- June 23rd, 2021, 2:35 pm
- Forum: Numerical Methods Forum
- Topic: Is it necessary to piecewisely fit power-law distribution in empirical data?
- Replies:
**2** - Views:
**531**

In AARON CLAUSET’s paper power-law distributions in empirical data: ‘’In practice, few empirical phenomena obey power laws for all values of x. More often the power law applies only for values greater than some minimum x_min. In such cases we say that the tail of the distribution follows a power law...

- December 10th, 2020, 2:44 pm
- Forum: Technical Forum
- Topic: Is there any mathematical tool to solve this HJB-like problem?
- Replies:
**0** - Views:
**1543**

In typical stochastic control problems, the control variables in HJB equation are usually continuous or discrete values in action space. However, what if the action space is constructed by unknown continuous functions? For example, in market making problem, the Avellaneda-Stoikov model calculates th...

- 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:
**5** - Views:
**4298**

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.

- 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:
**5** - Views:
**4298**

I find the queues are frequently long enough. A market making strategy for large tick asset may be suitable.

- 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:
**5** - Views:
**4298**

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 ...

- April 23rd, 2020, 2:16 am
- Forum: Numerical Methods Forum
- Topic: How to solve this quasi-variational inequality (QVI) numerically?
- Replies:
**6** - Views:
**3975**

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...

- April 22nd, 2020, 5:52 am
- Forum: Numerical Methods Forum
- Topic: How to solve this quasi-variational inequality (QVI) numerically?
- Replies:
**6** - Views:
**3975**

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...

- April 22nd, 2020, 1:05 am
- Forum: Numerical Methods Forum
- Topic: How to solve this quasi-variational inequality (QVI) numerically?
- Replies:
**6** - Views:
**3975**

Sorry,I have updated with a full link

- April 21st, 2020, 10:58 am
- Forum: Numerical Methods Forum
- Topic: How to solve this quasi-variational inequality (QVI) numerically?
- Replies:
**6** - Views:
**3975**

<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 lim...

- April 13th, 2020, 10:11 am
- Forum: Numerical Methods Forum
- Topic: How to solve this ODE?
- Replies:
**9** - Views:
**3631**

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...

- March 30th, 2020, 10:06 am
- Forum: Numerical Methods Forum
- Topic: How to solve this ODE?
- Replies:
**9** - Views:
**3631**

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...

- March 27th, 2020, 12:35 pm
- Forum: Numerical Methods Forum
- Topic: How to solve this ODE?
- Replies:
**9** - Views:
**3631**

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 ...

- February 18th, 2020, 10:02 am
- Forum: Technical Forum
- Topic: Is Ito’s lemma applicable to a diffusion process with transition probability?
- Replies:
**2** - Views:
**4336**

- 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:
**4336**

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...

- November 21st, 2019, 12:28 am
- Forum: General Forum
- Topic: How to make ito's forumula for jump-diffusion a martingale
- Replies:
**3** - Views:
**2852**

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.