applicable to Lebesgue-Stieltjes integrals. Does someone have a statement of ...
0
votes
1answer
162 views

How to solve this SDE?

Suppose we have the stochastic equation $dX_t=-\frac{1}{1-t}X_tdt+dW_t$ with $X_0=0$. I have to prove that exist soma function $f=f(t)$ such that the following occurs: ...
2
votes
3answers
648 views

Exchange integral and conditional expectation

I know that if we have $E[\int_0^1 |X_t|dt] < \infty$ we may apply Fubini's theorem and compute $E[\int_0^1 X_tdt] = \int_0^1 E[X_t]dt$. Is there a similar version that allows the exchange of ...
0
votes
1answer
121 views

Application of Fokker-Planck equation in Ito calculus

In http://markov.uc3m.es/2009/02/ito-calculus-for-the-rest-of-us/, is derived. But I don't get this: after all, the process is defined as - which means that $f(X,t)$ in this context is zero (or am I ...
0
votes
1answer
86 views

Basic partial derivative calculus and Ito Calculus

In http://markov.uc3m.es/2009/02/ito-calculus-for-the-rest-of-us/, after some statements about processes, it says that Now I am not getting how this is resulted. Can anyone explain this? This seems ...
3
votes
3answers
185 views

When is a stochastic process defined via a SDE Markovian?

I was wondering when a stochastic process defined via a SDE is Markovian? The SDE may involved Ito integral, Lebesgue integral, jump component, and any other things. The reason I ask this question is ...
1
vote
1answer
89 views

Calculating the average of $\sin^2$ of a stochastic process

I have a random process $\phi_t$ which evolves according to the SDE $$d \phi_t = \mu dt+ \sigma \sin \phi_t \,dW_t$$ with $\mu$ and $\sigma$ constants and $W_t$ a Wiener process. The initial condition ...
2
votes
1answer
188 views

Distribution of the integral of a diffusion process

Suppose $X(t)$ is a diffusion process with $E[X(t)]=0$ and variances $\sigma^2_t$ concave in time. If $X$ is also a Brownian motion, then the distribution of $\int_0^T X(t) dt$ is known to be ...
3
votes
1answer
111 views

Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the habitual ...
4
votes
2answers
2k views

How to solve this differential equation? (Steady State Solution of Forward Kolmogorov Equation)

Here's the full question and my attempt at answering it by solving the differential equation. Consider the following SDE $$ d\sigma = a(\sigma,t)dt + b(\sigma,t)dW $$ The Forward Equation (FKE) is ...
1
vote
0answers
65 views

Stochastic Exponential: $dZ=-\lambda Z dM + dL$ to $dZ=-\lambda Z dM + Zd\tilde{L}$ while $\tilde{L}$ is still orthogonal to $M$

I have a question concerning the paper http://www.researchgate.net/publication/228648002_No_arbitrage_and_the_growth_optimal_portfolio, Lemma 6.3, which is based on ...
2
votes
1answer
468 views

Partial derivative respect to random variable - How does one compute this?

CLARIFICATION: If someone could please help me understand the following: When examining the expected value in this specific situation, how is the distribution of $\theta$ relevant? What ...
0
votes
1answer
59 views

Is independence preserved in this special setting under a change of measure?

This is a question due to the answer of Did in this post Independent increments of $X_t:=\int_0^t\phi(s) dW_s$. Precisely, we assume that the dynamics of a stock prices follows ...
1
vote
1answer
133 views

Wiener process and joint distribution of $M_t$ and $W_t$

Why is $f_{M_t,W_t}(m,w) = \frac{2 ( 2 m - w)}{t\sqrt{2 \pi t}} e^{-\frac{(2m-w)^2}{2t}}, m \ge 0, w \leq m$ ? I now know what running maximum is, but unsure why joint distribution goes as above ...
1
vote
1answer
244 views

