# Tagged Questions

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.

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### Continuity of expected payoff from a diffusion

Fix a discount rate $r>0$, and let $m,v,f:\mathbb{R} \rightarrow \mathbb{R}$ be bounded measurable functions of locally bounded variation, with $v$ globally bounded below by some strictly positive ...
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### Time Change of a Brownian motion

We know that for if $X$ is a stochastic integral of the form below - $X_t = \int_0^t v(s,\omega) db(s,\omega)$. then we can use time change formula to claim that $X_t = W_{\alpha(t)}$ where $W$ is ...
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### Strong law of large number for semimartingale

I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion. Thanks
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### Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
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### 'Nonclassical' abstract Wiener space

Is it possible to construct an abstract Wiener space $(W,H,\mu)$ such that $C^{0,\frac{1}{2}}(\Omega)\subset H$ and $W$ is a normed function space such that the convergence in norm implies convergence ...
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### integrability of Brownian motion stopped at some stopping time

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion starting at zero and denote by $S=(S_t)_{t\ge 0}$ its running maximum, i.e. $S_t=\sup_{0\le s\le t}B_s$. Given a fixed number $p>1$, define the ...
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### About the boundary conditions of the Black-Scholes-Merton PDE [closed]

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
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### Systems of stochastic differential equations with non-Lipschitz coefficients

I am looking for references to any literature which might consider the existence / behavior / regularity of solutions to systems of stochastic differential equations with non-Lipschitz coefficients. ...
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### Existence of martingales given some constraint on laws

Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple ...
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### The existence of stationary measures for certain Markov process

My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...
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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 ...
I have Ito integral $X=\int_0^T f(t) dW(t)$ and I would like to proof that $P(X>K)>0$ for all $K$ provided $f(t) > \epsilon > 0$. My idea was $\int_0^T f(t) dW(t) \sim \int_0^T \epsilon dW(... 0answers 105 views ### Asymptotics of Variable Drift Ornstein–Uhlenbeck Process The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:$dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t$where$\theta>0$,$\mu$and$\sigma>0$are ... 0answers 61 views ### Deriving HJB equation (why$\frac{dZ_t}{dt}=0$?) I am trying to derive the HJB equation in a stochastic setting. Let me exemplify my problem with the simplest case where there is no control, just one state variable. Assume the payoff is given by $$... 0answers 54 views ### Strong Markov Property of the joint process (B_t,L_t)_{t\ge 0} Let B=(B_t)_{t\ge 0} be a Brownian motion and L=(L_t)_{t\ge 0} be its local time in zero. Given two strictly increasing functions \phi_1, \phi_2: \mathbb R_+\to\mathbb R such that \phi_1(0)=\... 0answers 177 views ### Joint law of a standard Brownian motion and its local time at a nonzero level Let B_t be the standard Brownian motion and L_t^a be the local time at level a. It is known that the joint-density of (L_t^0,B_t) is$$ P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\... 0answers 53 views ### Recursive parameter estimation for partially observed Ito SDEs I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ... 1answer 190 views ### Stochastic differential equation associated with an optimal control problem We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process$X_t$is controlled up until it is stopped at a stopping time$\...
Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in D$....