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In probability theory, a real valued [[stochastic process|process]] ''X'' is called a '''semimartingale''' if it can be decomposed as the sum of a [[local martingale]] and an adapted finite-variation process. | |||
Semimartingales are "good integrators", forming the largest class of processes with respect to which the [[Itō integral]] and the [[Stratonovich integral]] can be defined. | |||
The class of semimartingales is quite large (including, for example, all continuously differentiable processes, [[Wiener process|Brownian motion]] and [[Poisson process]]es). [[Martingale_(probability_theory)#Submartingales_and_supermartingales|Submartingales]] and [[Martingale_(probability_theory)#Submartingales_and_supermartingales|supermartingales]] together represent a subset of the semimartingales. | |||
==Definition== | |||
A real valued process ''X'' defined on the [[filtration (mathematics)#Measure theory|filtered probability space]] (Ω,''F'',(''F''<sub>''t''</sub>)<sub>''t'' ≥ 0</sub>,P) is called a '''semimartingale''' if it can be decomposed as | |||
:<math>X_t = M_t + A_t</math> | |||
where ''M'' is a [[local martingale]] and ''A'' is a [[càdlàg]] [[adapted process]] of locally [[bounded variation]]. | |||
An '''R'''<sup>''n''</sup>-valued process ''X'' = (''X''<sup>1</sup>,…,''X''<sup>''n''</sup>) is a semimartingale if each of its components ''X''<sup>''i''</sup> is a semimartingale. | |||
==Alternative definition== | |||
First, the simple [[predictable process]]es are defined to be linear combinations of processes of the form ''H''<sub>''t''</sub> = ''A''1<sub>{''t'' > ''T''}</sub> for stopping times ''T'' and ''F''<sub>''T''</sub> -measurable random variables ''A''. The integral ''H'' · ''X'' for any such simple predictable process ''H'' and real valued process ''X'' is | |||
:<math>H\cdot X_t\equiv 1_{\{t>T\}}A(X_t-X_T).</math> | |||
This is extended to all simple predictable processes by the linearity of ''H'' · ''X'' in ''H''. | |||
A real valued process ''X'' is a semimartingale if it is càdlàg, adapted, and for every ''t'' ≥ 0, | |||
:<math>\left\{H\cdot X_t:H{\rm\ is\ simple\ predictable\ and\ }|H|\le 1\right\}</math> | |||
is bounded in probability. The Bichteler-Dellacherie Theorem states that these two definitions are equivalent {{Harv|Protter|2004|p=144}}. | |||
==Examples== | |||
* Adapted and continuously differentiable processes are finite variation processes, and hence are semimartingales. | |||
* [[Wiener process|Brownian motion]] is a semimartingale. | |||
* All càdlàg [[Martingale (probability theory)|martingales]], submartingales and supermartingales are semimartingales. | |||
* [[Itō calculus#Itō processes|Itō processes]], which satisfy a stochastic differential equation of the form ''dX'' = ''σdW'' + ''μdt'' are semimartingales. Here, ''W'' is a Brownian motion and ''σ, μ'' are adapted processes. | |||
* Every [[Lévy process]] is a semimartingale. | |||
Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case. | |||
* [[Fractional Brownian motion]] with Hurst parameter ''H'' ≠ 1/2 is not a semimartingale. | |||
==Properties== | |||
* The semimartingales form the largest class of processes for which the [[Itō calculus|Itō integral]] can be defined. | |||
* Linear combinations of semimartingales are semimartingales. | |||
* Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the [[Stochastic calculus|Itō integral]]. | |||
* The [[quadratic variation]] exists for every semimartingale. | |||
* The class of semimartingales is closed under [[stopped process|optional stopping]], [[Stopping time#Localization|localization]], [[change of time]] and absolutely continuous [[Absolutely_continuous#Absolute_continuity_of_measures|change of measure]]. | |||
* If ''X'' is an '''R'''<sup>''m''</sup> valued semimartingale and ''f'' is a twice continuously differentiable function from '''R'''<sup>''m''</sup> to '''R'''<sup>''n''</sup>, then ''f''(''X'') is a semimartingale. This is a consequence of [[Itō's lemma]]. | |||
* The property of being a semimartingale is preserved under shrinking the filtration. More precisely, if ''X'' is a semimartingale with respect to the filtration ''F''<sub>t</sub>, and is adapted with respect to the subfiltration ''G''<sub>t</sub>, then ''X'' is a ''G''<sub>t</sub>-semimartingale. | |||
