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by EdisonCruise
June 15th, 2020, 7:37 am
Forum: Trading Forum
Topic: How to interpret the switching between price and time priority of the limit order?
Replies: 4
Views: 1990

Re: How to interpret the switching between price and time priority of the limit order?

I have no good way to handle the wide spread and manipulation issues now. Maybe those are real challenges and I have to expect a long process of trial and error.
by EdisonCruise
June 12th, 2020, 2:17 pm
Forum: Trading Forum
Topic: How to interpret the switching between price and time priority of the limit order?
Replies: 4
Views: 1990

Re: How to interpret the switching between price and time priority of the limit order?

I find the queues are frequently long enough. A market making strategy for large tick asset may be suitable.
by EdisonCruise
June 11th, 2020, 3:32 pm
Forum: Trading Forum
Topic: How to interpret the switching between price and time priority of the limit order?
Replies: 4
Views: 1990

How to interpret the switching between price and time priority of the limit order?

It is known that small tick size assets are more volatile, and the queues on limit order price are relatively short, so that they are filled mainly based on price-priority. For large tick size, the queues are usually much longer and time priority dominant. However, I find a future on cryptocurrency ...
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: 2208

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: 2208

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: 2208

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: 1992

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: 1992

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: 1992

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: 3036

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: 2046

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: 2285

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: 2046

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...
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