I learned an interesting continuous fourth dimension play tricks recently. The context is a depository fiscal establishment annotation , "The frail benefits of endowment destruction" that I wrote amongst John Campbell , most how to extend our habit model to jumps inwards consumption. The indicate hither is to a greater extent than interesting than that detail context.
Suppose ane fourth dimension serial \(x\) , which follows a diffusion , drives some other \(y\). In the simplest illustration , \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our illustration , the 2nd equation describes how habits \(y\) respond to consumption \(x\). The same form of construction mightiness depict how invested wealth \(y\) responds to property prices \(x\) , or how selection prices \(y\) respond to stock prices \(x\).
Now , suppose nosotros desire to extend the model to handgrip jumps inwards \(x\) , \[dx_t = \sigma dz_t + dJ_t.\] What hit nosotros hit most the 2nd equation? \(y_t\) forthwith tin saltation too. On the right mitt side of the 2nd equation , should nosotros role the left bound , the right bound , or something inwards between?
The park respond is to role the left limit. We generalize the model to jumps this way: \[dx_t = \sigma dz_t+ dJ_t \] \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] where \(y_{t_{-}}\) denotes the left limit.
That approach has some weird properties however. Suppose \(y_{t_-}=1\) , together with \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose in that place are 2 jumps of size 1/2 , ane at fourth dimension \(t\) together with ane at fourth dimension \(t+\varepsilon\). Now \(y\) jumps upwardly to 1.5 afterwards the outset saltation , together with and hence jumps some other \(1.5 \times 0.5 = 0.75\) , ending upwardly at \(y_{t+\varepsilon} =2.25\). Two one-half jumps lead hold a unlike answer than ane total jump.
Suppose instead nosotros extend the master copy model to jumps past times taking the saltation bound of a continuous process. Imagine that nosotros detect realizations of \(\{dz_t\}\) that acquire closer together with closer to a saltation inwards \(dx_t\) , together with let's break what happens to \(y_t\). The full general solution to the outset gear upwardly of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] hence , inwards the bound \(\Delta \rightarrow 0\) that \(x_t\) takes a saltation of size \(dJ_t\) , the jump-limit of a continuous motion is \[ dy_{t} \equiv y_t -y_{t_-} = y_{t_-}(e^{dx_{t}}-1) = y_{t_-}\sigma dz_t + y_{t_-}e^{dJ_t}\] rather than \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] So , the left-limit method produced a answer to a saltation that was unlike from the answer to a continuous procedure arbitrarily to a greater extent than or less a jump. For illustration , the left-limit approach tin hit a negative \(y_t\) , but this method , similar the diffusion procedure , cannot autumn below zero. This method also produces a answer to 2 one-half jumps that is the same every bit the answer to a total jump.
As y'all tin run across , the divergence is whether the soil variable \(y_t\) gets to modify during the jump. In the left-limit approach , the same \(y_{t_-}\) gets applied to the whole jump. In the continuous-limit version , \(y_t\) implicitly gets to deed spell the saltation inwards \(x_t\) is moving.
H5N1 nonlinear part of a saltation is a niggling new , but there's nada incorrect amongst it , together with it exists inwards the continuous fourth dimension literature. We don't run across it that ofttimes , because when you're exclusively studying ane serial it's easier to only modify the distribution of the saltation procedure instead. This interrogation occurs when y'all tin run across both serial x together with y together with y'all desire to model the human relationship betwixt them.
Which is right?
Which extension to jumps is correct? Both are mathematically correct. There is nada incorrect amongst writing downwards a model inwards which the answer to a saltation is unlike from the answer to continuous movements arbitrarily to a greater extent than or less jumps. The respond depends on the economical situation.
For illustration , consider models amongst bankruptcy constraints. Agents who tin continuously arrange their investments may ever avoid bankruptcy inwards a diffusion setting. If nosotros extend such a model to jumps amongst the continuous bound approach , implicitly preserving the investor's powerfulness to merchandise every bit fast every bit property prices modify fifty-fifty inwards the saltation bound , nosotros volition save bankruptcy avoidance inwards confront of a saltation inwards prices. However , if nosotros model portfolio adjustment to jumps amongst the left-limit generalization , agents may live on forced inwards to bankruptcy for cost jumps.
Sometimes , ane introduces jumps exactly to model a province of affairs inwards which prices tin deed faster than agents tin arrange their portfolios , hence agents may live on forced to bankruptcy. Then the left-limit generalization is correct. But if ane wants to extend a model to jumps for other reasons , spell avoiding bankruptcy , negative consumption , negative marginal utility (consumption below null or below habits) , violations of budget constraints , feasibility weather condition , borrowing constraints , together with hence forth , together with hence ane should lead a generalization inwards which the saltation gives the same outcome every bit the continuous limit.
