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by EdisonCruise
Yesterday, 12:35 pm
Forum: Numerical Methods Forum
Topic: How to solve this ODE?
Replies: 0
Views: 54

How to solve this ODE?

The following ODE is obtained from the Ornstein-Uhlenbeck process. I read it from the paper Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit. $$ \frac{\sigma^2}{2} \frac{d^2u(x)}{dx^2}+\mu(\theta-x) \frac{du(x)}{dx}=ru(x) $$ The paper does not provide the boundary conditions ...
by EdisonCruise
February 14th, 2020, 8:15 am
Forum: Technical Forum
Topic: Is Ito’s lemma applicable to a diffusion process with transition probability?
Replies: 2
Views: 910

Is Ito’s lemma applicable to a diffusion process with transition probability?

I want to model a continuous variable \(X_t\) by a stochastic process. With probability \(1-q(X_t)dt\) at an infinitesimal period \(dt\), it is a diffusion process. However, with probability \(q(X_t)dt\), \(X_t\) may jump to \(Y_t\). The probability density function of \(Y_t\) is \(p(Y_t)\). If I am...
by EdisonCruise
November 21st, 2019, 12:28 am
Forum: General Forum
Topic: How to make ito's forumula for jump-diffusion a martingale
Replies: 3
Views: 1498

Re: How to make ito's forumula for jump-diffusion a martingale

Thank you so much Alan. That's my also my understanding of this problem before reading page 26 of http://people.ucalgary.ca/~aswish/JumpProcesses.pdf, based on which after vanishing the \(dt\) term, there should be an additional \( -E [Z_{N_t}] \eta_t f'(X_t) \)  term in the \(f(t,x) \) PIDE.
by EdisonCruise
November 20th, 2019, 7:41 am
Forum: General Forum
Topic: What’s Ito’s lemma for Poisson process in this function?
Replies: 6
Views: 1796

Re: What’s Ito’s lemma for Poisson process in this function?

Thank you Alan. Maybe I confuse it with something. I have made a new thread for this problem.
by EdisonCruise
November 20th, 2019, 7:40 am
Forum: General Forum
Topic: How to make ito's forumula for jump-diffusion a martingale
Replies: 3
Views: 1498

How to make ito's forumula for jump-diffusion a martingale

\(dY_t=Z_{N_t} dN_t \) is a compound Poisson process with intensity \(\lambda\) and \( Z_{N_t} \) is a random variable for the jump size. \(dW_t \) is Brownian motion. The jump diffusion process \(X_t\) is defined as $$ dX_t=\nu_t dt+u_t dW_t + \eta_t dY_t$$ So the Ito's lemma for this jump diffusio...
by EdisonCruise
November 19th, 2019, 1:17 am
Forum: General Forum
Topic: What’s Ito’s lemma for Poisson process in this function?
Replies: 6
Views: 1796

Re: What’s Ito’s lemma for Poisson process in this function?

Thank you. I have corrected the typo. However, in \(dY_t=Z_{N_t} dN_t\), \(Z_{N_t}\) is a random variable, indicating the random jump size. I think it makes sense to include an expectation w.r.t.  \(Z_{N_t} \) in the drift term, as in page 26 of  http://people.ucalgary.ca/~aswish/JumpProcesses.pdf ...
by EdisonCruise
November 18th, 2019, 8:53 am
Forum: General Forum
Topic: What’s Ito’s lemma for Poisson process in this function?
Replies: 6
Views: 1796

Re: What’s Ito’s lemma for Poisson process in this function?

Thank you Alan. May I ask one more question? In page 645 of your given file, there is an Ito formula for the compensated compund Poisson process: $$f(Y_t)=f(0)+\int_0^t \!(f(Y_s)-f(Y_s-))(dN_s-\lambda ds)+\lambda \int_0^t \!(f(Y_s)-f(Y_s-))ds$$ where \(dY_t=Z_{N_t} dN_t \) is a compound Poisson proc...
by EdisonCruise
November 16th, 2019, 10:25 am
Forum: General Forum
Topic: What’s Ito’s lemma for Poisson process in this function?
Replies: 6
Views: 1796

What’s Ito’s lemma for Poisson process in this function?

If \(N_t\) is a Poisson process with intensity \(\lambda \),and \(dX_t=\delta dN_t\), \(q_t=-N_t\), then the Ito's lemma for function \(H(X_t)\) should be $$ dH(X_t)=[H(X_t+\delta)-H(X_t)]dN_t$$ For the function   \(H(X_t,q_t)\), why it is not something like this? $$ dH(X_t,q_t)=[H(X_t+\delta,q_t)-H...
by EdisonCruise
October 15th, 2019, 3:10 am
Forum: Technical Forum
Topic: Why is Bellman Equation solved by backwards?
Replies: 32
Views: 9234

Re: Why is Bellman Equation solved by backwards?

Thank you katastrofa. I think your explanation make sense.
by EdisonCruise
October 10th, 2019, 1:35 am
Forum: Technical Forum
Topic: Is there any stochastic control literature about a linear combination of variables?
Replies: 0
Views: 5070

Is there any stochastic control literature about a linear combination of variables?

Suppose I have a portfolio of N options and they are affected by M Brownian motions, where M<<N. I hope to maximize the expected return of this portfolio, which is a linear combination of each stock’s return. Then how to formulate the HJB equation? Is there any classical literature/material about th...
by EdisonCruise
July 24th, 2019, 2:33 am
Forum: Technical Forum
Topic: Why is Bellman Equation solved by backwards?
Replies: 32
Views: 9234

Re: Why is Bellman Equation solved by backwards?

Thank you Alan. I can understand the option pricing problem. The BS pde must be solved by backwards, only because the terminal condition, i.e., the option pay off, is well defined. The initial option price is unknown(or cannot be difined), so need to be solved. This is not because of the data is non...
by EdisonCruise
July 23rd, 2019, 1:13 am
Forum: Technical Forum
Topic: Why is Bellman Equation solved by backwards?
Replies: 32
Views: 9234

Re: Why is Bellman Equation solved by backwards?

Thank you all, but I what I cannot understand is the real reason that Bellman equation is ususally solved by backwards. Can any one give an exmaple in which both intitial and terminal conditions are well defined, but the Bellman equation can only be solved by backwards?
by EdisonCruise
July 22nd, 2019, 8:30 am
Forum: Technical Forum
Topic: Why is Bellman Equation solved by backwards?
Replies: 32
Views: 9234

Re: Why is Bellman Equation solved by backwards?

Thank you katstrofa. I aslo think the Bellman equation can be solve by forwards with well defined and NON-stationary data. But that's in contrast to Nicole Bäuerle and Ulrich Rieder' book. I cannot understand why the stationarity of data is related to forward/backward solution.
by EdisonCruise
July 22nd, 2019, 1:18 am
Forum: Technical Forum
Topic: Why is Bellman Equation solved by backwards?
Replies: 32
Views: 9234

Re: Why is Bellman Equation solved by backwards?

I cannot make a specific example, because I read that in a book. The followingsa are the images I took from Nicole Bäuerle and Ulrich Rieder' book. Maybe I can rephase the question, if the initial condition is well defined and the data is NON-stationary, can the Bellman equation be solved by forward...
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