I learned an interesting continuous fourth dimension play tricks recently. The context is a note, "The frail benefits of endowment destruction" that I wrote alongside John Campbell, almost 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 1 fourth dimension serial \(x\), which follows a diffusion, drives some other \(y\). In the simplest example, \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our example, the minute equation describes how habits \(y\) respond to consumption \(x\). The same sort of construction mightiness clit 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 grip jumps inwards \(x\), \[dx_t = \sigma dz_t + dJ_t.\] What arrive at nosotros arrive at almost the minute equation? \(y_t\) instantly tin saltation too. On the right paw side of the minute equation, should nosotros purpose the left limit, the right limit, or something inwards between?
The commons respond is to purpose 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 \(y_{t_-}=1\), too \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose at that spot are 2 jumps of size 1/2, 1 at fourth dimension \(t\) too 1 at fourth dimension \(t+\varepsilon\). Now \(y\) jumps upward to 1.5 afterward the showtime jump, too thence jumps some other \(1.5 \times 0.5 = 0.75\), ending upward at \(y_{t+\varepsilon} =2.25\). Two one-half jumps receive got a dissimilar reply than 1 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 too closer to a saltation inwards \(dx_t\), too let's reveal what happens to \(y_t\). The full general solution to the showtime ready of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] so, inwards the bound \(\Delta \rightarrow 0\) that \(x_t\) takes a saltation of size \(dJ_t\), the jump-limit of a continuous displace 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 reply to a saltation that was dissimilar from the reply to a continuous procedure arbitrarily only about a jump. For example, the left-limit approach tin create a negative \(y_t\), but this method, similar the diffusion process, cannot autumn below zero. This method also produces a reply to 2 one-half jumps that is the same equally the reply to a total jump.
As you lot tin see, the divergence is whether the state 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 piece the saltation inwards \(x_t\) is moving.
Influenza A virus subtype H5N1 nonlinear business office of a saltation is a piffling novel, but there's cypher incorrect alongside it, too it exists inwards the continuous fourth dimension literature. We don't encounter it that often, because when you're exclusively studying 1 serial it's easier to only modify the distribution of the saltation procedure instead. This inquiry occurs when you lot tin encounter both serial x too y too you lot desire to model the human relationship betwixt them.
Which is right?
Which extension to jumps is correct? Both are mathematically correct. There is cypher incorrect alongside writing downward a model inwards which the reply to a saltation is dissimilar from the reply to continuous movements arbitrarily only about jumps. The respond depends on the economical situation.
For example, consider models alongside bankruptcy constraints. Agents who tin continuously accommodate their investments may ever avoid bankruptcy inwards a diffusion setting. If nosotros extend such a model to jumps alongside the continuous bound approach, implicitly preserving the investor's powerfulness to merchandise equally fast equally property prices modify fifty-fifty inwards the saltation limit, nosotros volition save bankruptcy avoidance inwards confront of a saltation inwards prices. However, if nosotros model portfolio adjustment to jumps alongside the left-limit generalization, agents may move forced inwards to bankruptcy for cost jumps.
Sometimes, 1 introduces jumps exactly to model a province of affairs inwards which prices tin deed faster than agents tin accommodate their portfolios, thence agents may move forced to bankruptcy. Then the left-limit generalization is correct. But if 1 wants to extend a model to jumps for other reasons, piece avoiding bankruptcy, negative consumption, negative marginal utility (consumption below null or below habits), violations of budget constraints, feasibility conditions, borrowing constraints, too thence forth, thence 1 should select a generalization inwards which the saltation gives the same outcome equally the continuous limit.
Similarly, when extending selection pricing models to jumps, 1 may desire to model the saltation inwards such a agency that investors cannot accommodate portfolios fast enough. Then the left-limit extension is appropriate, too investors must concord the saltation risk. But 1 may wishing to accommodate jumps inwards property prices to improve lucifer property cost dynamics piece maintaining investor's powerfulness to dynamically hedge. Then the nonlinear extension is appropriate, maintaining the equivalence betwixt jumps too the limiting diffusion.