Similarly , when extending selection pricing models to jumps , ane may desire to model the saltation inwards such a means that investors cannot arrange portfolios fast enough. Then the left-limit extension is appropriate , together with investors must concord the saltation risk. But ane may want to accommodate jumps inwards property prices to ameliorate stand upwardly for property cost dynamics spell maintaining investor's powerfulness to dynamically hedge. Then the nonlinear extension is appropriate , maintaining the equivalence betwixt jumps together with the limiting diffusion.
A niggling to a greater extent than full general treatment
H5N1 niggling to a greater extent than mostly , suppose \[ dx_t = g dt + \sigma dz_t \] \[dy_t = \mu(y_t) dt + \lambda(y_t)dx_t.\] We desire to add together \(dJ_t\) to the outset equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If in that place is a saltation \(dJ_t\) , \(y\) moves past times an amount \[\frac{1}{\lambda(y_{t_-})}dy_t \equiv \frac{1}{\lambda(y_{t_-})}(y_t - y_{t_-}) = dx_t .\] The bound of a continuous motion solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again , y'all run across the crucial divergence , whether the soil variable gets to deed "during" the jump. We tin write this every bit a differential , past times writing the solution to this concluding differential equation every bit \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] together with and hence \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]
So , y'all don't have to extend the model to jumps amongst the left-limit approach , together with y'all don't lead hold to swallow the watch that a saltation has a unlike answer than an arbitrarily closed continuous-sample-path movement. The concluding equation shows y'all how to modify the model to include jumps inwards a means that preserves the belongings that the saltation has the same outcome every bit its continuous limit.
The point
Why a weblog shipping service on this? I asked a few continuous-time gurus , together with none of them had seen this number before. If person knows where this has all been worked out amongst proper is dotted together with ts crossed , I would similar to know together with mention it properly. (I would mean value the literature on selection pricing amongst jumps had done it , but I couldn't break a reference.) Or mayhap it hasn't been done together with person wants to hit it. I'm non skilful plenty at the technical aspects of continuous fourth dimension to write this amongst the right precision together with generality.
And it's a cool play tricks that may live on useful to person exterior of the narrow context that nosotros had for it.
Update:
Perhaps the right application is stock prices together with selection prices. When stock prices saltation , person must lead hold studied the instance that selection prices deed past times the same amount the Black-Scholes formula gives for the same size stock cost movement. Does anyone lead hold a citation to that case?
Suppose ane fourth dimension serial \(x\) , which follows a diffusion , drives some other \(y\). In the simplest illustration , \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our illustration , the 2nd equation describes how habits \(y\) respond to consumption \(x\). The same form of construction mightiness depict how invested wealth \(y\) responds to property prices \(x\) , or how selection prices \(y\) respond to stock prices \(x\).
Now , suppose nosotros desire to extend the model to handgrip jumps inwards \(x\) , \[dx_t = \sigma dz_t + dJ_t.\] What hit nosotros hit most the 2nd equation? \(y_t\) forthwith tin saltation too. On the right mitt side of the 2nd equation , should nosotros role the left bound , the right bound , or something inwards between?
The park respond is to role the left limit. We generalize the model to jumps this way: \[dx_t = \sigma dz_t+ dJ_t \] \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] where \(y_{t_{-}}\) denotes the left limit.
That approach has some weird properties however. Suppose \(y_{t_-}=1\) , together with \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose in that place are 2 jumps of size 1/2 , ane at fourth dimension \(t\) together with ane at fourth dimension \(t+\varepsilon\). Now \(y\) jumps upwardly to 1.5 afterwards the outset saltation , together with and hence jumps some other \(1.5 \times 0.5 = 0.75\) , ending upwardly at \(y_{t+\varepsilon} =2.25\). Two one-half jumps lead hold a unlike answer than ane total jump.
Suppose instead nosotros extend the master copy model to jumps past times taking the saltation bound of a continuous process. Imagine that nosotros detect realizations of \(\{dz_t\}\) that acquire closer together with closer to a saltation inwards \(dx_t\) , together with let's break what happens to \(y_t\). The full general solution to the outset gear upwardly of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] hence , inwards the bound \(\Delta \rightarrow 0\) that \(x_t\) takes a saltation of size \(dJ_t\) , the jump-limit of a continuous motion is \[ dy_{t} \equiv y_t -y_{t_-} = y_{t_-}(e^{dx_{t}}-1) = y_{t_-}\sigma dz_t + y_{t_-}e^{dJ_t}\] rather than \[ dy_t = y_{t_-} dx_t = y_{t_-} \sigma dz_t + y_{t_-}dJ_t \] So , the left-limit method produced a answer to a saltation that was unlike from the answer to a continuous procedure arbitrarily to a greater extent than or less a jump. For illustration , the left-limit approach tin hit a negative \(y_t\) , but this method , similar the diffusion procedure , cannot autumn below zero. This method also produces a answer to 2 one-half jumps that is the same every bit the answer to a total jump.