* (Jacod's Countable Expansion) The property of being a semimartingale is preserved under enlarging the filtration by a countable set of disjoint sets. Suppose that ''F''<sub>t</sub> is a filtration, and ''G''<sub>t</sub> is the filtration generated by ''F''<sub>t</sub> and a countable set of disjoint measurable sets. Then, every ''F''<sub>t</sub>-semimartingale is also a ''G''<sub>t</sub>-semimartingale. {{Harv|Protter|2004|p=53}} | |||
==Semimartingale decompositions== | |||
By definition, every semimartingale is a sum of a local martingale and a finite variation process. However, this decomposition is not unique. | |||
===Continuous semimartingales=== | |||
A continuous semimartingale uniquely decomposes as ''X'' = ''M'' + ''A'' where ''M'' is a continuous local martingale and ''A'' is a continuous finite variation process starting at zero. {{Harv|Rogers|Williams|1987|p=358}} | |||
For example, if ''X'' is an Itō process satisfying the stochastic differential equation d''X''<sub>t</sub> = σ<sub>t</sub> d''W''<sub>t</sub> + ''b''<sub>t</sub> dt, then | |||
:<math>M_t=X_0+\int_0^t\sigma_s\,dW_s,\ A_t=\int_0^t b_s\,ds.</math> | |||
===Special semimartingales=== | |||
A special semimartingale is a real valued process ''X'' with the decomposition ''X'' = ''M'' + ''A'', where ''M'' is a local martingale and ''A'' is a predictable finite variation process starting at zero. If this decomposition exists, then it is unique up to a P-null set. | |||
Every special semimartingale is a semimartingale. Conversely, a semimartingale is a special semimartingale if and only if the process ''X''<sub>t</sub><sup>*</sup> ≡ sup<sub>''s'' ≤ ''t''</sub> |X<sub>''s''</sub>| is [[Stopping time#Localization|locally integrable]] {{Harv|Protter|2004|p=130}}. | |||
For example, every continuous semimartingale is a special semimartingale, in which case ''M'' and ''A'' are both continuous processes. | |||
===Purely discontinuous semimartingales=== | |||
A semimartingale is called purely discontinuous if its quadratic variation [''X''] is a pure jump process, | |||
:<math>[X]_t=\sum_{s\le t}\Delta X_s^2</math>. | |||
Every adapted finite variation process is a purely discontinuous semimartingale. A continuous process is a purely discontinuous semimartingale if and only if it is an adapted finite variation process. | |||
Then, every semimartingale has the unique decomposition ''X'' = ''M'' + ''A'' where ''M'' is a continuous local martingale and ''A'' is a purely discontinuous semimartingale starting at zero. The local martingale ''M'' - ''M''<sub>0</sub> is called the continuous martingale part of ''X'', and written as ''X''<sup>c</sup> ({{Harvnb|He|Wang|Yan|1992|p=209}}; {{Harvnb|Kallenberg|2002|p=527}}). | |||
In particular, if ''X'' is continuous, then ''M'' and ''A'' are continuous. | |||
==Semimartingales on a manifold== | |||
The concept of semimartingales, and the associated theory of stochastic calculus, extends to processes taking values in a [[differentiable manifold]]. A process ''X'' on the manifold ''M'' is a semimartingale if ''f''(''X'') is a semimartingale for every smooth function ''f'' from ''M'' to '''R'''. {{Harv|Rogers|1987|p=24}} Stochastic calculus for semimartingales on general manifolds requires the use of the [[Stratonovich integral]]. | |||
==See also== | |||
*[[Sigma-martingale]] | |||
==References== | |||
*{{Citation|last=He|first=Sheng-wu|last2=Wang|first2=Jia-gang|last3=Yan|first3=Jia-an|year=1992|title=Semimartingale Theory and Stochastic Calculus|publisher=Science Press, CRC Press Inc.|isbn= 0-8493-7715-3}} | |||
*{{Citation|last=Kallenberg|first=Olav|year=2002|title=Foundations of Modern Probability|edition=2nd|publisher=Springer|isbn=0-387-95313-2}} | |||
*{{Citation|last=Protter|first=Philip E.|year=2004|title=Stochastic Integration and Differential Equations|publisher=Springer|edition=2nd|isbn=3-540-00313-4}} | |||
*{{Citation|last=Rogers|first=L.C.G.|last2=Williams|first2=David|year=1987|volume=2|title=Diffusions, Markov Processes, and Martingales|publisher=John Wiley & Sons Ltd|isbn=0-471-91482-7}} | |||
{{Stochastic processes}} | |||
[[Category:Stochastic processes]] | |||
[[Category:Martingale theory]] | |||
Revision as of 19:00, 14 September 2013
In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itō integral and the Stratonovich integral can be defined.
