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.
377 questions with no upvoted or accepted answers
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Conditions ensuring that conditional law of a process belongs to a given exponential family
Let $(X_t,Y_t)_{t\geq 0}$ be a pair of $\mathbb{R}^n$-(resp. $\mathbb{R}^m$)-valued stochastic processes on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$, ...
- 77
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79
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Laplace transform of pdf of hitting time for square root diffusion
Consider the SDEs
\begin{align}
dX_t&=\alpha X_tdt+\sqrt{v_t}X_tdB_t \\
dv_t&=\eta(\theta-v_t)dt+\xi \sqrt{v_t}dW_t
\end{align}
where $\alpha,\eta,\theta,\xi$ are constants and $\rho$ is the ...
- 255
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0
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76
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Gronwall type lemma for an Ito process
For all $t\in \mathbb{R}$ let $h_t = \frac{1}{2} + \int_0^t v_s\cdot dB_s$ be an Itô process, where $B_s$ is a standard Brownian of $\mathbb{R}^d$ and $v_t$ an $\mathbb{R}^d$ valued adapted process, ...
- 245
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124
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L2-closure of absolutely continuous stochastic processes?
Assume we have a possibly multidimensional Brownian motion on a probability space $(\Omega,\mathcal F,\mathbb P)$ where $(\mathcal F_t)_{t\in[0;T]}$ is the Brownian standard filtration. Let $\Vert X\...
- 335
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108
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Decomposition of reversed processes
Consider a reversed filtration $(\mathcal{F}_k)_{k \geq 0} $ $(\mathcal{F}_{k+1} \subset\mathcal{F}_k),$ $(X_k)_{k \geq0}$ is a processes in $L^1,\mathcal{F}_k$-adapted.
Is it possible to decompose $...
- 249
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80
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Almost supermartingale and a.s convergence
After reading a paper on the convergence of almost supermartingale, the following result appeared:
If $(X_k)_k,(Y_k)_k,(W_k)_k$ are three $(\mathcal{F}_k)$-adapted processes taking values in $\mathbb{...
- 249
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78
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If $(\alpha_t)$ is $\mathbb{F}^X$-progressive for a continuous process $(X_t)$, can we write $\alpha_t = \tilde{\alpha}(t,X)$?
Let $X = (X_t)_{t \geq 0}$ be a continuous, real-valued process defined on some probability space $(\Omega,\mathcal{F},P)$, and let $\mathbb{F}^X = (\mathcal{F}_{t}^X)_{t \geq 0}$ be the filtration ...
- 309
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44
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Normalization of exponential in the context of Feynman integrals from a White noise perspective
I apologize in advance if this question is not suitable for MO (please let me know), but the fact is that since I am not familiar with the theory of Feynman integrals I don't know whether this is a ...
- 515
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68
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Differentiable approximation of Brownian diffusion with unbounded volatility
Let $\{W_t\}_{t\in[0;T]}$ be a one-dimensional Brownian motion and let $\{\mathcal F_t\}_{t\in[0;T]}$ be the augmented filtration generated by this Brownian motion. Let $\{\sigma_t\}_{t\in[0;T]}$ be ...
- 335
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245
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Expectation of first exit time of a bounded set by a time-homogeneous Ito diffusion is finite
This is a question concerning Remark(i) under Theorem 7.4.1(Dynkin's formula) on Page 124, $\textit{SDE}$, by Oksendal.
It says that if $dX_t=\mu(X_t)dt+\sigma(X_t)dB_t$ is an $n$-dimensional time-...
- 715
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222
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Is my quadratic variation derivative bounded?
Let $\{W_t\}_{t\in[0;T]}$ be a Brownian motion, let $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ be continuous, bounded and Lipschitz continuous in the second argument, let $X$ be the unique ...
- 335
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158
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Translation of Dellacherie's Capacités et Processus Stochastiques
I have been studying the Strasbourg school's general theory of processes from Dellacherie and Meyer's Probabilities and Potential, and I really like it. I have heard very good reviews about another ...
- 141
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80
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Continuity of the random variable defining the occupation measure of a continuous Gaussian process
Suppose $Z:\Omega \times [0,1] \to \mathbb{R}$ is a continuous Gaussian process with mean $\mu(t)$ and covariance kernel $C(t,s)$. Consider the random variable
$$
X_\alpha = \lambda( \{t \; : \; Z(t) &...
