All Questions
405 questions
5
votes
1
answer
336
views
Joint distribution of drawdown time and value of geometric Brownian motion
Let $X$ be a geometric Brownian motion, satisfying the SDE
$$dX_t = \sigma X_t \, dW_t, X_0 = 1.$$
for $W$ a standard one dimensional Brownian motion, and $\sigma > 0$ a constant.
Define the ...
- 6,195
2
votes
1
answer
469
views
Textbook definition for "path measure" or "probability measure over paths"
I need a formal definition for the path measure for stochastic differential equations.
Which textbook or paper should I consult?
3
votes
0
answers
201
views
Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?
In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
- 177
1
vote
1
answer
247
views
How to rigorously prove that this sequence of stochastic processes converges to a deterministic process?
Assume that for each $n\in\mathbb{N}$, there's a stochastic function $f_n$ of type $\mathbb{R}^{m}\to\Delta\mathbb{R}^{m}$, and for each $x\in\mathbb{R}^{m}$, the distributions $\frac{f_n(x)-x}{\frac{...
- 353
5
votes
1
answer
531
views
Riemannian metric induced by a stochastic differential equation
Following this paper, a diffusion process in $\mathcal{R}^d$
$$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$
with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be ...
1
vote
1
answer
604
views
Is there an inverse Lamperti transformation for diffusions?
The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion.
For multidimensional processes there are some conditions on the ...
2
votes
1
answer
204
views
Comparing diffusion processes in different metrics
I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$.
Is there a way to apply ...
1
vote
0
answers
121
views
Stratonovich version of Girsanov
One version of Girsanov says that, that if $\mu_0$ is the law of a Brownian motion as a Borel measure on the space of continuous functions and we define the density
$$\frac{d\mu}{d\mu_0}:=\exp\left(\...
- 1,904
2
votes
1
answer
163
views
Does the time of maximum of a diffusion process admit a continuous density?
Let $W$ be a standard one dimensional Brownian motion, and consider the solution $X$ to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $X_0 = 0$ a.s., and where $\mu, \sigma: \mathbb R \...
- 6,195
0
votes
0
answers
75
views
Regularity of solutions to forward-backward stochastic differential equations
Suppose $X_t$, $P_t$ and $Z_t$ are one dimension random processes and satisfy
$$
\left\{
\begin{aligned}
d X_t
&= aP_t dt +bdB_t;\\
X_0
&= x_0;\\
d P_t
&=cP_t dt + c^*Z_t dB_t;
\\
P_T
&...
- 54
4
votes
1
answer
343
views
Convergence of a continuous time stochastic gradient descent algorithm
Let $f: \mathbb R \to \mathbb R$ be a $C^1$ convex function, satisfying the growth conditions
$$\lim_{x \to -\infty} \nabla f(x) = -\infty, \lim_{x \to \infty} \nabla f(x) = \infty.$$
and let $\...
- 6,195
4
votes
1
answer
509
views
What work has been done on SDE with diffusion coefficients of bounded variation in $\mathbb R^d$?
Consider the $d$-dimensional SDE, $d > 1$,
$$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t$$
where $W$ is a standard $d$-dimensional Brownian motion.
I am interested in the case where $\sigma: \mathbb ...
- 6,195
7
votes
1
answer
249
views
Onsager-Machlup functional when drift is time-dependent
Let $X(t)$ be a diffusion process on $\mathbb{R}^d$ generated by
\begin{align}
\mathcal{D} = \nabla^2 + \sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i},
\end{align}
where $b_i(x) \in \mathcal{C}_b^2(...
- 203
1
vote
0
answers
82
views
Uniqueness of global solution
I am reading Section 3.3 of this paper, and trying to understand the proof of uniqueness of a global solution to the following equation defined on the Torus $\mathbb{T}^3$
\begin{align*}
\mathrm{d} \...
- 101
5
votes
1
answer
1k
views
Correlated Brownian motions across different times and representation with independent processes
This is a more wide-net question of Two increasingly correlated Brownian motions and Williams decomposition.
In our problem we have two correlated Brownian motions $B^1,B^2$ (starting at time $t=0$ ...
- 5,474
1
vote
0
answers
235
views
Two increasingly correlated Brownian motions and Williams decomposition
The Williams decomposition is
Let $(B_t-\nu t)_{t\geq 0}$ be a Brownian motion with negative drift $\nu>0$ and let $M_\infty^{-\nu}:=\sup_{t\in [0,\infty]}(B_t-\nu t)$. Then conditionally on $M_\...
- 5,474
1
vote
0
answers
156
views
Fokker-Planck equation for a 3D Bessel bridge
Consider a 3D Bessel bridge $\rho_t$ connecting $(x,t)=(0,0)$ and $(x,t)=(0,T)$, whose SDE is given by
$$d\rho_t = \left(\frac{1}{\rho_t} - \frac{\rho_t}{T-t}\right)dt + dB_t,$$
where $B_t$ is a ...
