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Church
Topic Author
Posts: 175
Joined: September 4th, 2007, 10:27 am

### Stochastic bridge in combination with stochastic volatility and jumps

Hello,For a risk management purpose (so real-world, no risk-neutral) I need a stochastic bridge such as the Brownian Bridge for the Brownian Motion, but then representing a process with stochastic volatility and jumps.The specifics of the stochastic volatility process and jumps are not the most important (although the resulting process should be realistic of course).Are there any (preferably practical) ways to construct such a bridge?Thanks

Alan
Posts: 10033
Joined: December 19th, 2001, 4:01 am
Location: California
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### Stochastic bridge in combination with stochastic volatility and jumps

Interesting question. Looking in my library, there is a short discussion of general Markov process bridgesin Sec. 2.VI.14 ('Tied Down Markov Processes') in Doob's 'Classical Potential Theory and Its Probabilistic Counterpart'. Ignoring the jumps, the generator for a general diffusion bridge is known; see for example (4.4) here (O. Papaspiliopoulos and G. Roberts)This shows that, at a minimum, you will need to start with an SV process with a known joint transition density. There actually aren't too many of those,but see my forthcoming book for the joint transition density for the Heston model and 3/2-model. If you want to add jumps,and keep a known (or at least tractable) joint transition density, one possibility is to use an independent stochastic time change. How to do the time change is also discussed in my book. (The book's table of contents may be found in this thread).How to do the time change *and* preserve your bridge conditioning information is *not* discussed in my book -- it's an interesting problem, left to you, since I don't know. Another approach to adding jumps is to try an affine jump-diffusion and see if you can construct the joint transition density.Whether either approach will be realistic enough for whatever your application is, is pretty hard to say in the abstract.I have found some decent results using the first (time-change) method for some joint SPX/VIX time series modeling. Anyway, those are some ideas on how to proceed or at least get started.
Last edited by Alan on December 3rd, 2015, 11:00 pm, edited 1 time in total.

Alan
Posts: 10033
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Stochastic bridge in combination with stochastic volatility and jumps

Church, I wonder if you can elaborate some more about the risk management purpose?Also, what will be your proxy for the starting and ending instantaneous volatility? Finally, I think I would start with the easier case of Merton's jump-diffusion and try to solve that one first.Googling the key phrase 'Levy bridge process' turns up at the top of the results this paper by Mijatovic, Pistorius, & Stolte, which looks on-topic.outrun, some sort of Markov chain approach crossed my mind, too. It would seem likely a (finite-state) Markov chain conditioned to end up in a particular state would have a knowntransition matrix in terms of the original one. If so, then the problem reduces to creatinga (convergent, finite-state) Markov chain discretization for the target (SVJ or SVJJ) process. Some general theory:Markovian bridges (Fitzsimmons, Pitman, & Yor)
Last edited by Alan on December 5th, 2015, 11:00 pm, edited 1 time in total.

Church
Topic Author
Posts: 175
Joined: September 4th, 2007, 10:27 am

### Stochastic bridge in combination with stochastic volatility and jumps

The purpose is as follows: given an externally given set of realisations (scenarios) at t=1, generated with a 1-year time step, determine the impact of (amongst others) a dynamic hedging program that is managed on a day-to-day basis.My guess is that something along the lines that outrun suggests would probably be the most practical in the short term.

eh
Posts: 494
Joined: March 2nd, 2010, 9:26 am

### Stochastic bridge in combination with stochastic volatility and jumps

For jumps, you could use a Levy bridge. The VG bridge is easy to simulate.

Church
Topic Author
Posts: 175
Joined: September 4th, 2007, 10:27 am

### Stochastic bridge in combination with stochastic volatility and jumps

Are these easy to expand to the multidimensional case? All the literature on this focuses on the univariate case.

Lapsilago
Posts: 259
Joined: October 15th, 2004, 7:36 am
Location: Germany

### Stochastic bridge in combination with stochastic volatility and jumps

Hi Church,I think you could use a Brownian Motion in d dimensions with an appropriated subordinator. This might work for multi dimensional modelling. But never tried that...Best Lapsi

eh
Posts: 494
Joined: March 2nd, 2010, 9:26 am

### Stochastic bridge in combination with stochastic volatility and jumps

In general, multidimensional Levy bridges are quite tricky beasts.Following on from Lapsilago's suggestion, there is a construction of a VG bridge that uses a Brownian bridge with a gamma bridge subordinator. The Brownian bridge could be replaced with a multidimentional Brownian bridge, and the same gamma bridge subordinator could be used across all coordinates. This certainly makes sense for positive dependence.

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