A piffling to a greater extent than full general treatment
Influenza A virus subtype H5N1 piffling to a greater extent than generally, 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 showtime equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If at that spot 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 displace solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again, you lot encounter the crucial difference, whether the state variable gets to deed "during" the jump. We tin write this equally a differential, past times writing the solution to this concluding differential equation equally \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] too thence \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]
So, you lot don't have to extend the model to jumps alongside the left-limit approach, too you lot don't receive got to swallow the persuasion that a saltation has a dissimilar reply than an arbitrarily closed continuous-sample-path movement. The concluding equation shows you lot how to modify the model to include jumps inwards a agency that preserves the belongings that the saltation has the same outcome equally its continuous limit.
The point
Why a weblog postal service on this? I asked a few continuous-time gurus, too none of them had seen this number before. If somebody knows where this has all been worked out alongside proper is dotted too ts crossed, I would similar to know too holler it properly. (I would intend the literature on selection pricing alongside jumps had done it, but I couldn't reveal a reference.) Or mayhap it hasn't been done too somebody wants to arrive at it. I'm non adept plenty at the technical aspects of continuous fourth dimension to write this alongside the right precision too generality.
And it's a cool play tricks that may move useful to somebody exterior of the narrow context that nosotros had for it.
Update:
Perhaps the right application is stock prices too selection prices. When stock prices jump, somebody must receive got 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 receive got a citation to that case?
Suppose 1 fourth dimension serial \(x\), which follows a diffusion, drives some other \(y\). In the simplest example, \[dx_t = \sigma dz_t \] \[ dy_t = y_t dx_t. \] In our example, the minute equation describes how habits \(y\) respond to consumption \(x\). The same sort of construction mightiness clit 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 grip jumps inwards \(x\), \[dx_t = \sigma dz_t + dJ_t.\] What arrive at nosotros arrive at almost the minute equation? \(y_t\) instantly tin saltation too. On the right paw side of the minute equation, should nosotros purpose the left limit, the right limit, or something inwards between?
The commons respond is to purpose 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 \(y_{t_-}=1\), too \(dJ_t=1\). Then \(y_t\) jumps to \(y_t=2\). But suppose at that spot are 2 jumps of size 1/2, 1 at fourth dimension \(t\) too 1 at fourth dimension \(t+\varepsilon\). Now \(y\) jumps upward to 1.5 afterward the showtime jump, too thence jumps some other \(1.5 \times 0.5 = 0.75\), ending upward at \(y_{t+\varepsilon} =2.25\). Two one-half jumps receive got a dissimilar reply than 1 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 too closer to a saltation inwards \(dx_t\), too let's reveal what happens to \(y_t\). The full general solution to the showtime ready of equations is \[ y_{t+\Delta} = y_t e^{(x_{t+\Delta}-x_t - \frac{1}{2}\sigma^2\Delta)}\] so, inwards the bound \(\Delta \rightarrow 0\) that \(x_t\) takes a saltation of size \(dJ_t\), the jump-limit of a continuous displace 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 reply to a saltation that was dissimilar from the reply to a continuous procedure arbitrarily only about a jump. For example, the left-limit approach tin create a negative \(y_t\), but this method, similar the diffusion process, cannot autumn below zero. This method also produces a reply to 2 one-half jumps that is the same equally the reply to a total jump.
As you lot tin see, the divergence is whether the state 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 piece the saltation inwards \(x_t\) is moving.
Influenza A virus subtype H5N1 nonlinear business office of a saltation is a piffling novel, but there's cypher incorrect alongside it, too it exists inwards the continuous fourth dimension literature. We don't encounter it that often, because when you're exclusively studying 1 serial it's easier to only modify the distribution of the saltation procedure instead. This inquiry occurs when you lot tin encounter both serial x too y too you lot desire to model the human relationship betwixt them.
