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5 votes
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markov property for stochastic differential equation

Here, we assume that \begin{align*} g(t, x) = \mathbb{E}\left(h(X_T) \mid X_t = x \right). \end{align*} Note that, by Shiryaev, $g(t, x)$ is a Borel measurable function such that, for any Borel ...
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4 votes
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Scaling of a transition matrix

You are right, the rules to time-scale a T-years transition matrix $M_T$ are: $M_{k·T} = M_T^k$ $M_{T/k} = \sqrt[k]{M_T}$ The root of a matrix M can be obtained using the spectral decomposition: $M ...
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4 votes
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Pricing a piece of asset whose dividend stream following a Markovian matrix

Your equations are for cum-dividend prices, i.e. the price plus dividend today. The paper refers to ex-dividend prices. The correct two equations for investor group $a=1$ are \begin{align} p^1(0) =&...
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4 votes

Probability density function of simple equation, compound Poisson noise

I don't think you can have an explicit form. Let $Y_t= e^{at}X_t$ then : $$ Y_t -Y_0 =\sum_{i=1}^{N_t}e^{aT_i} $$ where $(T_i)_{i=1...N_t}$ are the jump times of your poisson process. then $$P(Y_t\...
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4 votes
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Regime Switching for Dynamic Correlations

The clearest and most intuitive article I have seen so far is Kritzman et al., Regime Shifts: Implications for Dynamic Strategies in FAJ (May / June 2012) It not only shows how you can use HMM for ...
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3 votes
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When predicting Forex price using HMM what, typically, are the states and what are the observations?

Your decision. You can define the states to be "positive return" or "negative return". So if the return is negative, then its state would be that "negative return". Or, you could even model them ...
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3 votes

Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$

This is a corollary of Feynman-Kac theorem. For self-containedness, I re-produce the proof as follows. Assume that there exists a $C^{1,2}$-function $F=F(t,x)$ defined on $[0,T]\times\mathbb{R}$ that ...
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3 votes
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backward Kolmogorov equations - Markov properties

The initial condition for the backward Kolmogorov PDE is that $$ u(0,x) = g(x) $$ for all $x$ in the relevant domain and not just at a particular point. So if your functions $f$ and $g$ agree only at ...
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2 votes
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Why Markov Functional Models (Hunt 2000) are not yet so popular?

it's difficult to say that they are not popular. Some people definitely use them for live pricing. I'd say the real question is "why are they not popular in the academic literature"? One answer would ...
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2 votes
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Non-recombining lattice in non-markovian models

I think I might have found the solution to my own question. The Markov property as stated above has no direct relation with the recombination of the approximating lattice. However, if we consider the &...
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2 votes
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Markov switching regime for stock returns

Two things to note: First you are assuming that stock returns follow some type of AR(1) which I do not think is a reasonable model; Casting consideration (1) aside, you can estimate what you want by ...
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1 vote
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Observed rating migration matrix to derive the generator matrix

In case an answer is still useful for you after 11 months -> No, a generator matrix can be directly derived from observed rating transition data from which a transition probability matrix can be ...
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1 vote

How can I 'quantize' a time-series in 'groups' exhibiting similar patterns?

Regime detection with hidden Markov model: http://scikit-learn.sourceforge.net/stable/modules/hmm.html
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  • 194
1 vote
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Why does the Weak Form of Market Efficiency and Markov Property hold?

The weak form of the efficient market hypothesis (EMH) just says that the market is efficient to all prior information contained within price. By definition, the weak form of EMH obeys the Markov ...
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1 vote
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Solving a backwards heat equation using stochastic calculus

Based on the form of your equation, we can consider the SDE \begin{align*} dX_t = \sigma dW_t, \end{align*} where $W$ is a standard Brownian motion. Since, for $0 \leq t \leq T$, \begin{align*} X_T =...
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1 vote

Why Markov Functional Models (Hunt 2000) are not yet so popular?

In context of Bermudan Options, I believe that since the model determines everything exogenously, calibrating to swaptions may give you cases where the implied forward rate is negatively correlated to ...
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