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.
376 questions with no upvoted or accepted answers
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Ito's formula for jump diffusions
Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$...
- 11
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79
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Stochastic Control with Stochastic Cost-functional
Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also?
That is, let $X_t^u$ is the solution to a controlled SDE
$$
dX_t = \mu(t,u_t,X_t^u)dt ...
- 5,405
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0
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159
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Construction of the quadratic variation process in infinite dimensions
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
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331
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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 processes.
- 111
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159
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Brownian motions under different filtrations, quadratic covariation, convergence of Riemann sum
I'm interested in knowing if there are any results of the following type:
Let $(B_t^i,\mathcal F_t^i)$, $i=1,2$ be two Brownian motions. Suppose the filtration $\{\mathcal F_t^1\}$ is contained in ...
- 325
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infinite dimensional funtional ito calculus
I've been reading into functional Ito calculus and everything I've come across deals with processes generated by finite dimensional semimartingales. In Dupire's 2009 landmark paper he speaks about ...
- 5,405
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Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$
Let
$U$ and $V$ be separable $\mathbb R$-Hilbert spaces
$\iota:U\to V$ be a Hilbert-Schmidt embedding
$Q:=\iota\iota^\ast$
$(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$
$(\Omega,\mathcal A,\...
- 167
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Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE
In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories.
More concretely, I want to obtain a SDE of type as ...
- 167
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86
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Brownian motion in perturbed (asymptotically flat) metric
Let $g_{\mathbb{R}^n}$ denote the usual Euclidean metric on $\mathbb{R}^n$ and let $B_g(t)$ denote the Brownian motion associated to a complete metric $g$ on $\mathbb{R}^n$. Consider a Brownian motion ...
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System of stochastic equations
I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$
$$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
- 31
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A problem on Markov chains and Dirichlet forms
Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy
$$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$
$$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$
$$c(x,x)=0\text{ for ...
- 369
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Limit (convergence) of stopping times
Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
- 1,835
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Are the elementary predictable processes dense in $L^2([M])$ for $M$ a local martingale?
The question is the one from the title. I know this is true when $M$ is an $L^2$ bounded martingale (which is often used in the classical approach to the construction of the stochastic integral) but I'...
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Does the martingale property holds after changing filtration?
Let $\Omega$ be the space of continuous real-valued functions $\omega=(\omega_t)_{t\ge 0}$ starting at zero, i.e. $\omega_0=0$. Let $\Lambda=\Omega\times \mathbb R_+$ and denote by $\lambda=(\omega,\...
- 1,835
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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
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Expected value of product of Ito integrals
Assume that we have a process $F(t,T)$ that fulfills the following SDE.
$$
dF(t,T) = \sigma(t,T)F(t,T)dW(t)
$$
where $t$ is the running time and $T>t$ is called the delivery-time. $\sigma(t,T)$ is ...
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Definition of Ito Integral
In Karatzas and Shreve, the integral for Bounded Progressively measurable processes is defined first. Then, for Bounded measurable and adapted processes ($f(t,\omega)$), the authors say that there ...
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251
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Inflated independent samples for Monte Carlo estimation
In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
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question related to Tanaka Formulae
Supposse $X=(X_t)$ is a cadlag martingale taking values in $\mathbb{R}$. If $f:\mathbb{R}\to\mathbb{R}$ is a convex function, then we have Tanaka Formulae. Now let $g: \mathbb{R}_+\times\mathbb{R}\to\...
- 1,835
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101
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a question about the modification of a supermartingale
Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process
$$X_{t}(\...
- 1,835
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218
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question about Doob-Meyer decomposition
Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition:
$$V_t=V_0+\int_0^...
- 1,835
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168
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Attractors and solutions to these generalized Ornstein–Uhlenbeck processes?
This is a question about generalized Ornstein–Uhlenbeck processes I asked on MSE, but I haven't received replies about their attractors and solutions yet. So I would appreciate if someone could give ...
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What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?
There is a theorem as follows:
Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto \...
- 6,287
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86
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Maximal principle for stochastic heat equation
Consider $\partial_{t}u=\partial_{xx}u$ with Neumann boundary condition
$u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$.
Then up to time $T$, the maximal value of $u$ should be ...
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93
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Potentials of class D
A potential $\pi_t$ is a positive supermartingale with the condition that $\mathbb{E}[\pi_t]\rightarrow 0$ as $t \rightarrow 0$. What are the necessary/sufficient conditions for a potential to be of ...
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132
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stochastic volatility valuation equation
I'm trying to derive the valuation equation under a general stochastic volatility model. What one can read in the litterature is the following reasonning:
One consider a replicating self-financing ...
- 21
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466
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Tanaka stochastic differential equation and Kolmogorov equation
Given Tanaka sde
$$dX_t=[a{\rm sign}(X_t)+b]dW_t$$
is there associated a diffusion process and so a Kolmogorov (Fokker-Planck) equation? What is this equation?
References answering the question are ...
- 1,687
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Inverse Skorokhod Embedding Problem
The Skorokhod Embedding Problem is well known and has many documented solutions in the literature.
Now if we are given a Brownian stochastic basis (satisfying usual hypothesis), a diffusion $X_t$ (...
- 1,334
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283
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Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
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Integration of independent Brownian motions
I am wondering if the following integral of stochastic Brownian motions has an analytical solution?
$$
\int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau}
$$
where $\tilde{...