- 19
2
votes
1
answer
549
views
A question related to Girsanov’s theorem
I’ve recently realised there is a subtlety in Girsanov’s theorem that I don’t really understand.
Consider a standard one dimensional Brownian motion $W$, and consider the SDE
$$dZ_t = \mu(t, Z_t) \, ...
- 6,195
1
vote
0
answers
89
views
Comparison of the numbers of particles surviving forever
Consider two $N\text{-}$particle systems as follows : for $1\le i\le N$,
$$X^i_t=1+\int_0^t(b+\phi^i_s) \, ds+W^i_t \quad\mbox{and} \quad Y^i_t=1+ct+W^i_t,\quad \forall t\ge 0,$$
where $c>b>0$ ...
- 1,334
2
votes
2
answers
416
views
Short time limits for SDE
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \sigma(X_t) \, dW_t \;, \quad X_0 = x_0\;.$$
where $\sigma:\mathbb R \to \mathbb R$ is a ...
- 6,195
2
votes
1
answer
296
views
Large noise limit for SDE with general volatility coefficients
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \sigma(X_t) \, dW_t \;, \quad X_0 = 1 \;.$$
where $\sigma:\mathbb R \to \mathbb R$ is a ...
- 6,195
2
votes
1
answer
493
views
Is the solution to this SDE always positive?
Let $W$ be a standard one dimensional Brownian motion, and consider the SDE
$$dX_t = \sigma(X_t) \, dW_t, \, \, \, X_0 = 1 \, \text {a.s.}$$
Assume $\sigma$ is regular enough that the above SDE admits ...
- 6,195
1
vote
1
answer
197
views
Construction of SDEs that admit more than one solution
I look for examples of SDEs (stochastic differential equations) s.t. the uniqueness of the solution fails, i.e.
$$dX_t = B(t,X_t)dt + \Sigma(t,X_t)dW_t,\quad \forall t\ge 0.$$
More precisely, the ...
- 1,334
1
vote
1
answer
201
views
A comparison principle for SDE
Let $W$ be a standard one dimensional Brownian motion, and $\mathcal F_t$ its natural filtration. Consider the SDE
$$dX_t = \mu_X (t, \omega) \, dt + \sigma_X (t, \omega) \, dW_t$$
$$dY_t = \mu_Y (t, \...
- 6,195
2
votes
1
answer
179
views
Solution of SDE with time power law singular diffusion
I was wondering if anything could be said at all about the well-psedness of the following time-inhomogeneous singular diffusion SDE:
\begin{align}d X_t&=\sigma(X_t,t ) d W_t , \qquad t\geq 0, ...
- 135
0
votes
0
answers
466
views
The relationship between measurability and weak measurability
For a Banach-valued random mapping $f:\Omega\rightarrow X$, there are three kind of measurability: strong measurability (can be approximated by sequence of simple
functions, measurability (the ...
- 113
1
vote
2
answers
240
views
Solution to SDE conditional on high maxima of driving Brownian motion
Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = X_t \, dW_t \;, \quad X_0 = 1 \;.$$
For every $\varepsilon > 0$, let $A_\varepsilon$ denote ...
- 6,195
1
vote
0
answers
157
views
The stochastic parallel transport as a limit of piecewise geodesic parallel transports
Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$,...
- 5,407
4
votes
1
answer
218
views
Schauder basis of the Hardy space of semi-martingales
Fix $p\in [1,2]$, a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_t,\mathbb{P})$, and let $\mathcal{H}_{\mathscr{S}}^p$ denote the space of semimartingales $X$ such that the norm
$$
\...
- 195
0
votes
2
answers
182
views
Distribution of local martingale is absolutly continuous to that of the Brownian motion?
Let $B(t, \omega)$ be a Brownian motion defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, adapted to a filtration $\{\mathcal{F}_t\}$. Let $\phi(t, \omega)$ be a $\{\mathcal{F}_t\}$-...
- 227
2
votes
1
answer
361
views
Is $g(t)=\mathbb P[\inf_{0\le s\le t}X_s>0]$ differentiable with respect to $t$?
Consider the SDE
$$dX_t =b(t)dt + a(t)dW_t,\quad \forall t>0,$$
with $X_0>0$ has a density function $\rho:\mathbb R_+\to\mathbb R_+$. Consider the probability $g(t):=\mathbb P[\inf_{0\le s\le t}...
user478492
1
vote
0
answers
34
views
Regime switching stochastic systems references
I'm looking for some good references discussing regime switching stochastic systems (Stochastic systems with markovian jump process) and their solutions.
Given a Continuous-time Markov Chain $\xi$ ...
