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6 votes

References for Stochastic Control for finance

Peter Forsyth of UWaterloo is my favourite author on this topic (one of my top three in MathFin!) Personal Homepage with Lots of Papers Optimal allocation under wealth goals, optimal decumulation ...
James Spencer-Lavan's user avatar
5 votes

Code examples of solving Stochastic Optimal Control problems

FWIW, I implemented one such control solution for my course project. See here
Danny's user avatar
  • 514
4 votes
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example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$

We assume that \begin{align*} \frac{dX_t}{X_t} &= (r+\pi Y_t)dt + \pi\sigma dW_t,\tag{1}\\ dY_t &= -\lambda Y_t + dB_t.\tag{2} \end{align*} From $(2)$, \begin{align*} Y_t = Y_0 e^{-\lambda t}...
Gordon's user avatar
  • 21.1k
3 votes

How do your solve for trader's optimal demand in market similar to Kyle's model?

Generic knowledge about this kind of models Let me try to get your model close to elements that are known: Time continuous Kyle's model is something that is solved in Çetin, Umut, and Albina Danilova....
lehalle's user avatar
  • 12.2k
3 votes

References for Stochastic Control for finance

Look into Huyên Pham, Continuous-time Stochastic Control and Optimization with Financial Applications; Salvatore Federico, Giorgio Ferrari, Luca Regis (Editors). Applications of Stochastic Optimal ...
Dimitri Vulis's user avatar
3 votes
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Question on derivation step in portfolio replication under different borrowing and lending rates

Noting that $$ B= V -\alpha S = V - (\alpha S)^+ + (\alpha S)^- $$ $$ = (V - (\alpha S)^+)^+ - (V - (\alpha S)^+)^- + (\alpha S)^-,$$ a clearer way to write the dynamics of the funding costs (funding ...
ir7's user avatar
  • 5,043
3 votes

How to, from various hypotheses on the P&L, get known models (BS, Heston etc ...)

You can characterize local volatility (LV) models by the existence of a delta-hedging strategy which reduces the variance of P&L to zero, together with an assumption that P&L is a continuous ...
q.t.f.'s user avatar
  • 1,885
2 votes

Understanding the HJM drift condition's dimensions

Your issue is that you misinterpreted the NA criterion, it reads: $$ a_t(x) \triangleq \sum_{i=1}^{\infty} \left(b^i_t(x) \int_0^x (b_t^i(u))^T du\right)e_i, $$ where $b^i_t$ denotes the $i^{th}$ ...
ABIM's user avatar
  • 373
2 votes
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Merton portfolio allocation problem proportions/weights >1 or <0?

Your statement should be correct, the weights into the risky asset are not bounded between $0$ and $1$. Essentially, by setting $r=0$ you omit the term which shows that your weights always sum up to ...
Stefan Voigt's user avatar
  • 1,466
2 votes

How do you actually solve a stochastic HJB equation in practice?

An HJB is made of two components: a "core component" that corresponds to applying an optimal control the "dynamics" of the value function (that is surrounding this optimizattion). ...
lehalle's user avatar
  • 12.2k
2 votes

Game theory and stochastic calculus

The field you have in mind is covered with differential game theory, and it game birth to Mean Field Games (MFG), the book posted in a comment is certainly the reference: Probabilistic Theory of Mean ...
lehalle's user avatar
  • 12.2k
2 votes

References for Stochastic Control for finance

it is a bit late but the 2007 book "Applied Stochastic Control of Jump Diffusions" from Oksendal and Sulem is quite good too.
pierrot's user avatar
  • 96
2 votes

Closed-form solution to optimal single assset position sizing with predicted returns

As you have defined the Sharpe ratio, it is independent of your position. You have $$ \mathrm{E}[rf(z_1)] = \alpha z_1 f(z_1) $$ and $$ \mathrm{Var}[rf(z_1)] = \sigma_2^2f(z_1)^2 $$ and hence $$ \frac{...
Chris Taylor's user avatar
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2 votes
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Is stochastic control with the HJB equation used in market making/algo trading at institutions?

I would say that there are two things that we can talk about: Research purpose Real Trading Those models are a good thing to start when you try to build something that has to have characteristics of ...
ltrd's user avatar
  • 501
1 vote

Beta Weighting Deltas: What happens to the non-correlation part?

"At Beta=1 the underlying is expected to be as volatile as the index as well as move (more or less) together with the index." is not right. Beta has nothing to do with volatility, at-least ...
Arshdeep's user avatar
  • 2,222
1 vote
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American option pricing formulation

In explicitly wording my own question yesterday and naming my doubts, I think I may have stumbled upon the explanation: On the one hand, indeed we have $$ {\text{ess}\sup}_{s\in[0,T]}\mathbb{E}\left[...
Martin K's user avatar
1 vote
Accepted

non-Markovian/path-dependent optimal log utility and HJB-PDE

"Has anyone used this functional calculus to generalize the HJB-PDE from Markovian SDEs to non-Markovian SDEs? Can we simply write down the HJB-PDE with his new notions of derivatives and obtain ...
GeekBoy's user avatar
  • 26
1 vote

non-Markovian/path-dependent optimal log utility and HJB-PDE

This answer will provide somewhat of an educated guess, but is by no means rigorous or exact. It is based on my preliminary readings/understanding of subsections 5.4.1 and 5.4.2 of Applied Stochastic ...
Nap D. Lover's user avatar
1 vote

Ansatz and HJB equation

derivatives had a wrong sign (had to be γ(-f) instead of γf for example)
sle's user avatar
  • 121
1 vote

Stochastic optimization and mean field games : textbooks

I answer my own question. A starting point would be : the summer school on mean field games, provided by the University of Chicago. summer school
zestiria's user avatar
  • 111
1 vote

The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market

Such relationships are commonly covered in statistical mechanics, so any decent statsictal mechanics book should help. Here is an article that gives a very nice summary of the key concepts: https://...
Magic is in the chain's user avatar
1 vote
Accepted

optimal strategy problem (using Jensen's inequality)

I assume that the problem is $$\max_{\pi} E\left(\ln Z_T^{\Pi} \right).$$ Note that $\ln Z_t^{\Pi} = \ln X_t^{\Pi} -\ln X_t^{\rho}$. Moreover, \begin{align*} d\ln Z_t^{\Pi} &= d\ln X_t^{\Pi} -d\ln ...
Gordon's user avatar
  • 21.1k
1 vote
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trading strategy problem - initial capital x buys S over time [0,T] at the constant rate of x/T euros per unit of time

Equations (1) to (3) are correct. Your investment strategy is then, $\forall t > 0$ $$ X_t = \theta _ t S_t $$ Provided you use this strategy as part of self-financing portfolio you can write the P&...
Quantuple's user avatar
  • 14.7k
1 vote
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investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$

Let $Y_t := 2 S_t^1 S_t^2 $. Applying (multivariate) Itô to the function $f(t,S_t^1,S_t^2)=2 S_t^1 S_t^2$ yields a stochastic differential equation for $Y_t$ $$ \frac{dY_t}{Y_t} = \frac{dS_t^1}{S_t^1}...
Quantuple's user avatar
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