- 11
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0
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42
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Bound on the radon-nikodym derivative between two stochastic processes at a time point
I have two stochastic differential equations on $\mathbb{R}^d$ adapted to the same filtration evolving for finite time $t\in [0, T]$ at the same start distribution:
\begin{align*}
dX_t &= b(t, X_t)...
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When we should integrate on both side over a SDE?
Maybe I am quite stupid, I am quite confused about, when we should use ito formula to solve SDE and when it is appropriate to integrate directly to get the solution?
Specifically, let us consider the ...
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Interpretation of Lévy process with signed Lévy measures
Suppose that I have a non-decreasing, pure jump Lévy process of finite variation $X$ with Lévy measure $\pi$. The Lévy measure is then supported on $(0,+\infty)$. Suppose that the Lévy measure is a ...
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101
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Simulation of Markov processes with exponential timestepping
Let $(Y_t)_{t\ge0}$ be a time-homogeneous Markov process with transition semigroup $(\kappa_t)_{t\ge0}$. Numerical simulation of $(Y_t)_{t\ge0}$ can be done in the following way:
Choose an initial ...
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Can the optimal stopping problem be expressed in another form by strong Markov property?
$X_t$ is a strong Markov process in $(\Omega, \mathcal{F},\mathcal{F}_t,\mathbb{P})$. $\tau$ is a stopping time, $T>0, \mathbb{E}_x(\cdot)=\mathbb{E}(\cdot|X_0=x)$. By Markov property, $\mathop{\rm{...
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White noise: a tempered distribution version of the stochastic convolution
Let $\xi$ be a space-time white noise, that is a centered Gaussian process with covariance $E[\xi_{f}\xi_h]=\int_{\mathbb{R}_+ \times \mathbb{R}^d}fh,$ for $f,h\in L^2(\mathbb{R}_+ \times \mathbb{R}^d)...
- 573
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Martingale defined by an integral
Consider a probability space $(\Omega,\mathcal{F},P).$ Let $f \in C^{\infty}_{c}(\mathbb{R}^d,\mathbb{R}),p \geq 2.$ $(X_r^{y})_{(r,y) \in \mathbb{R}_+ \times \mathbb{R}^d}$ is a stochastic process ...
- 573
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101
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Integration with respect to $B_H(t) B_H(s) - \mathbb{E} \{ B_H ( t ) \, B_H ( s) \}$
The time-derivative $\frac{dB_H}{dt}$ of the fractional Brownian motion may be interpreted as a random Schwartz distribution acting on a test function by
$$
\left\langle \frac{dB_H}{dt}, f \right\...
- 620
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122
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Laplace transform of a stochastic process
Let $R := (R_1, R_2)$ be a two-dimensional diffusion process defined by the following SDE:
$$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t} \, \mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \, \mathrm{d}W_t$...
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120
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Predictability of the mild solution of a SPDE
Consider the following theorem (picture below) taken from Pardoux's lecture notes: Stochastic partial differential equations available at scholar google: https://scholar.google.ca/scholar?q=etienne+...
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255
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Distribution of "occupation times" of Brownian Motion
Let $B_t(\omega)$ be a standard Brownian motion and let $A\in\mathcal{B}(\mathbb R)$ be a Borel set.
I would like to find the distribution of $$Y_A(\omega):=\lambda(\{t\in[0,1]:B_t(\omega)\in A\})=\...
- 245
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0
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75
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Regularity of solutions to forward-backward stochastic differential equations
Suppose $X_t$, $P_t$ and $Z_t$ are one dimension random processes and satisfy
$$
\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
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0
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152
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How is the definition of the stochastic integral w.r.t. an Ito process consistent?
I have the following definition for an
Ito process:
For $a(\omega, t), b(\omega, t)$ real valued, adapted stochastic processes
that respectively satisfy the conditions
$$
P(\int_0^t \vert a(\omega, s)...
- 101
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0
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466
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The relationship between measurability and weak measurability
For a Banach-valued random mapping $f:\Omega\rightarrow X$, there are three kind of measurability: strong measurability (can be approximated by sequence of simple
functions, measurability (the ...
- 113
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0
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131
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Does this book use non-mainstream stochastic analysis constructions and is thus perhaps not a good start?
I'm attempting to read a book on stochastic calculus by D.H. Fremlin, which is the 6th volume of his treatise on measure theory encompassing all kinds of topics related it.
Before I make a very ...
- 41
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0
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58
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The different quotient of the first exit time with respect to the initial state
Let $B_t$ be a 1-dimensional Brownian motion and $t \in [0,T]$. Suppose we have a diffusion process $X_t$ such that
$$ dX_t = u(X_t,t) dt + v(X_t,t) dt \hspace{10pt} \text{ and } \hspace{10pt} X_0= x \...
- 54
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0
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176
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A convergence question in $L^2$ construction of Brownian motion
I feel confused with a particular step in the $L^2$ consturction of Brownian motion.
Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...
- 227
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0
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236
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Canonical Markov process and abstract Markov process
I have the following question:
Why do some books work with the canonical Markov process instead of just the abstract one, as from my point of view they both share exactly the same properties in terms ...
0
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0
answers
121
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Martingale representation of a stopped Brownian motion
This question follows from the previous post Question on the martingale representation theorem which has not been answered. I consider thus a particular case. Let $(B_t)_{t\ge 0}$ be a standard ...
- 1,334
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0
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70
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Where does the "mixing" occur in convex combination of Girsanov measures?
In this post, Ofer says that taking the convex combination of two Girsanov measures yields a drift $BF_1+(1-B)F_2$ where $B$ is a Bernoulli random variable with parameter $\lambda$, independent of the ...
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