Solving Forward Equation

I've currently started reading 'Lectures on Partial Differential Equations' by Faris. Page 44 he states the following forward equation: $$J=a(y)p-\frac{1}{2}\frac{\partial \sigma(y)^2p}{\partial y} = ...
3
votes
1answer
536 views

Multidimensional infinitesimal generator of a jump-diffusion

Let $X=\{X_t\}_{t\geq0}$ be an $n$-dimensional Markov process, defined by the SDE $$dX_t = \mu(t, X_t) \, dt + \sigma(t,X_t) \, dB_t+\beta(t-,X_{t-}) \, dN_t,$$ where $\mu, \sigma$ and $\beta$ are ...
0
votes
1answer
46 views

Stochastic Processes Question

Give an example of a stochastic process $X_{n}$ that is not a Markov chain, such that $P_{y}(N(y)=\infty)=0$ but $E_{y}N(y)=\infty$
3
votes
2answers
422 views

Can we prove directly that $M_t$ is a martingale

Suppose we define the stochastic process $$M_t:=e^{\int_0^t\phi_s dW_s -\frac{1}{2}\int_0^t\phi_s^2ds}$$ where $\phi\in L^2[0,T]$, $t\in [0,T]$. Note that $M_t$ is just the stochastic exponential of ...
1
vote
1answer
106 views

Convergence to Brownian motion integral

Let $X_i$ be i.i.d with $\mathbb{E}(X_i) = 0$ and $Var(X_i) =1, \, S_n = \sum_{i=1}^n X_i$. I would like to show that $\sum_{i=1}^n \frac{f(S_i/\sqrt{n})}{n}$ converges to $\int_0^1 f(B_t)dt$ in ...
4
votes
0answers
177 views

Brownian motion integral

Let $(B_t)$ be a standard Brownian motion, $f$ a continuous function and $X_t = \int_0^t f(s)B_s ds$. I was able to prove that $(X_t)$ is a Gaussian process with zero mean and trying to find the ...
2
votes
1answer
185 views

Conditional expectation of a stochastic exponential

Suppose we have a process $(X_t)$ such that under a measure $Q$ we know that $$X_t=\mathcal{E}\left(\int_0^t\lambda(s) \, dW(s)\right)$$ for a deterministic function $\lambda(s)$ and a Brownian ...
1
vote
1answer
44 views

Why $\mathbb E \left[ \exp (u X_t )\right] < \infty$ if and only if $\int_{|z|>1} e^{uz} ~\nu(dz) < \infty$ for a Lévy process?

Consider a Lévy process $\left( X_t\right)_{t\geq 0} $ whose Lévy measure is $\nu$. Why have we for all $u \in \mathbb R$ the following result? $\mathbb E \left[ \exp (u X_t )\right] < \infty$ ...
2
votes
1answer
583 views

Stochastic Calc

(a) Consider the process $$ \mathrm d\sqrt{v} = (\alpha - \beta\sqrt{v})\mathrm dt + \delta \mathrm dW $$ Here $\alpha, \beta,$ and $\delta$ are constants. Using Ito's Lemma show that $$ \mathrm dv = ...
2
votes
2answers
1k views

Ito Isometry and quadratic variation

Here is a confusion regarding stochastic integrals. Let $Y_t=\int_0^tW_sds$ where $W_t$ is a Brownian Motion. Now $dY_t=W_tdt$. So from this expression one can conclude that $dY_t \cdot ...
1
vote
1answer
132 views

Show that $Z_t = Z_0 \exp\left( \mu t + X_t \right)$ is well defined where $X_t$ is a Jump process (Lévy process)

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$ \nu \left( dx\right) = A ...
3
votes
1answer
305 views

linear combination of two Wiener processes

I have a question concerning the linear combination of two Wiener processes (please see http://en.wikipedia.org/wiki/Wiener_process for a definition). Let $W$ and $\tilde{W}$ be two Wiener processes ...
1
vote
1answer
67 views

Continuous Everywhere

If a stochastic process $X_{t}$ $\sim N(0,t^3/3)$ and $Y_{t}$ is defined as follows: $Y_{t} = X_{t}/t$, if $t>0$ and $ Y_{t} = 0$, if $t=0$, then how can I show that $Y_{t}$ is continuous in ...
5
votes
4answers
183 views

Bell Numbers: How to put EGF $e^{e^x-1}$ into a series?

