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      Reference request in optimal stopping [closed]

      I am given the following task. Distributed over a trading day, I am supposed to buy a certain quantity of a good. The price of this good changes during the day. The goal is to buy the required ...
      1
      vote
      0answers
      62 views

      Non-diagonalizable matrix in a discretized Ornstein-Uhlenbeck process

      I am attempting to implement a pairs trading algorithm for two securities by approximating a discretized version of the Ornstein-Uhlenbeck process: \begin{equation*} d\mathbf{S}_t = \mathbf{\kappa}(\...
      1
      vote
      0answers
      124 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. ...
      3
      votes
      2answers
      192 views

      Large deviation bound for O-U process

      Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of $$ d X_t = -\alpha X_t dt + \sigma dB_t $$ Is there an exponential bound (large-deviation bound) for $$ P\left( \max_{t\le T} |X_t| \ge z \...
      1
      vote
      0answers
      65 views

      Extending risk neutral measure to insurance/mortality filtration

      In insurance mathematics, one often models the underlying of an insurance policy with a Black Scholes model on a filtered probability space $(\Omega,\mathbb{Q},\mathcal{F},\mathbb{F}=(\mathcal{F}_{t}))...
      1
      vote
      1answer
      284 views

      Is it safe to work on a Cadlag modification of a Feller process?

      Let $f$ be a continuous bounded function. $X$ is a Feller process, and $\hat X$ is its Cadlag modification. By the definition of the modification, one can write $$\mathbb E[f(X_t)] = \mathbb E[f(\hat ...
      1
      vote
      2answers
      112 views

      Is zero a regular point for a drifted $\alpha$-stable process?

      We consider 1-d process of the form $Y_{t} = bt + M_{t}^{\alpha}$, where $M_{t}^{\alpha}$ is $\alpha$-stable process for some $\alpha \in (0,2)$ with its levy symbol $\eta(u) = - |u|^{\alpha}.$, and $...
      2
      votes
      0answers
      191 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 ...
      4
      votes
      0answers
      157 views

      compactness of a probability set

      I have a question about the compactness of a set of martingale measures. Let $\Omega=\mathcal{C}[0,1]$ be the space of continuous functions on $[0,1]$ and $\mathcal{M}_{\Omega}$ be the family of ...
      5
      votes
      3answers
      1k views

      One can earn nothing on the Brownian motion, true ?

      Consider any discrete time stochastic process $p(n)$ (price) with independent increments $\xi_k$ and $E(\xi_k)=0$. E.g. Brownian motion (i.e. $\xi_k = N(0,1)$). Consider some "trading strategy" ...
      0
      votes
      1answer
      442 views

      Mathematical properties of financial prices

      Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes. What is known about their mathematical properties ? I know ...
      4
      votes
      1answer
      351 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: $$\bar{s}^2_T+4\sum_{k=0}^{T-1}\bar{s_k}...
      1
      vote
      1answer
      574 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 ...
      1
      vote
      2answers
      215 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 ...
      1
      vote
      0answers
      127 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 ...

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