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Questions tagged [stochastic-differential-equations]

Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.

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119 views

Ito lemma for manifold semimartingales

I'm looking for a generalization of the usual Ito lemma to manifolds $M$, preferably not under the assumption that $M$ is embedded in $\mathbb{R}^d$. Unfortunately any reference I've found either ...
2
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1answer
93 views

Modified square root process

I am dealing with the following stochastic differential equation (SDE) $ \begin{cases} dS_t &= \mu S_t dt + \sigma_1 S_tdW^1_t\\dG_t &= kS_t(\alpha - G_t)dt + \sigma_2\sqrt{G_tS_t}dW^2_t \end{...
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0answers
65 views

stochastic differential equations via Hida distributions?

Can one prove the solvability of stochastic Ito differential equations via Hida distributions?
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1answer
113 views

Probability that SDE visits any point

This is a reference request question. Statement: I am interested in an SDE of the form \begin{equation}\fbox{1}~~~ {\rm d}X_t = f(X_t)\,{\rm d} t + g(t) \, {\rm d} B_t \end{equation} Where we ...
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0answers
24 views

Generalized polynomial chaos with 2D independent uniform variables for first order equation

I'm trying to transform the first order differential equation $\dot{x} = k x(t)$ into the corresponding set of stochastic differential equations. I have two independent uniformly distributed ...
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0answers
49 views

Matching Numbers in Ito McKean

Matching numbers are the basics Ito and McKean use to build out a bunch of stuff, like singular points and shunts. The four maching numbers $e_1, e_2, e_3, e_4$ are defined as $e_1 = \lim_{b \...
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1answer
231 views

Any modern/recent version of Ito & McKean?

This's a wonderful book[1] but the latest edition I have is dated 1973. Is there recent book(s)/rewrite(s) that covers the same subjects and elucidate with more explicit arguments and details of their ...
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0answers
35 views

first eigenvalue and relaxation time for ergodic diffusions

As a prototype example of ergodic diffusion, consider a real valued Ornstein Uhlenbeck process of the form $$ d X_t = -\alpha X_t d t + \sigma d W_t, \quad X_0 = x $$ where $\alpha, \sigma > 0$ ...
2
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1answer
218 views

Weak convergence of sum of log normal random variables

Let $S_t$ be the Geometric Brownian Motion, we know that $$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$ and the distribution of $S_t$ is known explicitly. Please see the ...
2
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1answer
649 views

Variance of Multi-Dimensional Ornstein Uhlenbeck process

I am trying to compute the asymptotic variance of OU process $$ d X_t = - H X_t dt + S dW_t $$ where $X_t$ takes value in $R^d$. $H$ and $S$ are $d\times d$ matrices that does not have $HS = SH$ in ...
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1answer
154 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/...
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1answer
87 views

inhomogeneous Ornstein-Ulhenbeck process / invariant probability measure

Let $\gamma$ be a continuous function from $\mathbb{R}^+$ to $\mathbb{R}^+$ and consider the real valued inhomogeneous Ornstein-Uhlenbeck process satisfying $$ d X_t = -\gamma_t X_t d t + d W_t, \...
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0answers
85 views

Gaussian Processes and Paths

As we know, the only stationary Gaussian Markov process with continuous autocorrelation function is the stationary OU process (Doob's theorem). To show this, one shows that the autocorrelation ...
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0answers
30 views

numerical scheme for SDE and empirical estimation of rate of convergence

Consider $\{X_t , t \geq 0 \}$ real valued diffusion satisfying $$ d X_t = b(X_t) d t + \sigma (X_t) d W_t, \quad X_0 = x \in \mathbb{R} $$ where $b, \sigma$ are well-behaved functions and $W$ is a ...
5
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2answers
310 views

Reference for Feynman-Kac

I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's ‎An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...
1
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1answer
111 views

Non-commutative Ito Formula

Does there exist a formula of the Ito lemma for matrix valued processes under matrix multiplication? That is where $$ \Delta X_{t+\Delta t} \neq X_{t+\Delta t} - X_t $$ but instead $$ \Delta X_t = ...
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2answers
104 views

A solution to stochastic PDE $du(t)= a(t)u(t)\,dt +s(t)\,dz$

Is there a general (integral) solution to $du(t)= -a(t)u(t)\,dt +\sigma(t)\,dz$? Is the following $u(t)=e^{-\int_{t_0}^{t} \alpha(s) \, ds}u(t_0)+\int_{t_0}^t \sigma(v) e^{-\int_v^t a(s) \, ds} \, dz(...
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0answers
48 views

