All Questions
7 questions
2
votes
0
answers
85
views
Can an SDE be made to follow the flow lines of a vector field?
Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE
$$dX_t = V(X_t) \, dW_t,$$
where we identify $V(X_t) \in \mathbb R^n$ with ...
- 6,155
1
vote
1
answer
923
views
Solutions to linear SDE with many noise sources
It is well known how to solve the linear stochastic ODEs with one source of noise
$$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$
See, for instance, https://math.stackexchange.com/questions/1788853/...
- 749
2
votes
3
answers
469
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
2
votes
1
answer
593
views
General solution to system of stochastic linear differential equations
Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...
- 749
2
votes
0
answers
98
views
Non-existence for a sort of probability measures
We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...
- 257
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
1
vote
1
answer
208
views
Finding a stochastic differential equation as limit of a discrete stochastic equation
I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability $...
- 11