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by EdisonCruise
April 23rd, 2020, 2:16 am
Forum: Numerical Methods Forum
Topic: How to solve this quasi-variational inequality (QVI) numerically?
Replies: 6
Views: 733

Re: How to solve this quasi-variational inequality (QVI) numerically?

Yes, i t is a minimisation problem at each time level. I hope this is a non-trival example: $$ 0=\min[ \frac{\partial{h(t,s)}}{\partial{t}}+ \min \limits_{a\in A} (\mathbb{E}(h(t,s+\xi a)-h(t,s-\xi a))) ;  h(t,s)-b; -h(t,s)-b]$$ where \(\mathbb{E} \) is the expectation operator with respect to the B...
by EdisonCruise
April 22nd, 2020, 5:52 am
Forum: Numerical Methods Forum
Topic: How to solve this quasi-variational inequality (QVI) numerically?
Replies: 6
Views: 733

Re: How to solve this quasi-variational inequality (QVI) numerically?

Maybe the QVI above can be simplified as below: $$ 0=min( \frac{\partial{h}}{\partial{t}}+\phi_1 h; \phi_2 h; \phi_3 h) $$ where \( \phi_1,\phi_2\), and \(\phi_3  \) indicate some combinations of  operators (e.g. differential operator or expectation operator) on an unknown function \( h \). Is there...
by EdisonCruise
April 21st, 2020, 10:58 am
Forum: Numerical Methods Forum
Topic: How to solve this quasi-variational inequality (QVI) numerically?
Replies: 6
Views: 733

How to solve this quasi-variational inequality (QVI) numerically?

<a href= https://i.ibb.co/zbWQyPg/QVI.jpg " /> I found this equation from Eq (10) in the paper Hedge and Speculate: Replicating Option Payoffs with Limit and Market Orders by Alvaro Cartea, Luhui Gan, and Sebastian Jaimungal.   I think I can solve the first equation which is for the use of limit or...
by EdisonCruise
April 13th, 2020, 10:11 am
Forum: Numerical Methods Forum
Topic: How to solve this ODE?
Replies: 9
Views: 1014

Re: How to solve this ODE?

Thank you so much for your suggestions. I think I can solve this ODE numerically with (1) A psuedo time term is added to the equation as below: $$ min[rV(x)-\frac{\sigma^2}{2} \frac{d^2V(x)}{dx^2}-\mu(\theta-x) \frac{dV(x)}{dx}+\frac{dV(x)}{dt}, V(x)-(x-c)]=0 $$ (2) boundary conditions: \(\frac {\pa...
by EdisonCruise
March 30th, 2020, 10:06 am
Forum: Numerical Methods Forum
Topic: How to solve this ODE?
Replies: 9
Views: 1014

Re: How to solve this ODE?

Thank you all for your suggestions. Following Alan’s suggestions to split the integration limits, I can do the integration by a change of variable method to go around the singularity point. However, I am still not sure on the below two questions: (1)      Boundary conditions It seems that as \(x\rig...
by EdisonCruise
March 27th, 2020, 12:35 pm
Forum: Numerical Methods Forum
Topic: How to solve this ODE?
Replies: 9
Views: 1014

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: 1627

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: 1911

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: 2173

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: 1911

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: 2173

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: 2173

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: 2173

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...
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