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1
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0answers
77 views

Supermartingale inequality on a particular event

Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a ...
2
votes
0answers
103 views

Lyapunov function of exponential growth for existence of a solution of an SDE

Let $$dX_t = a(X_t) dt + b(X_t) dW_t$$ be a one-dimensional stochastic differential equation, where the coefficients $a,b: \mathbb{R} \rightarrow \mathbb{R}$ satisfy for every ball $B_R$ the following ...
4
votes
0answers
155 views

Malliavin calculus w.r.t $G$-Brownian motion

I wonder if it is possible to define a Malliavin calculus w.r.t $G$-Brownian motion defined on a Sublinear Expectation Space available on this link. G–Brownian motion has a very rich and interesting ...
2
votes
1answer
239 views

Solving a SDE with quadratic drift

I am wondering whether the following SDE can be solved explicitly? $$ d X_t = X_t^2 d t + X_t d B_t $$ where $B_t$ is a standard Brownian motion. If not, can we say some thing about the moments of ...
2
votes
0answers
96 views

Existence of predictable quadratic covariation for a special pair of local martingales

In Limit theorems for stochastic processes, by Jacod and Shiryaev we have the existence of a predictable quadratic covariation process stated as the following theorem $\mathbf{Theorem}$ To each ...
0
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1answer
135 views

Colored noise in SDE

I want to numerically study the behavior of a system described by a set of differential equations in the presence of colored noise. It seems that the standard procedure is to use the Langevin ...
2
votes
1answer
180 views

Conditional law of an Ito's stochastic integral

Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real ...
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0answers
45 views

Is it possible to define a mixed normal having conditional variance almost everywhere null?

I'm trying to proving the stable limit of a martingale M_n(t). When I calculate the limit in probability of its quadratic variation, I find that it is always null except for a point. It seems to me ...
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0answers
25 views

Tail for the integrale of a diffusion process

hello, I would like to compute the following tail $ \mathbb{P}\left( \left[ \int_{0}^{T} f(X_t)dt \right] >x\right) $ if $ \mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x) $ X is a Diffusion ...
5
votes
0answers
360 views

When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb ...
1
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0answers
238 views

How is Kolmogorov forward equation derived from the theory of semigroup of operators?

In Lamperti's Stochastic Processes, given a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$, a ...
1
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0answers
226 views

Iterated Ito Integral, Gaussian Volterra Process

Let me define $$ J^f_{n}(t) = \, \int_0^t \int_0^{t_1} \ldots \int_0^{t_{n-1}} f(t, t_1, \ldots, t_n) \; dB_{t_n} ...dB_{t_1} $$ where $f:[0,1]^{n+1} \to \mathbb{R}$ is a nice deterministic ...
0
votes
1answer
236 views

Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...
1
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0answers
92 views

Time integral of a diffusion

Define $\bar\sigma^2_t=\frac{1}{t}\int_0^t\sigma^2(X_s)ds$ where $\sigma(x)\geq0$ is a measurable function and $X_t$ a diffusion process defined by \begin{equation} ...
1
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0answers
80 views

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 ...
1
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0answers
148 views

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 ...
0
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2answers
578 views

When are two operators simultaneously diagonalisable?

I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...
0
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0answers
116 views

Application of Stochastic Calculus

As you know, one of the application of stochastic integral is to find a solution for a partial differential equation(PDE) with boundary condition. I can refer to the well known formula called ...
2
votes
3answers
265 views

Ito formulae for stochastic processes with finite cubic, quartic … n-tic variation

Many stochastic processes that you encounter are kind of well-behaved, i.e. have infinite variation, yet finite quadratic variation. My question revolves around stochastic processes that have ...
1
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1answer
347 views

Good books on stochastic partial differential equations?

I have a system of 2 PDEs, one with a probabilistic right side, and kind of stuck on what to read about those things.. Any good books around? Both analytical (if any) and numerical methods are ...
2
votes
1answer
131 views

Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...
1
vote
1answer
219 views

On martingale representation theorem

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(\mathcal{F_{t}})_{0\le t\le T}$ a filtration generated by standard Brownian motion $W_t$. Let $f(x)$ be $C^1$ function such that $|f'(x)| ...
4
votes
1answer
342 views

Converse to Girsanov's theorem?

Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...
2
votes
1answer
169 views

Limit of a Wiener integral

How to show that $$ \lim_{\alpha \rightarrow \infty} \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.}$$ where $\left (B_s ...
0
votes
1answer
85 views

Concerning Jump process (Lévy process)

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$ \nu \left( dx\right) = A ...
0
votes
1answer
178 views

Compute the expected value of the product between a Lebesgue–Stieltjes type integral and an Ito integral

Hi, I have the following expected value to compute $E[ \int_{o}^{T} f(t) dt \int_{o}^{T} H(s) dW(s)]$, where $f(t)$ and $H(s)$ are two stochastic processes adapted to the filtration generated by the ...
1
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0answers
51 views

Maximal Principle for stochastic heat equation

Consider $\partial_{t}u=\partial_{xx}u$ with Neumannboundary conditon $u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$. Then up to time $T$, the maximal value of $u$ should be ...
2
votes
0answers
134 views

Cameron-Martin like RKHS

Hello, I know that $k(x,y)=min(x,y)$ is the reproducing kernel of the Cameron Martin space of all i.i.d. RVs of Brownian motion at different times, with the $cov$ inner product. What is the RKHS ...
1
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1answer
119 views

Can we express a one-dimensional raised Bessel Bridge as a function of a single Brownian Motion?

