# Tagged Questions

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votes

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60 views

### Girsanov theorem with Geometric Brownian Motion

I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...

**3**

votes

**1**answer

108 views

### Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...

**2**

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**0**answers

63 views

### Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...

**2**

votes

**1**answer

104 views

### weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

**0**

votes

**1**answer

148 views

### On the superior of generalized Ornstein-Uhlenbeck process

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde:
$dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $
$\partial_x X_t(0) = \partial_x X_t(1) = 0, $
$X_0 = 0, $
...

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vote

**0**answers

206 views

### Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...

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**0**answers

190 views

### Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail.
Here is what I mean exactly. ...

**3**

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**0**answers

105 views

### Expectation of running maximum of diffusion processes

Let $X$ be a one-dimensional Ito diffusion $$X_t=x+ \int_0^t b(X_s)ds + \int_0^t \sigma(X_s)dW_s,$$ where $b,\sigma$ satisfy the usual Lipschitz continuity and linear growth conditions. Define the ...

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123 views

### Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type
$$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$
converges to zero (in probability/distribution) for $h ...

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vote

**2**answers

110 views

### Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation
\begin{equation}
dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0,
\end{equation}
where ...

**0**

votes

**1**answer

71 views

### Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent.
1): $X$ is $L^p$-integrable.
2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...

**4**

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**0**answers

223 views

### Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...

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vote

**0**answers

75 views

### question about Doob-Meyer decomposition

Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:
...

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votes

**1**answer

358 views

### Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$:
$$dX_t = f(X_t)\,dt + dW_t,$$
where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...

**1**

vote

**0**answers

91 views

### On the solution of a stochastic partial differential equation

Consider a simple SPDE as follows:
$\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$,
$u(t,0)=u(t,1)=0$,
$u(0,x)=v(x)$,
where $V$ is a bounded, smooth ...

**7**

votes

**2**answers

296 views

### Can every discrete martingale be embedded in a continuous martingale?

Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...

**1**

vote

**1**answer

71 views

### The jump and the left martingale of semimartingale

Let $X_{t}$ be a semimartingale. Define
$\Delta X_{t} = X_{t}- X_{t-}$.
For fixed $s> 0$, $\Delta X_{s}$ and $X_{s-}$ are two random variable. Are they independent to each other? I think the ...

**3**

votes

**1**answer

107 views

### The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$.
This property is called the ...

**3**

votes

**0**answers

74 views

### The distribution of Jump gaps of Levy process

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...

**3**

votes

**1**answer

117 views

### Can this two-dimensional process self intersect?

I would like to know more about the two-dimensional processes derived from Brownian motion by the following stochastic differential equation (in the Ito sense)
$$dX_t = f(X_t) dt + ...

**0**

votes

**1**answer

121 views

### a dominated convergence theorem for martingale (II)

The question is presented in
a dominated convergence theorem for martingale
Let $\{(X_1^n, X_2^n)\}_n$ be a sequence of martingales defined some probability space. (which means ...

**3**

votes

**1**answer

118 views

### Domino Shuffling and Warren's process

In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the ...

**3**

votes

**1**answer

118 views

### a $L^1$ convergence for backward martingale

I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ...

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**2**answers

147 views

### On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients

Consider the SPDE $$\frac{\partial}{\partial t}u_t(x) = \frac{\kappa}{2}\frac{\partial^2}{\partial x^2}u_t(x) + u_t(x)(K-u_t(x)) + \sigma u_t(x) \xi(t,x),$$
where $(t,x)\in {\mathbb R}_+\times ...

**2**

votes

**0**answers

101 views

### On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form

Denote $E = C([0, 1])$. I am consider a 1-dimentional stochastic heat equation on $h$:
$\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x)$, for all $(t, x) \in (0, ...

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votes

**1**answer

117 views

### On the expected value of a random integral:

Is it possible to find the expected value of $u(t)$ in terms of the following information:
$$u(t)=\int_{0}^{t}(t-s)(f(s)+(T-s)Y)X_sds$$
where:
$X_s$ is a wide sense stationary process with known ...

**5**

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74 views

### Stochastic calculus for several inputs

In "On the Gap Between Deterministic and Stochastic Ordinary Differential Equations," The Annals of Probability, Vol. 6, No. 1 (Feb., 1978), pp. 19-41, Hector J. Sussmann showed that a stochastic ...

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votes

**1**answer

84 views

### Multiplicative version of Novikov inequality for Ito integral

It is clear that Ito isometry
$E(∫^t_0fdW)^2=E(∫^t_0f^2dt)$
can be written in the multiplicative form as
$E(∫^t_0fdW\cdot∫^t_0gdW)=E(∫^t_0f⋅gdt).$
Is it possible to obtain the multiplicative version ...