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766
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Derivative of the function of random variable
Suppose we have a function $\phi(X)$ of random variable $X$. Suppose both of $\phi(X)$ and $X$ are random variables. If $\phi$ is differentiable, how to calculate the derivative of $\phi(X)$ w.r.t. $...
- 195
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746
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Local martingale but not martingale
For a 3-dimensional Brownian motion $B = (B_t, t ≥ 0)$ and $x ∈ \mathbb{R}^3 \backslash \{0\}$ define the process
$Y = (Y_t, t ≥ 0)$ via $Y_t =\frac{1}{|B_t+x|}$ how come this is a continuous local ...
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94
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Generator of a Hilbert space valued Wiener process from the solution of a martingale problem
Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
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276
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Path dependent Markov property
Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded
\begin{align*}
\Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty)
\end{align*}
Then my question is:...
- 159
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33
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Condtions for a stochastic process to be locally non-factorizable
Given a stochastic process $X=(X_t)_{t\in I}$ on $\mathbb{R}^d$ with continuous sample paths supported on a prob. space $(\Omega, \mathscr{F}, \mathbb{P})$ and such that each pair $(X_s, X_t)$, with $(...
- 463
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55
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Stochastic Analysis: proof using integral approximation
Let $t_1<t_2 \in [0,T]$, $f(t)\ge0$ and $Z_t$ an increasing process with $Z_0= 0$.
We have clearly that
\begin{equation}
\int_{[0,t_2[}f(s)dZ_s \ge \chi_{t_1<T}(Z_{t_2} - Z_{t_1})\min_{t_1\le s \...
- 23
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185
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Ito's Lemma (CVF) on product of Poisson processes
I have the following stochastic differential equation:
$da(t)=\{r(t)a(t)+w(t)−pc(t)\}dt+βa(t)dq(t)$,
with $q(t)$ a Poisson process with arrival rate $λ$ and its increment $dq(t)$ is denoted by:
$dq(t)...
- 11
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80
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Large deviations estimate for arbitrary continuous function
Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...
- 5,405
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237
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On the level of measure theory, what does it mean for a drift to be deterministic?
Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...
- 11
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73
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conditional expected value and in Stochastic differential equations
Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
- 159
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59
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Representation of optimal controls as diffusions
In reading this post I couldn't help but wonder the following question:
Let $\sigma>0$ and suppose, as in the motivational post, we are given a stochastic optimal control problem:
$$
\begin{...
- 5,405
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97
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Applications of Kazamaki Conditions
I'm interested in applications of this theorem by Sekiguchi Kazamaki:
"Continuous Exponential Martingales and BMO" - Theorem 1.12:
Let $M$ be a continuous local martingale and $Z(M):= \exp(M-\frac{1}{...
- 21
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265
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Wiener isometry for semimartingales
Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying
$$
\mathbb{E}\left[
\int_0^{\...
- 5,405
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62
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Distances between up and down crosses in Gaussian Processes
Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...
- 91
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43
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Reference for tensor multiplication and derivatives from a computational / concrete standpoint
I am looking for a reference for some fairly elementary definitions and calculations about "tensor-valued" functions, i.e. functions of the form $A : \mathbb R^d \to \mathbb R^{d^{n\times}}$.
For ...
- 677
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56
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About a class of expectations
Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
- 2,246
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59
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Matching Stochastic Flows
Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the ...
- 19
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59
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Existence and uniqueness of the asymptotic distribution of $x(k+1) = Ax(k) + v(k)$
Consider the linear discrete-time stochastic systems:
\begin{equation}
x_{k+1} = Ax_k + v_k,
\end{equation}
with time-instants $k \in \mathbb{N}$, state $x_k \in \mathbb{R}^n$, stochastic process $v_k ...
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26
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Singular direction of a particle system
Consider a system of n-sdes in $\mathbb{R}$ ( the formula is not important).
The corresponding particle system $X(t)=(X_{1}(t),X_{2}(t),...,X_{n}(t))$ lives in $\mathbb{R}^{n}$ and assume that ...
- 165
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0
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371
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Locally Lipschitz sufficiently implies a Gronwall inequality?