The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes). Submartingales and supermartingales together represent a subset of the semimartingales.
Definition
A real valued process X defined on the filtered probability space (Ω,F,(Ft)t ≥ 0,P) is called a semimartingale if it can be decomposed as
where M is a local martingale and A is a càdlàg adapted process of locally bounded variation.
An Rn-valued process X = (X1,…,Xn) is a semimartingale if each of its components Xi is a semimartingale.
Alternative definition
First, the simple predictable processes are defined to be linear combinations of processes of the form Ht = A1{t > T} for stopping times T and FT -measurable random variables A. The integral H · X for any such simple predictable process H and real valued process X is
This is extended to all simple predictable processes by the linearity of H · X in H.
A real valued process X is a semimartingale if it is càdlàg, adapted, and for every t ≥ 0,
is bounded in probability. The Bichteler-Dellacherie Theorem states that these two definitions are equivalent Template:Harv.
Examples
- Adapted and continuously differentiable processes are finite variation processes, and hence are semimartingales.
- Brownian motion is a semimartingale.
- All càdlàg martingales, submartingales and supermartingales are semimartingales.
- Itō processes, which satisfy a stochastic differential equation of the form dX = σdW + μdt are semimartingales. Here, W is a Brownian motion and σ, μ are adapted processes.
- Every Lévy process is a semimartingale.
Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case.
- Fractional Brownian motion with Hurst parameter H ≠ 1/2 is not a semimartingale.
Properties
- The semimartingales form the largest class of processes for which the Itō integral can be defined.
- Linear combinations of semimartingales are semimartingales.
- Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the Itō integral.
- The quadratic variation exists for every semimartingale.
- The class of semimartingales is closed under optional stopping, localization, change of time and absolutely continuous change of measure.
- If X is an Rm valued semimartingale and f is a twice continuously differentiable function from Rm to Rn, then f(X) is a semimartingale. This is a consequence of Itō's lemma.
- The property of being a semimartingale is preserved under shrinking the filtration. More precisely, if X is a semimartingale with respect to the filtration Ft, and is adapted with respect to the subfiltration Gt, then X is a Gt-semimartingale.
- (Jacod's Countable Expansion) The property of being a semimartingale is preserved under enlarging the filtration by a countable set of disjoint sets. Suppose that Ft is a filtration, and Gt is the filtration generated by Ft and a countable set of disjoint measurable sets. Then, every Ft-semimartingale is also a Gt-semimartingale. Template:Harv
Semimartingale decompositions
By definition, every semimartingale is a sum of a local martingale and a finite variation process. However, this decomposition is not unique.
Continuous semimartingales
A continuous semimartingale uniquely decomposes as X = M + A where M is a continuous local martingale and A is a continuous finite variation process starting at zero. Template:Harv
For example, if X is an Itō process satisfying the stochastic differential equation dXt = σt dWt + bt dt, then
Special semimartingales
A special semimartingale is a real valued process X with the decomposition X = M + A, where M is a local martingale and A is a predictable finite variation process starting at zero. If this decomposition exists, then it is unique up to a P-null set.
Every special semimartingale is a semimartingale. Conversely, a semimartingale is a special semimartingale if and only if the process Xt* ≡ sups ≤ t |Xs| is locally integrable Template:Harv.
For example, every continuous semimartingale is a special semimartingale, in which case M and A are both continuous processes.
Purely discontinuous semimartingales
A semimartingale is called purely discontinuous if its quadratic variation [X] is a pure jump process,
Every adapted finite variation process is a purely discontinuous semimartingale. A continuous process is a purely discontinuous semimartingale if and only if it is an adapted finite variation process.
Then, every semimartingale has the unique decomposition X = M + A where M is a continuous local martingale and A is a purely discontinuous semimartingale starting at zero. The local martingale M - M0 is called the continuous martingale part of X, and written as Xc (Template:Harvnb; Template:Harvnb).
In particular, if X is continuous, then M and A are continuous.
Semimartingales on a manifold
The concept of semimartingales, and the associated theory of stochastic calculus, extends to processes taking values in a differentiable manifold. A process X on the manifold M is a semimartingale if f(X) is a semimartingale for every smooth function f from M to R. Template:Harv Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.
See also
References
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010