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votes

**3**answers

302 views

### When is a continuous path stochastic process be representable as diffusion or Ito process?

When can a continuous path (Markovian) stochastic process in one dimension be represented as an Ito or a diffusion process? What are the examples when it can not be?

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vote

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64 views

### Time change for non-homogeneous Markov processes

Background: Let $C$ be the space of continuous function on $[0,T]$, $f, \sigma \in C$ bounded with $\sigma^2 \geq \varepsilon > 0$ and let $X=(X_t)_{t\in [0,T]}$ be a diffusion process of ...

**2**

votes

**1**answer

174 views

### Stochastic integral with respect to discontinuous martingale

in my research, I need to deal with a stochastic integral with respect to a compensated poisson process, namely,
$ \int_0^t f(X_t) dM_t,$
where $M(t) = N(t) - \int_0^t \lambda(s)ds$.
The integrand ...

**5**

votes

**1**answer

420 views

### On the pathwise uniqueness of solutions of SDEs(Stochastic Differential Equations)

Suppose that $(\Omega,\mathscr{F},P)$ is a complete probability space equipped a filtration $\{\mathscr{F}_t\}$ satisfying the usual conditions. $B_t$ is a 1-dimentional Brownian motion with respect ...

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**0**answers

108 views

### Is there a theory of SDEs whose coefficients are themselves adapted processes (i.e. “may depend on the past”)?

Is there an existence and uniqueness theorem for SDEs of the following type:
$dW_{t}=d\tilde{W}_{t}+\mu\left(\left(W_{s}\right)_{0\le s\le t},t\right)dt$,
where $\tilde{W}_{t}$ is say ...

**0**

votes

**1**answer

238 views

### Markov Chain: state reduction

Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following:
Firstly we have a Markov chain $\{Y_k\}$ with finite ...

**2**

votes

**1**answer

247 views

### Upper bound on the maxima of ratio of expectation of quantities under Gaussian measure

Let $\lambda,\eta >0$ be given, and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued function. Define
$$\Delta(u)= \frac{\int u(h) \exp(-\eta ...

**2**

votes

**0**answers

111 views

### Lyapunov function of exponential growth for existence of a solution of an SDE

Let
$$dX_t = a(X_t) dt + b(X_t) dW_t$$
be a one-dimensional stochastic differential equation, where the coefficients $a,b: \mathbb{R} \rightarrow \mathbb{R}$ satisfy for every ball $B_R$ the following ...

**3**

votes

**2**answers

308 views

### Solving a SDE with quadratic drift

I am wondering whether the following SDE can be solved explicitly?
$$
d X_t = X_t^2 d t + X_t d B_t
$$
where $B_t$ is a standard Brownian motion. If not, can we say some thing about the moments of ...

**2**

votes

**1**answer

193 views

### Conditional law of an Ito's stochastic integral

Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real ...

**2**

votes

**0**answers

59 views

### Tail for the integral of a diffusion process

I would like to compute the following tail,
$$
\mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right),
$$
assuming
$$
\mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x),
$$
and $X$ is a diffusion ...

**5**

votes

**0**answers

370 views

### When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form
$$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$
for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb ...

**1**

vote

**1**answer

311 views

### Iterated Ito Integral, Gaussian Volterra Process

Let me define
$$
J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1}
$$
where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice deterministic ...

**0**

votes

**1**answer

317 views

### Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...

**1**

vote

**0**answers

98 views

### Time integral of a diffusion

Define $\bar\sigma^2_t=\frac{1}{t}\int_0^t\sigma^2(X_s)ds$ where $\sigma(x)\geq0$ is a measurable function and $X_t$ a diffusion process defined by
\begin{equation}
...

**1**

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**0**answers

163 views

### What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?

There is a theorem as follows:
Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto ...

**1**

vote

**1**answer

394 views

### Good books on stochastic partial differential equations?

I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are ...

**2**

votes

**1**answer

135 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 usuual ...

**4**

votes

**1**answer

368 views

### Converse to Girsanov's theorem?

Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...

**2**

votes

**1**answer

180 views

### Limit of a Wiener integral

How to show that
$$ \lim_{\alpha \rightarrow \infty} \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.}$$
where $\left (B_s ...

**0**

votes

**1**answer

95 views

### Concerning 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 ...

**0**

votes

**1**answer

203 views

### Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral

Hi, I have the following expected value to compute
$E[ \int_{o}^{T} f(t) dt \int_{o}^{T} H(s) dW(s)]$,
where $f(t)$ and $H(s)$ are two stochastic processes adapted to the filtration generated by the ...