A Bessel Bridge is a Brownian Motion, conditioned such that $B(0) = B(1) = 0$ and $B([0, 1]) \ge 0$. A raised Bessel Bridge is a generalization of this: it's a Brownian Motion conditioned such that ...
3
votes
1answer
136 views

log-likelihood of ito diffusion

Consider a diffusion process: $ \text{d}X_t = f(X_t)\text{d}t + \text{d}W_t$ I've seen it given that the log-likelihood of the path is proportional to the Onsager-Machlup functional $ \int_0^T ...
4
votes
1answer
270 views

Trajectorial version of Doob's $L^2$ inequality

In the paper http://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0154.pdf you can find a trajectorial version of Doob's inequality. It is given by: ...
0
votes
2answers
167 views

Properties of the Euler Discretization of a diffusion

Let $X$ be a continuous 1-d diffusion: $$ dX_t = a(X_t)dt + b(X_t)dW_t, X_0 = x. $$ W is a standard Brownian Motion and $a(\cdot)$ and $b(\cdot)$ can have nice regularity properties. Let ...
0
votes
1answer
75 views

Parameter Sensitivity of Stochastic Process

How do I compute the derivative \frac{\partial X_t}{\partial \sigma}? Where dX_t=\theta (\mu-X_t)dt+\sigma \sqrt{X_t}dZ_t
2
votes
0answers
119 views

Computing a density function for the integral of a stochastic process, given its transition function

$P$ is a one-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | ...
4
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0answers
126 views

Integrating a Bessel Bridge

Preliminaries An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. ...
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1answer
686 views

Bochner integral of stochastic process = path by path Lebesgue integral?

After some helpful comments, I realized that I had to repost this question in a more systematic way. On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
2
votes
1answer
136 views

contraction property for conditioned SDEs

Consider a strictly convex potential $U: \mathbb{R}^d \to \mathbb{R}$ and the Langevin diffusion $$dX = -\nabla U(X) dt + dW \qquad (*)$$ where $W$ is a standard Brownian motion. If $(X_t)_{t \geq 0}$ ...
1
vote
1answer
303 views

Solving an Ornstein-Uhlenbeck-like SDE $y(t,T)=H_t + \mathbb{E}[\int_t^T y(s-,T)dX_s|\mathcal{F}_t]$

I have asked a similar question involving some finance background some time ago here math.stackexchange, however no really good answer came up. I was able to find a solution at least for a special ...
0
votes
2answers
295 views

Representation theorem for continuous uniformly integrable martingales

For some time $u$ and positive continuous process $a_t$ adapted to $\mathcal{F}_t$ I have a (continuous-time) martingale defined as: $$M_t(u) = \mathbb{E}[a_u | \mathcal{F}_t]$$ for $t\leq u$. I ...
1
vote
1answer
201 views

SDE-removal of the diffusion coefficients

from math.stackexchange I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_{loc}$. If I have \begin{align} dX_t=b(X_t)dt+\sigma dW_t, \end{align} ...
1
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0answers
71 views

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 ...
1
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2answers
175 views

market completion in stochastic volatility model

Hi all, Consider a stochastic volatility model. As there are two sources of risk and one asset only, this is an imcomplete market. One can complete the market by considering a derivative V1 used to ...
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0answers
98 views

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 ...
2
votes
1answer
275 views

A wrong proof of Squared Bessel process

The squared Bessel process with $\delta$-dimension for $\delta>0$, denoted by $BESQ^\delta(y)$, is given by $$d Y_t = \delta t + 2 \sqrt{Y_t} d B_t, \ Y_0 = y\ge 0$$ where $B_t$ is BM under ...
2
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0answers
175 views

Is this process strictly positive?

Let $W_t$ is standard Brownian motion under probability measure $P$. Consider 1-D stochastic differential equation $$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$ We assume $\sigma(0) = 0$, and ...
2
votes
0answers
93 views

Does this series stopping times marching forward?

Let $W_t$ is standard Brownian motion under probability measure $P$. Consider stochastic differential equation $$ dY_t = dt + Y_t dW_t, \ Y_0 = 0.$$ Note that, the above SDE has a strong non-negative ...
0
votes
1answer
205 views

Stochastic processes with random matrices

I am currently working on complex networks. I consider a matrix $\cal N$ with random entries $\delta_{ik}$. These entries are varying randomly in time and so I have a sequence of random matrices that ...
0
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1answer
153 views

References/Papers on analytic solutions to SDEs

Does anyone know of any good references/research papers on finding analytic solutions to stochastic differential equations and/or finding approximating solutions to such a system? I am particularly ...
10
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0answers
574 views

surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
2
votes
1answer
109 views

change the sign of volatility

Assume the time inhomogeneous SDE $dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$ has a solution $X(t)$. If we replace $\sigma$ with its absolute value, does the new SDE ...