I'm working on exponential generating functions, especially on the EGF for the Bell numbers $B_n$. I found on the internet the EGF $f(x)=e^{e^x-1}$ for Bell numbers. Now I tried to use this EGF to ...
3
votes
1answer
55 views

two r.v sharing the same law

I have a question: Let $X=B^{+}$ or $X=|B|$ where $B$ is the standard Brownian motion. Set $$J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$$ where $p>1$ and $q$ its conjugate ...
1
vote
1answer
97 views

Brownian motions identical distributions

Let $(B_t)_t$ be a standard Brownian motion, and $$ A = \sup\{t\leq 1\mid B_t =0 \},\qquad B = \inf\{ t\geq 1\mid B_t =0 \}. $$ I would like to show that $A$ and $B^{-1}$ are identically distributed ...
-1
votes
1answer
243 views

How can I prove it is a martingale when there is a jump process

Let $N_t$ be a Possion process, $M_t=N_t - \lambda t$ we can easily show that $M_t$ is a martingale. Now $\int_0^t\Phi_udM_u=.....=\sum_{i=1}^{N(t)}\Phi_{\tau_i}-\lambda\int_0^t \Phi(u)du $ $\tau_i$ ...
0
votes
2answers
98 views

One stochastic integrability problem

On a lecture notes, there is a following arguement: To make $\int_0^T \pi_t dW_t$ well-defined, (maybe it means to make $\int_0^T \pi_t dW_t<\infty \ \ a.s.$) we only need $\int_0^T \pi_t^2 ...
1
vote
1answer
1k views

Expectation and variance of this stochastic process

I am trying to compute the expectation and variance of the following stochastic process: $$ Z_t = \exp \left( \frac{1}{2} \int_0^t W_s \, dW_s \right) $$ where $W_t$ is a standard Brownian motion. I ...
10
votes
1answer
416 views

Why do people simulate with Brownian motion instead of “Intuitive Brownian Motion”?

I have just recently begun studying Brownian motion and stochastic calculus at the level of an undergraduate or beginning graduate student of applied mathematics. (Textbooks I've looked at are by ...
2
votes
1answer
109 views

Are coordinate projections in the Skorokhod space continuous?

I was wondering whether coordinate projections in the Skorokhod space $D[0,1]$ are actually continuous (and, if so, how can this be proven)? many thanks for any comments/ideas. cheers!
7
votes
1answer
286 views

Strictly positive martingales

Does the following property for martingales hold? Given a continuous martingale $(X_t)_{t\leq T}$ that is almost surely strictly positive at time T, i.e. $\mathbb{P}(X_T >0)=1$, we have $P(X_t > ...
5
votes
1answer
325 views

Computing the limit of the expectation of a function of a stochastic process (phew!)

I state my problem in a few lines then describe what I have already done. I have a quite simple stochastic differential equation (SDE): $dx=-2x \, dt+\sqrt{1-x^2} \, dW$ with $W$ a brownian. I ...
2
votes
1answer
558 views

Exponential Martingale in the BS-model with time depending parameters

Suppose we have the BS model with time depending coefficient. This means we have to processes $dX_t=rX_tdt$ and $dU_t=U_t(\mu(t) dt +\sigma(t) dW_t)$, where $ W$ is a Brwonian Motion. Moreover we ...
4
votes
1answer
97 views

How to show that the following process is a submartingale

Suppose we have a filtration $(\mathcal{F}_t)$ satisfying the usual conditions. Let $W$ be a Brownian Motion with respect to that filtration. We define the two processes $X_t:=W^2_t$ and ...