As y'all tin run across , the divergence is whether the soil variable \(y_t\) gets to modify during the jump. In the left-limit approach , the same \(y_{t_-}\) gets applied to the whole jump. In the continuous-limit version , \(y_t\) implicitly gets to deed spell the saltation inwards \(x_t\) is moving.
H5N1 nonlinear part of a saltation is a niggling new , but there's nada incorrect amongst it , together with it exists inwards the continuous fourth dimension literature. We don't run across it that ofttimes , because when you're exclusively studying ane serial it's easier to only modify the distribution of the saltation procedure instead. This interrogation occurs when y'all tin run across both serial x together with y together with y'all desire to model the human relationship betwixt them.
Which is right?
Which extension to jumps is correct? Both are mathematically correct. There is nada incorrect amongst writing downwards a model inwards which the answer to a saltation is unlike from the answer to continuous movements arbitrarily to a greater extent than or less jumps. The respond depends on the economical situation.
For illustration , consider models amongst bankruptcy constraints. Agents who tin continuously arrange their investments may ever avoid bankruptcy inwards a diffusion setting. If nosotros extend such a model to jumps amongst the continuous bound approach , implicitly preserving the investor's powerfulness to merchandise every bit fast every bit property prices modify fifty-fifty inwards the saltation bound , nosotros volition save bankruptcy avoidance inwards confront of a saltation inwards prices. However , if nosotros model portfolio adjustment to jumps amongst the left-limit generalization , agents may live on forced inwards to bankruptcy for cost jumps.
Sometimes , ane introduces jumps exactly to model a province of affairs inwards which prices tin deed faster than agents tin arrange their portfolios , hence agents may live on forced to bankruptcy. Then the left-limit generalization is correct. But if ane wants to extend a model to jumps for other reasons , spell avoiding bankruptcy , negative consumption , negative marginal utility (consumption below null or below habits) , violations of budget constraints , feasibility weather condition , borrowing constraints , together with hence forth , together with hence ane should lead a generalization inwards which the saltation gives the same outcome every bit the continuous limit.
Similarly , when extending selection pricing models to jumps , ane may desire to model the saltation inwards such a means that investors cannot arrange portfolios fast enough. Then the left-limit extension is appropriate , together with investors must concord the saltation risk. But ane may want to accommodate jumps inwards property prices to ameliorate stand upwardly for property cost dynamics spell maintaining investor's powerfulness to dynamically hedge. Then the nonlinear extension is appropriate , maintaining the equivalence betwixt jumps together with the limiting diffusion.
A niggling to a greater extent than full general treatment
H5N1 niggling to a greater extent than mostly , suppose \[ dx_t = g dt + \sigma dz_t \] \[dy_t = \mu(y_t) dt + \lambda(y_t)dx_t.\] We desire to add together \(dJ_t\) to the outset equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If in that place is a saltation \(dJ_t\) , \(y\) moves past times an amount \[\frac{1}{\lambda(y_{t_-})}dy_t \equiv \frac{1}{\lambda(y_{t_-})}(y_t - y_{t_-}) = dx_t .\] The bound of a continuous motion solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again , y'all run across the crucial divergence , whether the soil variable gets to deed "during" the jump. We tin write this every bit a differential , past times writing the solution to this concluding differential equation every bit \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] together with and hence \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]
So , y'all don't have to extend the model to jumps amongst the left-limit approach , together with y'all don't lead hold to swallow the watch that a saltation has a unlike answer than an arbitrarily closed continuous-sample-path movement. The concluding equation shows y'all how to modify the model to include jumps inwards a means that preserves the belongings that the saltation has the same outcome every bit its continuous limit.
The point
Why a weblog shipping service on this? I asked a few continuous-time gurus , together with none of them had seen this number before. If person knows where this has all been worked out amongst proper is dotted together with ts crossed , I would similar to know together with mention it properly. (I would mean value the literature on selection pricing amongst jumps had done it , but I couldn't break a reference.) Or mayhap it hasn't been done together with person wants to hit it. I'm non skilful plenty at the technical aspects of continuous fourth dimension to write this amongst the right precision together with generality.
And it's a cool play tricks that may live on useful to person exterior of the narrow context that nosotros had for it.
Update:
Perhaps the right application is stock prices together with selection prices. When stock prices saltation , person must lead hold studied the instance that selection prices deed past times the same amount the Black-Scholes formula gives for the same size stock cost movement. Does anyone lead hold a citation to that case?
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