# Questions tagged [stochastic-calculus]

Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

611 questions
Filter by
Sorted by
Tagged with
138 views

For the laws of two pure-jump Markov processes $\mu_1$ and $\mu_2$ on $\mathbb R^n$, which generators are $H_1f(x)=\int h(x,dy) (f(y)-f(x))$ and $H_2f(x)=\int e^{-g(x,y)} h(x,dy) (f(y)-f(x))$ (...
22 views

### Can one change the dimension of a Bessel process by a Girsanov change of measure?

Recall that a (squared) Bessel process $X_t$ with the dimension $\delta_0>0$ is the solution of the SDE $$d X_t = 2\,\sqrt{X_t}\,d W_t+\delta_0\,d t.$$ A naive application of the Girsanov Theorem ...
84 views

168 views

### Exponential or sub-exponential ergodicity?

Consider the one-dimensional stochastic differential equation $$d X(t) = -sgn(X(t))dt + dW(t),$$ where $W$ is a standard Brownian motion, and $sgn(x) = 1$ if $x > 0$ and $-1$ if $x\le 0$. It can be ...
27 views

### Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?

I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" ...
30 views

### Stopping times about Brownian motion with draft

Assumet $M(t) = B(t) + \mu t$ where $B(t)$ is a standard Brownian Motion. Denote: $$T_a := \inf \{ t \geq 0, \, M(t) = a\}, \quad T_b := \inf \{ t \geq 0, \, M(t) = b\}$$ The question asks to ...
62 views

### The weak convergence of finite dimensional distribution of Gaussian process does not imply the weak convergence in $C[0,1]$

In the study of weak convergence in $C[0,1]$, a common example is always being considered: $$X_{n}(t)=nt1_{[0,1/n]}(t)+(2-nt)1_{(1/n,2/n]}(t).$$ This example serves a counter-example to show that the ...
58 views

### Find a conditional expectation of a difference of two independent Poisson process

Consider two independent Poisson processes $N,M$ with rate $\lambda$, and define $$X(t):=x+\dfrac{1}{\sqrt{n}}[N(t)-M(t)].$$ From this formula we know that $X(0)=x$. Now, I want to compute the ...
21 views

### Stochastic differential equations with correlated Brownian Motions

let's consider an sde of this kind: \begin{equation} \label{eq:system} \begin{cases} dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t \\ X_0=x_0 \\ dY_t=B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^2 \\ Y_0=y_0 \end{...
16 views

### Bifractional Brownian motion admit a representation in the form of a stochastic integral?

good morning. You know the fractional Brownian motion, multifractional Brownian motion and sub-fractional Brownian motion, can be represented as a wiener integral ( moving average representation ). ...
60 views

### Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$

I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...
22 views

### Sufficient condition for weak existence of solution of a SDE

Please be adviced that I'm cross-posting this question from MSE since it's very likely it will remain unsolved, and I haven't been able to obtain an answer from my colleges/professors. It's a well ...
207 views

254 views

### Show that this process is not a martingale

I am cross-posting this question from MSE since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer. The most surprising ...
26 views

### Reference for an infinite system of SDEs

Consider system of the following form, \begin{align*} \mathrm{d} X_k(t) = \big(AX(t)\big)_k\mathrm{d}t + B_k(X_k(t))\mathrm{d}t+\mathrm{d}W_k(t),\quad k\in\mathbb{Z}, \end{align*} where $A$ is matrix, ...
22 views

### Step verification in derivation of Ito formula

At the page 125 (see 4.90) (see images below, this proof is from "Statistics of Stochastic processes" by Lipster and Shiryaev) we consider function $u(t,W_t)=f(t,at+bW_t)$ and $a(\omega), b(\omega)$ ...
45 views

150 views

Let $B$ be a Brownian motion. Definining a pathwise stochastic integral $I(f):=\int f~dB$ for certain classes of deterministic functions is straightforward: For instance if $f=\sum_ic_i1\{[t_i,t_{i+1})... 0answers 150 views ### Proving that$dX_t=a_tX_td\tilde{B}_t$is a martingale Fix$T>0. $Consider the probability space$(\Omega,F,Q)$and a Brownian motion$\{B_t\}_{t\leq T}$and filtration$\{F\}_t$generated by the Brownian paths. Suppose$a_t,\gamma_t$are random ... 0answers 84 views ### Malliavin derivative of Ito process Let$X_t= X_0 + \int_0^t \mu(s,X_s)ds + \int_0^t \sigma(s,X_s)dW_s$where$\mu$and$\sigma$are$C^1$functions satisfying the usual growth restriction and$W_t$is a$d$-dimensional Brownian motion. ... 2answers 208 views ### Weak convergence in Skorohod topology Let$D([0,T];R^d)$be the space of càdlàg functions endowed with the usual Skorohod topology.$X_t(\omega):=\omega(t)$denotes the usual canonical process. Assume that a family of probability ... 0answers 34 views ### Derivation of a differential equation from a SDE Suppose there is a non-homogeneous Markov process with state space$\mathbb{R}_{+}$driven by this McKean-Vlasov-tipe SDE: $$dY_t = a \mathbb{E}[Y_t]\ dt - b\ Y_t\ dt - Y_t\ dN_{aY_t}$$ where$...
I have a question on heat kernels and eigenfunctions of symmetric Markov processes. Let $X=(\{X_t\}_{t \ge0 },\{P_x\}_{x \in M})$ be a $\mu$-symmetric Markov process on a locally compact separable ...
Suppose I have a discrete stochastic process $\{ X_i \}_{i=1,\ldots..}$ defined as, $X_{i+1} = X_i - \eta \nabla f(X_i) + \sqrt{\eta} \xi_i$ where $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and \$\xi_i \...