All Questions
16 questions
3
votes
0
answers
60
views
Comparison theorem for SDEs driven by a continuous martingale
Consider the well-known comparison theorem for SDEs, versions of which appear in several textbooks, e.g., Karatzas and Shreve, Proposition 5.2.18, or Revuz and Yor, Theorem IX.3.7.
The result states ...
- 103
1
vote
0
answers
134
views
Generating realizations from $n$-dimensional geometric Brownian motion where the variables are constrained to sum to 1
Is there a way to simulate an $N$-dimensional geometric Brownian motion i.e. variable $$x_i, i \in [1, N] $$ is diffusing in log-space such that $$\log (x_i)$$ follows a Brownian motion with a given ...
- 21
1
vote
0
answers
100
views
Reference request: $d X_t = b(X_t) d t + f (p_t(X_t)) d W_t$ where $p_t$ is the p.d.f. of $X_t$
Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then
$$
d X_t = ...
- 657
3
votes
2
answers
271
views
For a SDE with smooth transition densities, if every point is "path-accessible", is every positive-measure set probabilistically accessible?
Suppose we have a $C^\infty$ manifold $M$ and $C^\infty$ vector fields $b,\sigma_1,\ldots,\sigma_k$ on $M$, and for convenience define the set of vector fields
$$ \mathcal{S} = \{b,\sigma_1,-\sigma_1,\...
- 708
4
votes
1
answer
262
views
Bounded density for diffusions with diffusion coefficients bounded away from $0$
Consider a diffusion given by
$$X_t=\int_0^t a(s,X_s)\,dW_s$$
for $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and $a$ is a smooth enough positive function bounded away from $...
- 128k
1
vote
1
answer
337
views
Bessel process conditioned to stay positive
This question has also been asked on https://math.stackexchange.com/questions/4174928/bessel-process-conditioned-to-stay-positive
Suppose the stochastic process $(X_t)_{t\ge 0}$ with start in $X_0:=x&...
- 284
2
votes
2
answers
381
views
Discrete random walk and SDEs
My advisor has some vague ideas about the relation between discrete random walks and SDEs, and advise me to read a little bit about them.
To be more precise, ( if I understand correctly what my ...
- 327
0
votes
1
answer
341
views
Hitting probability for mean-reverting stochastic process
I quote Delbaen and Shirakawa (2002).
Starting from a stochastic differential equation of the form:
$$dr_t=\alpha\left(r_{\mu}-r_t\right)dt+\beta\sqrt{\left(r_t-r_m\right)\left(r_M-r_t\right)}dW_t\...
1
vote
0
answers
62
views
Reference request for invariance principles
In various places, an example being
https://projecteuclid.org/download/pdf_1/euclid.aoap/1034625254,
the authors consider a discrete-time process (real-valued, say) $(X_n)_{n \in \mathbb{N}}$, define ...
- 111
4
votes
1
answer
306
views
How to make sense of recursively defined SPDE solutions, like in Hairer's "Solving the KPZ equation" paper?
In Martin Hairer's 2013 paper "Solving the KPZ equation", the process $X_\epsilon^\bullet$ is defined as the stationary solution to
$$
\partial_t X_\epsilon^{\bullet} = \partial_x^2 X_\epsilon^{\...
- 147
2
votes
2
answers
255
views
Process with covariance $E[Y_{t}Y_{s}]=a_{1}-a_{2}|t-s|$
We have a centered Gaussian process $X_{t}$ where we have exact equality $$E[X_{t}X_{s}]=a_{1}-a_{2}|t-s|$$ for $|t-s|<\epsilon_{0}\ll \frac{a_{1}}{a_{2}}$ and $a_{i}>0$.
Q: I am curious if ...
- 5,474
3
votes
2
answers
1k
views
Is the "hybrid" Black-Scholes Hull-White model arbitrage free?
Given a "hybrid" Black-Scholes Hull White (BSHW) model. That is, the stock price is modelled by a Black Scholes SDE:
\begin{equation} dS(t) = \mu(t)S(t)dt + \sigma_{S}(t)S(t)dW^{\mathbb{P}}_{S}(t)
\...
- 203
3
votes
0
answers
170
views
Feynman-Kac formula for *general* Sturm-Liouville operator
One way to state (omitting technical requirements) the Feynman-Kac formula that I am familiar with is as follows.
Let $u$ be a solution to the pde
$$u_t(x,t)=-\frac{\sigma^2(x,t)}2u_{xx}(x,t)-V(x,t)u(...
- 891
2
votes
3
answers
471
views
Existence of solution to SDE with perscribed initial and terminal conditions
The SDEs \begin{equation}
dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t
\end{equation} with prescribed initial conditions are well studied. My question came up in my research and I have not found much on ...
- 5,405
1
vote
0
answers
118
views
Full version of Soucaliuc's research announcement "Réflexion entre deux diffusions conjuguées"
Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes:
[1] F. Soucaliuc, Réflexion entre deux diffusions ...
- 43
1
vote
1
answer
482
views
Wong-Zakai smooth approximation in probabilty for stochastic differential equations
I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to an $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) ...
- 111