Time discretization of the (stochastic) Navier-Stokes equation

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be nonnempty and open $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_{L^2(\Lambda,\:\mathbb R^d)}$ I've found a thesis where ...
2
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0answers
84 views

Markov chain approximates a fractional diffusion

Let assume that $$ dX_t=\mu(X_t)dt+\sigma(X_t)dW_t^H, X_0\in \mathbb{R} $$ Where $\mu(.), \sigma(.)$ satisfy some conditions that guarantee $X_t$ exists, and $dW_t^H$ is a fractional Brownian motion ...
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0answers
87 views

Monte-carlo estimation on the drift of SDE

On any probability space $(\Omega, \mathcal{F} , \mathbb{P})$ with a Brownian motion $W$, we consider the following one-dimensional SDE: $$dX_t = F(t, X_t) \, dt + dW_t,$$ where $F(t,x) = \mathbb{...
5
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1answer
267 views

Reference: Stochastic Analysis on Hilbert Manifolds

I'm looking for a reference to a book which develops an It\^{o} lemma for semi-martingales with values in infinite dimensional Hilbert-Manifolds. I expect the techniques to be the same but still I ...
3
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2answers
398 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) \...
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0answers
115 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(...
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2answers
159 views

Reference: Non-smooth Ito Lemma

Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth? I've been looking but have not found much, any ...
6
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0answers
81 views

Reference Request: Vector-Valued Ito Formula

I know that there exist Ito formulae to understand $ f(X), $ where $f: H\rightarrow \mathbb{R}$ is sufficiently nice, $H$ is a Hilbert space and $X$ is an $H$-valued semi-martingale. However I'm ...
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0answers
52 views

Parametric distribution where the parameter follows a diffusion process

I'm looking for a distribution $P_{\theta}$ with pdf $f (t,\theta)$ over $\mathbb{R}^{+}$ such that there exists functions $\mu(\theta)$ and $\sigma(\theta)$ such that for all $t>0$: $$\mu(\theta)\...
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0answers
163 views

Mean and Variance of SDE

What is the mean and the variance of $y_t$, given the following SDE: $dy_t = -x_t y_t dt + \sigma_1 dW^1_t$ $dx_t = -\sigma_2 y_t dW^2_t$ $W^1$ and $W^2$ are (possibly correlated) Wiener ...
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1answer
869 views

Mixing the Ornstein-Uhlenbeck Process and Geometric Brownian Motion

The Ornstein-Uhlenbeck process with mean reversion level 0 is defined as follows: $$dX_t=a X_t dt + \sigma_1 d W_{1t}. \tag{1} $$ Geometric Brownian motion is defined as follows: $$dX_t= a X_t dt + ...
5
votes
1answer
355 views

Definition of the nonlinear part of the drift in a (stochastic) Navier-Stokes equation

Let $T>0$ $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be bounded and open $\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\overline{\mathcal V}^{\left\|\;\cdot\;\...
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0answers
56 views

If $(Φ^x)_{x∈ℝ}$ is a family of real-valued stochastic processes and $B$ is a Brownian motion, then $\int_0^tΦ^x_s\:dB_s=(\int_0^t\Phi_s\:dB_s)(x)$

Let $T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$ $B$ be a (standard, real-valued) $\mathcal F$...
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0answers
136 views

Transforming reaction-diffusion equations to random walk processes

I have a two species reaction-diffusion system which is a Turing-type (activator-inhibitor) equation. I am trying to transform my reaction-diffusion system into a system of multiple walkers on a ...
2
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1answer
303 views

On representing a continuous time Markov chain by a stochastic integral of a Poisson random measure

Let $Q=(q_{ij})$ be the transition rate matrix of a continuous time Markov chain $\{ X_t \}$ with countable state space $M$. Let $q_i = -q_{ii}=\sum_{j \neq i}q_{ij}$, and let $\Gamma_{ij}$ be defined ...
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0answers
89 views

Embedding a martingale by SDE

Let me reformulate my question. Let $(X_0,X_T)$ be a martingale on $\mathbb R$, then it is known that one has a SDE: $$Z_t=Z_0+\int_0^t\sigma(s,Z_s)dB_s, \mbox{ for all } t\in [0,T]~~~~~~~~~~~~~~(\...
2
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0answers
88 views

I've found a representation of the Itō-Stratonovich correction term and don't understand the used notion of a “trace”

Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a ...
3
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0answers
51 views

Perscribed/Inverting Conditional Expectation

I'm having difficulty finding papers which deal with the following inversion problem. Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...
2
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0answers
182 views