In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...
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141
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Meaning of $. \wedge t$ (. \wedge t) in stochastic analysis
In Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I they define (on page 4) a metric :
$${\bf d}_\infty ((t,\omega),(t',\omega')) := |t-t'| + \|\omega_{.\wedge t}-{\omega'...
- 13
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134
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Heat equation, free boundary and dynamic programming
I have a dynamic programming problem with an underlying diffusion $$ d X_t = \mu \, dt + d b_t$$
where $b_t$ is a standard brownian motion.
The HJB equation for the value function $v(x,t)$ I get is ...
- 553
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74
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Has this type of pathwise (S)DE been studied before?
I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before.
Let $(G,\ast)$ be an abelian $C^1$ Lie group....
- 708
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95
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Prove that a local martingale with spatial parameter is differentiable
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...
- 167
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0
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61
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Predictability of jumping times of increasing cad lag processes
The following is a remark that appears just at the beginning of the proof of Proposition 16.23 of the book Stochastic Processes of Richard Bass at page 121. I have not been able to prove yet that ...
- 69
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134
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Operator-valued stochastic integral and quadratic variation for operator-valued processes
Let $U$ be a separable $\mathbb R$-Hilbert space and $W$ be a $Q$-Wiener process on a complete and right-continuous filtered probability space. Let $H$ be a separable $\mathbb R$-Hilbert space and $X$ ...
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79
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Change of variables Levy process
Let $L$ be a Lévy process and define $M_t:=L_t-t\mathbb E(L(1)),$ then $M$ is a centred martingale.
Now consider the stochastic integral for $f$ a continuous process
$$\int_0^t f(t-s) \ dM_s,$$
is ...
- 83
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0
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88
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Questions about generalized Polynomial Chaos, book by Dongbin Xiu
I have some questions about Chapter 5 from the book Numerical Methods for Stochastic Computations, by Dongbin Xiu.
Theorem 5.7: Let $Y$ be a random variable and $\mathbb{E}[Y^2]<\infty$. Let $Z$ ...
- 31
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0
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63
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Martingale covariation operator in infinite-dimensions
Let
$(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space
$U,H$ be separable $\mathbb R$-Hilbert spaces
$(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$...
- 167
1
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0
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81
views
How can we show that the tensor-quadratic variation has locally bounded variation?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$
$U,H$ be infinite-dimensional ...
- 167
1
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0
answers
235
views
Associative law of the stochastic integral in Hilbert spaces
Let
$(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
$T>0$
$I:=(0,T]$
$(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$
...
- 167
1
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0
answers
84
views
Particle density in phase space normalization under proliferation
Consider $1,..,N$ indistinguishable particles in $\mathbb{R}^2$ and let them evolve according to a brownian motion and proliferation. Let $u: \mathbb{R}_+ \times \mathbb{R}^2 \rightarrow \mathbb{R}_0^+...
1
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0
answers
302
views
Unique EMM & completeness in the Black-Scholes model
Consider the Black-Scholes model
$$ dS(t) = \mu(t) S(t) dt + \sigma(t) S(t) dW^{\mathbb{P}}(t) $$
$$ dB(t) = r(t) B(t) dt$$
Steele shows now in "Stochastic Calculus & Financial Applications" (Ch. ...
- 203
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0
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106
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Domain of a reflected stochastic differential equation
I am currently investigating the domain of the infinitesimal generator of a reflected stochastic differential equation (for a smooth and bounded domain) with Lipschitz coefficients. Namely SDEs of the ...
1
vote
0
answers
100
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Convergence and boundedness in $L^\infty([0,T]\times \Omega)$ of Karhunen-Loeve expansion
Let $X:[0,T]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process in $L^2([0,T]\times\Omega)$. Consider the Karhunen-Loeve expansion of $X$:
$$ X(t,\omega)=\mu_X(t)+\sum_{n=1}^\infty \sqrt{\nu_n}\...
- 141
1
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0
answers
90
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Onsager-Machlup Function of a Killed Diffusion Process
Given a diffusion process $ X_t $ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$ \mathcal{L}(x,v) =...
- 213
1
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0
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340
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Construction of the quadratic variation for Hilbert space valued local martingales
Let
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a ...
- 167