Which is right?
Which extension to jumps is correct? Both are mathematically correct. There is cypher incorrect alongside writing downward a model inwards which the reply to a saltation is dissimilar from the reply to continuous movements arbitrarily only about jumps. The respond depends on the economical situation.
For example, consider models alongside bankruptcy constraints. Agents who tin continuously accommodate their investments may ever avoid bankruptcy inwards a diffusion setting. If nosotros extend such a model to jumps alongside the continuous bound approach, implicitly preserving the investor's powerfulness to merchandise equally fast equally property prices modify fifty-fifty inwards the saltation limit, nosotros volition save bankruptcy avoidance inwards confront of a saltation inwards prices. However, if nosotros model portfolio adjustment to jumps alongside the left-limit generalization, agents may move forced inwards to bankruptcy for cost jumps.
Sometimes, 1 introduces jumps exactly to model a province of affairs inwards which prices tin deed faster than agents tin accommodate their portfolios, thence agents may move forced to bankruptcy. Then the left-limit generalization is correct. But if 1 wants to extend a model to jumps for other reasons, piece avoiding bankruptcy, negative consumption, negative marginal utility (consumption below null or below habits), violations of budget constraints, feasibility conditions, borrowing constraints, too thence forth, thence 1 should select a generalization inwards which the saltation gives the same outcome equally the continuous limit.
Similarly, when extending selection pricing models to jumps, 1 may desire to model the saltation inwards such a agency that investors cannot accommodate portfolios fast enough. Then the left-limit extension is appropriate, too investors must concord the saltation risk. But 1 may wishing to accommodate jumps inwards property prices to improve lucifer property cost dynamics piece maintaining investor's powerfulness to dynamically hedge. Then the nonlinear extension is appropriate, maintaining the equivalence betwixt jumps too the limiting diffusion.
A piffling to a greater extent than full general treatment
Influenza A virus subtype H5N1 piffling to a greater extent than generally, 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 showtime equation. The left-limit approach is \[dy_t = \mu(y_{t_-}) dt + \lambda(y_{t_-})dx_t \] If at that spot 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 displace solves the differential equation \[\int_{y_{t_-}}^{y_t} \frac{1}{\lambda(\xi)}d\xi = dx_t\] Again, you lot encounter the crucial difference, whether the state variable gets to deed "during" the jump. We tin write this equally a differential, past times writing the solution to this concluding differential equation equally \[y_t-y_{t_-}=f(x_t-x_{t_-};y_{t_-})\] too thence \[dy_t = \mu(y_{t_-}) dt + f(dx_t;y_{t_-})=\mu(y_{t_-}) dt + \lambda(y_{t_-})\sigma dz_t+f(dJ_t;y_{t_-})\]
So, you lot don't have to extend the model to jumps alongside the left-limit approach, too you lot don't receive got to swallow the persuasion that a saltation has a dissimilar reply than an arbitrarily closed continuous-sample-path movement. The concluding equation shows you lot how to modify the model to include jumps inwards a agency that preserves the belongings that the saltation has the same outcome equally its continuous limit.
The point
Why a weblog postal service on this? I asked a few continuous-time gurus, too none of them had seen this number before. If somebody knows where this has all been worked out alongside proper is dotted too ts crossed, I would similar to know too holler it properly. (I would intend the literature on selection pricing alongside jumps had done it, but I couldn't reveal a reference.) Or mayhap it hasn't been done too somebody wants to arrive at it. I'm non adept plenty at the technical aspects of continuous fourth dimension to write this alongside the right precision too generality.
And it's a cool play tricks that may move useful to somebody exterior of the narrow context that nosotros had for it.
Update:
Perhaps the right application is stock prices too selection prices. When stock prices jump, somebody must receive got 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 receive got a citation to that case?