Definition of the Stratonovich integral in Hilbert spaces

Let $T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$ $B$ be a (standard, real-...
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2answers
456 views

Deriving the HJB equation for exponential utility

I would like to derive the HJB equation for the following stochastic optimal control problem: $ \Phi(t,x)=\sup_{h} E \left[\exp \left\{\gamma \int_t^T g(X_s,h(s);\gamma)\ ds \right\} \right]$ where ...
1
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2answers
163 views

Conditional Wiener measure continuous

consider a complete Riemannian manifold $M$ with heat kernel $p_M$ and let $U\subset M$ be an open set. Let $W_{x,t}^{y}$ be the Wiener measure associated to the Brownian motion starting at $x$ and ...
2
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0answers
89 views

Well-posedness of a stochastic differential equation in the Stratonovich sense

Let $T>0$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$ $(B_t)_{t\in[0,\:T]}$ be an $(\mathcal F_t)_{...
2
votes
1answer
280 views

Reflecting Brownian motion and its transition probability density

I have a question about reflecting Brownian motion on an unbounded domain. Let us consider the reflecting Brownian motion $\{X_t\}_{t \ge 0}$ on the following domain $\bar{D}$ of $\mathbb{R}^2$: \...
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0answers
142 views

Conditional Expectation of solution to SDE

Suppose $g \in C^{1,2,2}$ and $X_t$ and $Y_t$ solve some SDEs $$ dY_t = a(t,X_t,Y_t)dt +b(t,X_t,Y_t)dW_t $$ and $$ dX_t = c(t,X_t)dt + d(t,X_t)d\tilde{W}_t $$, then Iio's lemma implies $$ g(t,X_t,...
4
votes
1answer
79 views

Time Integral over the (positive) Innovations of a Stochastic Process

Consider the Itô-Process $$X(t) = X_{0} + \int_{0}^{t}\mu(s,X(s))\,ds + \int_{0}^{t}\sigma(s,X(s))\,dW(s)$$ where you can safely assume that the drift $\mu$ and the volatility $\sigma$ satisfy the ...
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0answers
24 views

Usually trivial Excursion-type process

How Can i construct a stochastic process $X_t$ which has the property that: $X_t \in [0,1]$ for all $t \in [0,T]$ and $m(\{t \in [0,T] : X_t>0 \})\leq \delta$, for some pre-chosen $\delta \in [0,T]...
2
votes
1answer
88 views

Joint distribution of integrals of diffusion and driving noise

Consider a generic diffusion of the form $$dX_t=f(t,X_t)dt+dB_t,$$ where $f$ is some nice function and $B_t$ is a standard Brownian motion. The marginal distributions of the integrals $$I:=\int_0^...
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0answers
61 views

Stochastic Inverse

Let $X_t$ be a semi-martingale and $H_t$ be a predictable process and $g$ be a measurable bijective function with measurable inverse. Does there exist a function $f(h,x)$ satisfying $$ \int_0^Tf(H_t,...
3
votes
0answers
89 views

When we integrate with respect to a $Q$-Wiener process on $U$, why do we restrict integrands to be operators on $Q^{1/2}U$ (instead of $U$)?

When we integrate with respect to a $Q$-Wiener process $(W_t)_{t\ge 0}$ ($Q$ being a bounded, linear, nonnegative and self-adjoint operator on a separable $\mathbb R$-Hilbert space $U$ with finite ...
4
votes
1answer
246 views

Variance and expectation of timed-change squared Bessel process

Let $X_t$ be a squared Bessel process satisfying the SDE: $$ dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t $$ and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...
2
votes
0answers
43 views

Minimizer of a class of SDEs

Setup Let $\mathscr{H}$ be a separable Hilbert space, $\mathcal{X}\triangleq \langle \Omega,\mathscr{F},\mathscr{F}_t,\mathbb{P}\rangle$ be a stochastic base and $X_t$ be an $H$-valued stochastic ...
2
votes
1answer
210 views

Generalizing HJB equation for a terminal stopping time

The following is one version of the Hamilton–Jacobi–Bellman (HJB) equation: Suppose we have a Brownian motion $W$ and a counting process $N$ with a stochastic intensity $\lambda$ on a time interval $[...
2
votes
1answer
214 views

Solving a matrix ODE

Consider the linear matrix differential equation $\def\diag{\mathrm{diag}}$ \begin{align} U(0) &= I\\ \frac{\mathrm{d}U}{\mathrm{d}t}(t) &= U(t) \phantom{.} Q(t) & & \quad(1) \end{...