- 141
0
votes
1
answer
493
views
Fokker-Planck: uniqueness and convergence to stationary distribution
Consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
- 305
1
vote
0
answers
87
views
Reference request : upper bound of marginal densities of a SDE with discontinuous coefficient
Consider the one-dimensional SDE
$$X_t = x+ \int_0^t\frac{\sigma(s,X_s)}{1+{\bf 1}_{\{b(s,X_s)>0\}}}dW_s,\quad \forall t\ge 0,$$
where $W_t$ is a standard BM and $b,\sigma$ are sufficiently regular ...
user478492
2
votes
1
answer
240
views
Uniqueness of the solution to some degenerate SDE
Consider the one-dimensional stochastic differential equation:
$$dX_t = {\bf 1}_{\{X_t>0\}}\big(b(t,X_t)dt + a(t,X_t)dW_t\big),\quad \forall t>0,$$
or equivalently
$$dX_t = b(t,X_t)dt + a(t,X_t)...
user478492
2
votes
0
answers
50
views
Continuation : Uniqueness of the solution to some SDE with discontinuous coefficient
Consider the SDE below
$$X_t=X_0+\int_0^t b(s)ds+\int_0^t\frac{dW_s}{1+m(s){\bf 1}_{\{b(s)>0\}}},\quad \forall t\ge 0,~~~~~~~~~~~~~~~(\ast)$$
where $X_0>0$ is square integrable, $b:\mathbb R_+\...
- 1,334
0
votes
1
answer
349
views
Probability that a geometric Brownian motion with additional determinstic drift ever hits zero
Let $W$ be a standard Brownian motion, and let $X_t$ be the solution to the following SDE
$$dX_t = (\mu X_t - Cke^{-kt}) \, dt + \sigma X_t \, dW_t$$
where $\mu, \sigma, C, k > 0$ are constants, ...
- 6,195
2
votes
0
answers
116
views
Is a Riccati BSDE explicitly solvable?
Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
- 335
1
vote
1
answer
183
views
Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?
Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE.
Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...
- 11
2
votes
0
answers
186
views
Can integrals with respect to time-changed Brownian motion be seen as integrals with respect to Brownian motion?
Let $X_t:=W_{t\wedge \tau}$ for $t\ge 0$, where $(W_t)_{t\ge 0}$ is a standard Brownian motion and $\tau:=\inf\{t\ge 0: |W_t|=1\}$. It holds
$$X_t=\int_0^t {\bf 1}_{\{|X_s|<1\}}dW_s,\quad \forall t\...
- 1,334
2
votes
0
answers
117
views
How does the probability of staying positive depend on the diffusion coefficient?
Let $X$ and $Y$ be two continuous martingales given as
$$X_t=z + \int_0^t a(s,X_s)\, dW_s,\quad \quad Y_t=z + \int_0^t b(s,Y_s) \, dW_s,$$
where $z>0$, $a,b$ are Lipschitz and bounded functions s....
- 21
5
votes
0
answers
400
views
Uniform bound for the occupation time of a diffusion
Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$.
Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions.
Suppose the ...
- 6,195
0
votes
1
answer
277
views
Autocorrelation function of Itô process
I'm working with a time independent (vector) Itô SDE such as:
$$
dX = a(X) dt + b(X) dW.
$$
I've looked (numerically) at several examples and it seems that the autocovariance function $r_{xx}(\Delta t)...
- 309
0
votes
0
answers
97
views
Uniqueness of the solution to some SDE of state-dependent coefficient
This is a continuation of my question posted in Uniqueness of the solution to some SDE
Consider
$$X_t=X_0 + t + \int_0^t \frac{\sigma(s,X_s)}{1+m(s)}dW_s,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$
...
- 1,334
0
votes
1
answer
260
views
Has this "stochastic differential equation" been studied?
Update: Thanks to GJC20's answer on the existence and uniqueness. Let me reformulate my questions 3/4 as follows: There exists a unique non-increasing and continuously differentiable function $f:\...
user420828
1
vote
0
answers
124
views
On the Lipschitz constant of $\Gamma$
Let $b: \mathbb R_+\times\mathbb R\times \mathbb R\to\mathbb R$ be a function as nice as possible, and $C^1([0,T])$ be the space of continuously differentiable functions $\alpha:[0,T]\to\mathbb R$ ...
- 1,334
1
vote
1
answer
133
views
What are the optimal times to sample a process?
Let $X$ be a one dimensional Ito diffusion given by
$$X_t = b \,W_t$$
where $b$ is a constant, and $W$ is a standard Brownian motion.
Let $B$ be another Brownian motion independent of $W$, and define ...
- 6,195
2
votes
1
answer
139
views
Search for conditions of the positive probability that a stochastic process never hits zero
Consider a stochastic process $X$ defined by
$$X_t:=1+\int_0^t b(s,X_s) \, ds+ W_t,\quad \forall t\ge 0,$$
where $(W_t)_{t\ge 0}$ is a standard Brownian motion. Suppose that $b:\mathbb R_+ \times \...
user420828
5
votes
1
answer
289
views
What is the formal definition of a stochastic PDE and a solution to a stochastic PDE?
While searching through this Wikipedia article, I have stumbled uopn the following 'stochastic' heat equation
$$\partial_tu=\Delta u+\xi,$$
where $\xi$ is the space-time white noise. However, I don't ...
- 153
1
vote
0
answers
91
views
When enlarging a filtration makes a stochastic processes into a solution to an SDE
Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...
- 5,405